the critical thinking principle of falsifiability

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A Primer on Critical Thinking and Business Ethics

ISBN : 978-1-83753-309-1 , eISBN : 978-1-83753-308-4

Publication date: 27 July 2023

Executive Summary

All of us seek truth via objective inquiry into various human and nonhuman phenomena that nature presents to us on a daily basis. We are empirical (or nonempirical) decision makers who hold that uncertainty is our discipline, and that understanding how to act under conditions of incomplete information is the highest and most urgent human pursuit (Karl Popper, as cited in Taleb, 2010, p. 57). We verify (prove something as right) or falsify (prove something as wrong), and this asymmetry of knowledge enables us to distinguish between science and nonscience. According to Karl Popper (1971), we should be an “open society,” one that relies on skepticism as a modus operandi, refusing and resisting definitive (dogmatic) truths. An open society, maintained Popper, is one in which no permanent truth is held to exist; this would allow counter-ideas to emerge. Hence, any idea of Utopia is necessarily closed since it chokes its own refutations. A good model for society that cannot be left open for falsification is totalitarian and epistemologically arrogant. The difference between an open and a closed society is that between an open and a closed mind (Taleb, 2004, p. 129). Popper accused Plato of closing our minds. Popper's idea was that science has problems of fallibility or falsifiability. In this chapter, we deal with fallibility and falsifiability of human thinking, reasoning, and inferencing as argued by various scholars, as well as the falsifiability of our knowledge and cherished cultures and traditions. Critical thinking helps us cope with both vulnerabilities. In general, we argue for supporting the theory of “open mind and open society” in order to pursue objective truth.

Mascarenhas, O.A.J. , Thakur, M. and Kumar, P. (2023), "Critical Thinking for Understanding Fallibility and Falsifiability of Our Knowledge", A Primer on Critical Thinking and Business Ethics , Emerald Publishing Limited, Leeds, pp. 187-216. https://doi.org/10.1108/978-1-83753-308-420231007

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Copyright © 2023 Oswald A. J. Mascarenhas, Munish Thakur and Payal Kumar. Published under exclusive licence by Emerald Publishing Limited

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Law of Falsifiability

The Law of Falsifiability is a rule that a famous thinker named Karl Popper came up with. In simple terms, for something to be called scientific, there must be a way to show it could be incorrect. Imagine you’re saying you have an invisible, noiseless, pet dragon in your room that no one can touch or see. If no one can test to see if the dragon is really there, then it’s not scientific. But if you claim that water boils at 100 degrees Celsius at sea level, we can test this. If it turns out water does not boil at this temperature under these conditions, then the claim would be proven false. That’s what Karl Popper was getting at – science is about making claims that can be tested, possibly shown to be false, and that’s what keeps it trustworthy and moving forward.

Examples of Law of Falsifiability

• Astrology – Astrology is like saying certain traits or events will happen to you based on star patterns. But because its predictions are too general and can’t be checked in a clear way, it doesn’t pass the test of falsifiability. This means astrology cannot be considered a scientific theory since you can’t show when it’s wrong with specific tests.
• The Theory of Evolution – In contrast, the theory of evolution is something we can test. It says that different living things developed over a very long time. If someone were to find an animal’s remains in a rock layer where it should not be, such as a rabbit in rock that’s 500 million years old, that would challenge the theory. Since we can test it by looking for evidence like this, evolution is considered falsifiable.

Why is it Important?

The Law of Falsifiability matters a lot because it separates what’s considered scientific from what’s not. When an idea can’t be tested or shown to be wrong, it can lead people down the wrong path. By focusing on theories we can test, science gets stronger and we learn more about the world for real. For everyday people, this is key because it means we can rely on science for things like medicine, technology, and understanding our environment. If scientists didn’t use this rule, we might believe in things that aren’t true, like magic potions or the idea that some stars can predict your future.

Implications and Applications

The rule of being able to test if something is false is basic in the world of science and is used in all sorts of subjects. For example, in an experiment, scientists try really hard to see if their guess about something can be shown wrong. If their guess survives all the tests, it’s a good sign; if not, they need to think again or throw it out. This is how science gets better and better.

Comparison with Related Axioms

• Verifiability : This means checking if a statement or idea is true. Both verifiability and falsifiability have to do with testing, but falsifiability is seen as more important because things that can be proven wrong are usually also things we can check for truth.
• Empiricism : This is the belief that knowledge comes from what we can sense – like seeing, hearing, or touching. Falsifiability and empiricism go hand in hand because both involve using real evidence to test out ideas.
• Reproducibility : This idea says that doing the same experiment in the same way should give you the same result. To show something is falsifiable, you should be able to repeat a test over and over, with the chance that it might fail.

Karl Popper brought the Law of Falsifiability into the world in the 1900s. He didn’t like theories that seemed to answer everything because, to him, they actually explained nothing. By making this law, he aimed to make a clear line between what could be taken seriously in science and what could not. It was his way of making sure scientific thinking stayed sharp and clear.

Controversies

Not everyone agrees that falsifiability is the only way to tell if something is scientific. Some experts point out areas in science, like string theory from physics, which are really hard to test and so are hard to apply this law to. Also, in science fields that look at history, like how the universe began or how life changed over time, it’s not always about predictions that can be tested, but more about understanding special events. These differences in opinion show that while it’s a strong part of scientific thinking, falsifiability might not work for every situation or be the only thing that counts for scientific ideas.

Related Topics

• Scientific Method : This is the process scientists use to study things. It involves asking questions, making a hypothesis, running experiments, and seeing if the results support the hypothesis. Falsifiability is part of this process because scientists have to be able to test their hypotheses.
• Peer Review : When scientists finish their work, other experts check it to make sure it was done right. This involves reviewing if the experiments and tests were set up in a way that they could have shown the work was false if it wasn’t true.
• Logic and Critical Thinking : These are skills that help us make good arguments and decisions. Understanding falsifiability helps people develop these skills because it teaches them to always look for ways to test ideas.

In conclusion, the Law of Falsifiability, as brought up by Karl Popper, is like a key part of a scientist’s toolbox. It makes sure that ideas need to be able to be tested and possibly shown to be not true. By using this rule, we avoid believing in things without good evidence, and we make the stuff we learn about the world through science stronger and more reliable.

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Falsifiability

Karl popper's basic scientific principle, karl popper's basic scientific principle.

Falsifiability, according to the philosopher Karl Popper, defines the inherent testability of any scientific hypothesis.

This article is a part of the guide:

• Inductive Reasoning
• Deductive Reasoning
• Hypothetico-Deductive Method
• Scientific Reasoning
• Testability

Browse Full Outline

• 1 Scientific Reasoning
• 2.1 Falsifiability
• 2.2 Verification Error
• 2.3 Testability
• 2.4 Post Hoc Reasoning
• 3 Deductive Reasoning
• 4.1 Raven Paradox
• 5 Causal Reasoning
• 6 Abductive Reasoning
• 7 Defeasible Reasoning

Science and philosophy have always worked together to try to uncover truths about the universe we live in. Indeed, ancient philosophy can be understood as the originator of many of the separate fields of study we have today, including psychology, medicine, law, astronomy, art and even theology.

Scientists design experiments and try to obtain results verifying or disproving a hypothesis, but philosophers are interested in understanding what factors determine the validity of scientific endeavors in the first place.

Whilst most scientists work within established paradigms, philosophers question the paradigms themselves and try to explore our underlying assumptions and definitions behind the logic of how we seek knowledge. Thus there is a feedback relationship between science and philosophy - and sometimes plenty of tension!

One of the tenets behind the scientific method is that any scientific hypothesis and resultant experimental design must be inherently falsifiable. Although falsifiability is not universally accepted, it is still the foundation of the majority of scientific experiments. Most scientists accept and work with this tenet, but it has its roots in philosophy and the deeper questions of truth and our access to it.

What is Falsifiability?

Falsifiability is the assertion that for any hypothesis to have credence, it must be inherently disprovable before it can become accepted as a scientific hypothesis or theory.

For example, someone might claim "the earth is younger than many scientists state, and in fact was created to appear as though it was older through deceptive fossils etc.” This is a claim that is unfalsifiable because it is a theory that can never be shown to be false. If you were to present such a person with fossils, geological data or arguments about the nature of compounds in the ozone, they could refute the argument by saying that your evidence was fabricated to appeared that way, and isn’t valid.

Importantly, falsifiability doesn’t mean that there are currently arguments against a theory, only that it is possible to imagine some kind of argument which would invalidate it. Falsifiability says nothing about an argument's inherent validity or correctness. It is only the minimum trait required of a claim that allows it to be engaged with in a scientific manner – a dividing line between what is considered science and what isn’t. Another important point is that falsifiability is not any claim that has yet to be proven true. After all, a conjecture that hasn’t been proven yet is just a hypothesis.

The idea is that no theory is completely correct , but if it can be shown both to be falsifiable  and supported with evidence that shows it's true, it can be accepted as truth.

For example, Newton's Theory of Gravity was accepted as truth for centuries, because objects do not randomly float away from the earth. It appeared to fit the data obtained by experimentation and research , but was always subject to testing.

However, Einstein's theory makes falsifiable predictions that are different from predictions made by Newton's theory, for example concerning the precession of the orbit of Mercury, and gravitational lensing of light. In non-extreme situations Einstein's and Newton's theories make the same predictions, so they are both correct. But Einstein's theory holds true in a superset of the conditions in which Newton's theory holds, so according to the principle of Occam's Razor , Einstein's theory is preferred. On the other hand, Newtonian calculations are simpler, so Newton's theory is useful for almost any engineering project, including some space projects. But for GPS we need Einstein's theory. Scientists would not have arrived at either of these theories, or a compromise between both of them, without the use of testable, falsifiable experiments. 

Popper saw falsifiability as a black and white definition; that if a theory is falsifiable, it is scientific , and if not, then it is unscientific. Whilst some "pure" sciences do adhere to this strict criterion, many fall somewhere between the two extremes, with  pseudo-sciences  falling at the extreme end of being unfalsifiable. 

Pseudoscience

According to Popper, many branches of applied science, especially social science, are not truly scientific because they have no potential for falsification.

Anthropology and sociology, for example, often use case studies to observe people in their natural environment without actually testing any specific hypotheses or theories.

While such studies and ideas are not falsifiable, most would agree that they are scientific because they significantly advance human knowledge.

Popper had and still has his fair share of critics, and the question of how to demarcate legitimate scientific enquiry can get very convoluted. Some statements are logically falsifiable but not practically falsifiable – consider the famous example of “it will rain at this location in a million years' time.” You could absolutely conceive of a way to test this claim, but carrying it out is a different story.

Thus, falsifiability is not a simple black and white matter. The Raven Paradox shows the inherent danger of relying on falsifiability, because very few scientific experiments can measure all of the data, and necessarily rely upon generalization . Technologies change along with our aims and comprehension of the phenomena we study, and so the falsifiability criterion for good science is subject to shifting.

For many sciences, the idea of falsifiability is a useful tool for generating theories that are testable and realistic. Testability is a crucial starting point around which to design solid experiments that have a chance of telling us something useful about the phenomena in question. If a falsifiable theory is tested and the results are significant , then it can become accepted as a scientific truth.

The advantage of Popper's idea is that such truths can be falsified when more knowledge and resources are available. Even long accepted theories such as Gravity, Relativity and Evolution are increasingly challenged and adapted.

The major disadvantage of falsifiability is that it is very strict in its definitions and does not take into account the contributions of sciences that are observational and descriptive .

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Martyn Shuttleworth , Lyndsay T Wilson (Sep 21, 2008). Falsifiability. Retrieved Apr 09, 2024 from Explorable.com: https://explorable.com/falsifiability

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5 Falsifiability

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• Deductive Logic
• Persuasive Reasoning and Fallacies
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Inquiry-based Activity:  Popular media and falsifiability

Introduction : Falsifiability, or the ability for a statement/theory to be shown to be false, was noted by Karl Popper to be the clearest way to distinguish science from pseudoscience. While incredibly important to scientific inquiry, it is also important for students to understand how this criterion can be applied to the news and information they interact with in their day-to-day lives. In this activity, students will apply the logic of falsifiability to rumors and news they have heard of in the popular media, demonstrating the applicability of scientific thinking to the world beyond the classroom.

Question to pose to students : Think about the latest celebrity rumor you have heard about in the news or through social media. If you cannot think of one, some examples might include, “the CIA killed Marilyn Monroe” and “Tupac is alive.” Have students get into groups, discuss their rumors, and select one to work with.

Note to instructors: Please modify/update these examples if needed to work for the students in your course. Snopes is a good source for recent examples.

Students form a hypothesis : Thinking about that rumor, decide what evidence would be necessary to prove that it was correct. That is, imagine you were a skeptic and automatically did not believe the rumor – what would someone need to tell or show you to convince you that it was true?

Students test their hypotheses : Each group (A) should then pair up with one other group (B) and try to convince them their rumor is true, providing them with the evidence from above. Members of group B should then come up with any reasons they can think of why the rumor may still be false. For example – if “Tupac is alive” is the rumor and “show the death certificate” is a piece of evidence provided by group A, group B could posit that the death certificate was forged by whoever kidnapped Tupac. Once group B has evaluated all of group A’s evidence, have the groups switch such that group B is now trying to convince group A about their rumor.

Do the students’ hypotheses hold up? : Together, have the groups work out whether the rumors they discussed are falsifiable. That is, can it be “proven?” Remember, a claim is non-falsifiable if there can always be an explanation for the absence of evidence and/or an exhaustive search for evidence would be required. Depending on the length of your class, students can repeat the previous step with multiple groups.

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Karl Popper: Theory of Falsification

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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Karl Popper’s theory of falsification contends that scientific inquiry should aim not to verify hypotheses but to rigorously test and identify conditions under which they are false. For a theory to be valid according to falsification, it must produce hypotheses that have the potential to be proven incorrect by observable evidence or experimental results. Unlike verification, falsification focuses on categorically disproving theoretical predictions rather than confirming them.

• Karl Popper believed that scientific knowledge is provisional – the best we can do at the moment.
• Popper is known for his attempt to refute the classical positivist account of the scientific method by replacing induction with the falsification principle.
• The Falsification Principle, proposed by Karl Popper, is a way of demarcating science from non-science. It suggests that for a theory to be considered scientific, it must be able to be tested and conceivably proven false.
• For example, the hypothesis that “all swans are white” can be falsified by observing a black swan.
• For Popper, science should attempt to disprove a theory rather than attempt to continually support theoretical hypotheses.

Theory of Falsification

Karl Popper is prescriptive and describes what science should do (not how it actually behaves). Popper is a rationalist and contended that the central question in the philosophy of science was distinguishing science from non-science.

Karl Popper, in ‘The Logic of Scientific Discovery’ emerged as a major critic of inductivism, which he saw as an essentially old-fashioned strategy.

Popper replaced the classical observationalist-inductivist account of the scientific method with falsification (i.e., deductive logic) as the criterion for distinguishing scientific theory from non-science.

All inductive evidence is limited: we do not observe the universe at all times and in all places. We are not justified, therefore, in making a general rule from this observation of particulars.

According to Popper, scientific theory should make predictions that can be tested, and the theory should be rejected if these predictions are shown not to be correct.

He argued that science would best progress using deductive reasoning as its primary emphasis, known as critical rationalism.

Popper gives the following example:

Europeans, for thousands of years had observed millions of white swans. Using inductive evidence, we could come up with the theory that all swans are white.

However, exploration of Australasia introduced Europeans to black swans.  Poppers’ point is this: no matter how many observations are made which confirm a theory, there is always the possibility that a future observation could refute it.  Induction cannot yield certainty.

Karl Popper was also critical of the naive empiricist view that we objectively observe the world. Popper argued that all observation is from a point of view, and indeed that all observation is colored by our understanding. The world appears to us in the context of theories we already hold: it is ‘theory-laden.’

Popper proposed an alternative scientific method based on falsification.  However, many confirming instances exist for a theory; it only takes one counter-observation to falsify it. Science progresses when a theory is shown to be wrong and a new theory is introduced that better explains the phenomena.

For Popper, the scientist should attempt to disprove his/her theory rather than attempt to prove it continually. Popper does think that science can help us progressively approach the truth, but we can never be certain that we have the final explanation.

Critical Evaluation

Popper’s first major contribution to philosophy was his novel solution to the problem of the demarcation of science. According to the time-honored view, science, properly so-called, is distinguished by its inductive method – by its characteristic use of observation and experiment, as opposed to purely logical analysis, to establish its results.

The great difficulty was that no run of favorable observational data, however long and unbroken, is logically sufficient to establish the truth of an unrestricted generalization.

Popper’s astute formulations of logical procedure helped to reign in the excessive use of inductive speculation upon inductive speculation, and also helped to strengthen the conceptual foundation for today’s peer review procedures.

However, the history of science gives little indication of having followed anything like a methodological falsificationist approach.

Indeed, and as many studies have shown, scientists of the past (and still today) tended to be reluctant to give up theories that we would have to call falsified in the methodological sense, and very often, it turned out that they were correct to do so (seen from our later perspective).

The history of science shows that sometimes it is best to ’stick to one’s guns’. For example, “In the early years of its life, Newton’s gravitational theory was falsified by observations of the moon’s orbit”

Also, one observation does not falsify a theory. The experiment may have been badly designed; data could be incorrect.

Quine states that a theory is not a single statement; it is a complex network (a collection of statements). You might falsify one statement (e.g., all swans are white) in the network, but this should not mean you should reject the whole complex theory.

Critics of Karl Popper, chiefly Thomas Kuhn , Paul Feyerabend, and Imre Lakatos, rejected the idea that there exists a single method that applies to all science and could account for its progress.

Popperp, K. R. (1959). The logic of scientific discovery . University Press.

Further Information

• Thomas Kuhn – Paradigm Shift Is Psychology a Science?
• Steps of the Scientific Method
• Positivism in Sociology: Definition, Theory & Examples
• The Scientific Revolutions of Thomas Kuhn: Paradigm Shifts Explained

The Concept of Falsifiability in Psychology: Importance and Applications

In the field of psychology, the concept of falsifiability plays a crucial role in promoting scientific rigor, testability, and critical thinking. By allowing hypotheses to be tested and findings to be reproducible, falsifiability aids in the advancement of psychological research.

This article explores how falsifiability is applied in psychological research, with examples from theories such as Freud’s Psychoanalytic Theory and Skinner’s Behaviorism. We also delve into the limitations of falsifiability in psychology and provide suggestions for improving its application, including the incorporation of qualitative data and promoting open science practices.

• Falsifiability is an essential concept in psychology that promotes scientific rigor and critical thinking.
• It allows for testability and reproducibility of research findings, leading to more reliable and valid conclusions.
• While it has limitations, incorporating qualitative data, utilizing multiple methods of data collection, and promoting open science practices can improve the application of falsifiability in psychological research.
• 1 What Is Falsifiability in Psychology?
• 2.1 Promotes Scientific Rigor
• 2.2 Allows for Testability and Reproducibility
• 2.3 Encourages Critical Thinking
• 3.1 Hypothesis Testing
• 3.2 Experimental Design
• 3.3 Statistical Analysis
• 4.1 Freud’s Psychoanalytic Theory
• 4.2 Maslow’s Hierarchy of Needs
• 4.3 Skinner’s Behaviorism
• 5.1 Overemphasis on Quantitative Data
• 5.2 Difficulty in Measuring Complex Psychological Phenomena
• 5.3 Potential for Bias and Misinterpretation
• 6.1 Incorporating Qualitative Data
• 6.2 Utilizing Multiple Methods of Data Collection
• 6.3 Promoting Open Science Practices
• 7.1 What is the concept of falsifiability in psychology and why is it important?
• 7.2 How does falsifiability impact the validity of psychological research?
• 7.3 Can you provide an example of falsifiability in psychology?
• 7.4 How does the concept of falsifiability impact the practice of psychology?
• 7.5 In what ways is falsifiability applied in psychological research?
• 7.6 How does the concept of falsifiability differ from the concept of verification?

What Is Falsifiability in Psychology?

Falsifiability in psychology refers to the principle proposed by Karl Popper that scientific theories must be capable of being tested and disproven through empirical observation and experimentation.

Karl Popper’s concept of falsifiability has played a significant role in distinguishing scientific theories from pseudoscience in psychology. By requiring that theories be testable and subject to potential refutation, falsifiability sets a high standard for scientific validity.

For example, in psychological research, this principle is applied when researchers design experiments to specifically test hypotheses that can be proven wrong. This ensures that the conclusions drawn are based on solid empirical evidence rather than unfalsifiable claims.

One famous application of falsifiability in psychology is in behaviorism, where theories like those by B.F. Skinner were subject to rigorous experimental testing, leading to the development of evidence-based practices.

Why Is Falsifiability Important in Psychology?

Falsifiability holds paramount importance in psychology as it ensures that theories and hypotheses are grounded in empirical evidence, fostering critical thinking and advancing towards scientific truth as advocated by Karl Popper.

Falsifiability not only serves as a cornerstone of scientific inquiry in psychology but also acts as a safeguard against pseudoscience and unsubstantiated claims.

By requiring that hypotheses be formulated in a way that they can be potentially proven wrong through observation or experimentation, falsifiability encourages researchers to test and refine their ideas rigorously.

This rigorous testing process helps differentiate between valid scientific theories and mere speculation, contributing to the credibility and reliability of psychological research.

Promotes Scientific Rigor

The principle of falsifiability promotes scientific rigor by establishing stringent criteria for theories to be substantiated through empirical methods and logical criteria, thereby fostering genuine scientific discovery.

One of the key standards that falsifiability sets for empirical validation is the requirement for theories to make predictions that can be tested and potentially proven false. This process ensures that scientific claims are not merely based on speculation but on concrete evidence that can either support or refute the proposed theory.

Falsifiability also emphasizes the importance of logical coherence in scientific theories. By demanding that theories are internally consistent and free from logical contradictions, this principle safeguards the integrity of scientific reasoning and prevents unfounded conclusions.

The application of falsifiability encourages scientists to constantly strive for genuine scientific breakthroughs. The willingness to subject theories to rigorous testing and potential falsification pushes researchers to explore new ideas, refine existing concepts, and ultimately advance the frontiers of knowledge in their respective fields.

Allows for Testability and Reproducibility

Falsifiability allows for the testability and reproducibility of scientific claims through rigorous experimental design and empirical tests, ensuring the reliability and validity of scientific experiments.

One of the key aspects of using falsifiability in scientific research is the ability to formulate testable hypotheses , which can be empirically examined and potentially proven wrong. By setting up experiments with clear parameters and measurable outcomes, researchers can systematically test their hypotheses and gather data to support or refute their claims. This process not only promotes transparency in the scientific method but also encourages replication studies by other scientists to validate the initial findings.

• Experimental methodologies such as randomized controlled trials, observational studies, and laboratory experiments play a crucial role in gathering empirical evidence to support scientific claims. These methodologies involve manipulating variables, collecting data, and analyzing results to draw meaningful conclusions.
• Validating hypotheses requires a rigorous approach to data collection and analysis. Utilizing statistical tools and peer review processes, researchers can ensure the credibility of their findings and mitigate biases that may affect the outcomes of their experiments.
• The peer review system serves as a quality control mechanism in the scientific community, where experts evaluate the methodology, results, and interpretations of a study before it gets published. This critical assessment helps uphold the standards of scientific research and enhances the overall reliability of scientific claims.

Encourages Critical Thinking

Falsifiability encourages critical thinking by fostering a logical relationship between theories and empirical observations, inviting critique and scrutiny to refine scientific understanding in psychology.

This fundamental concept, introduced by philosopher Karl Popper, underlines the significance of empirical evidence in testing the validity of hypotheses.

Psychologists leverage falsifiability to ensure their theories are subject to rigorous examination, advocating for testable assertions grounded in observable phenomena.

Through the lens of falsifiability, researchers navigate the complexities of psychological phenomena with a commitment to evidence-based reasoning and logical coherence.

How Is Falsifiability Applied in Psychological Research?

Falsifiability is applied in psychological research through rigorous hypothesis testing, meticulous experimental design, and the systematic analysis of empirical data to validate or invalidate theoretical propositions.

By adhering to the principle of falsifiability, researchers aim to formulate hypotheses that are precise, testable, and potentially refutable.

Experiment design plays a crucial role in this process, as it involves creating conditions that allow for the systematic manipulation of variables to observe their effects on the outcomes.

The collection and analysis of data are then conducted with meticulous attention to detail, utilizing statistical techniques to draw meaningful conclusions about the validity of the underlying psychological theories.

Hypothesis Testing

Hypothesis testing in psychology involves subjecting theoretical propositions to empirical scrutiny and logical verification, aligning with the principles of falsifiability to establish the validity of scientific claims.

When formulating hypotheses in psychological research, researchers first identify the variables they are interested in studying and propose a relationship between them. These hypotheses are often based on existing theories or empirical observations. The hypotheses are then tested through carefully designed experiments or observational studies. Data is collected, analyzed, and interpreted to determine whether the results support or refute the hypothesis.

Through this process, researchers aim to make inferences about the population from which the sample was drawn. Statistical tests are employed to assess the likelihood that the results are due to chance or if they reflect a true relationship. Validating hypotheses in psychology requires rigorous methodological approaches and adherence to ethical standards to ensure the reliability and validity of the findings.

Experimental Design

Experimental design in psychological research adheres to the principles of falsifiability by following the scientific method, gathering empirical evidence, and structuring experiments to test specific hypotheses.

By carefully constructing experiments, researchers aim to isolate the effects of certain variables on the outcome, ensuring that any observed changes can be attributed to the manipulated factor. Controlled variables play a crucial role in this process, as they help to eliminate extraneous influences that could confound the results. Through systematic manipulation and control of variables, researchers can establish cause-and-effect relationships, providing valuable insights into the intricate workings of the human mind.

Statistical Analysis

Statistical analysis plays a crucial role in psychological research by evaluating empirical data, testing theoretical predictions, and applying Bayesian inductive logic to draw valid conclusions in alignment with the principles of falsifiability.

Statistical analysis supports falsifiability in psychology through rigorous hypothesis testing. By establishing clear hypotheses and systematically collecting data, researchers can objectively evaluate and validate their theories. Statistical modeling allows for the assessment of complex relationships among variables, aiding in the thorough evaluation of psychological phenomena. The use of inferential statistics enables researchers to make generalizations about populations based on sample data, providing a basis for confident conclusions. By embracing statistical analysis in psychological research, scholars uphold the standards of scientific inquiry and contribute to the advancement of the field.

What Are Some Examples of Falsifiability in Psychology?

Examples of falsifiability in psychology include Freud’s Psychoanalytic Theory , Maslow’s Hierarchy of Needs , and Skinner’s Behaviorism , where empirical observations and experimental tests are used to validate or refute these theoretical frameworks.

Freud’s Psychoanalytic Theory posited that unconscious processes significantly influence human behavior and personality development, involving concepts like the id, ego, and superego. Through empirical testing, researchers have been able to explore the applicability of these constructs and their impact on individuals’ mental lives.

In Maslow’s Hierarchy of Needs, the framework suggests that human motivation progresses through distinct stages based on fulfilling fundamental needs before advancing to higher-level aspirations. This model has been subjected to empirical research, analyzing whether individuals indeed prioritize basic needs before pursuing self-actualization.

Skinner’s Behaviorism focuses on the impact of reinforcement and punishment on shaping behavior. Through experimental studies, researchers have investigated the effectiveness of operant conditioning in altering behavior patterns and determining its generalizability to various contexts.

Freud’s Psychoanalytic Theory

Freud’s Psychoanalytic Theory is subject to falsifiability through empirical observations and experimental tests, where the validity of concepts such as the Oedipus complex and unconscious motivations can be scrutinized for scientific rigor.

By applying the principle of falsifiability, researchers can design studies to test the predictions and hypotheses derived from Freud’s theory. For example, studies using neuroimaging techniques have provided insights into brain activity related to unconscious thoughts and desires, offering empirical support for Freud’s ideas.

Behavioral experiments and clinical observations have allowed psychologists to explore manifestations of defense mechanisms, offering an avenue to examine the inherent conflicts in the human psyche, as posited by Freud. This rigorous scientific approach distinguishes psychology as a discipline that continually seeks to refine its understanding of complex human behavior.

Maslow’s Hierarchy of Needs

Maslow’s Hierarchy of Needs presents an example of falsifiability in psychology, where the hierarchical structure of human needs can be empirically tested and validated, aligning with the principles of scientific inquiry.

Researchers have utilized various methodologies to scrutinize the concept of Maslow’s Hierarchy , conducting surveys, experiments, and case studies.

Through empirical testing, psychologists can analyze whether the hierarchy accurately reflects human behavior across different cultures and contexts. By observing individuals’ responses to different levels of needs fulfillment, researchers can assess the model’s reliability and predictive power.

This process highlights the importance of empirical evidence in shaping psychological theories and understanding human motivation.

Skinner’s Behaviorism

Skinner’s Behaviorism exemplifies falsifiability in psychology by emphasizing observable behaviors, experimental analysis, and the application of scientific methods to test and validate behavioral principles through empirical evidence.

Skinner believed that the key to understanding human behavior lied in studying observable actions rather than diving into unobservable mental processes. This focus on external behaviors transformed psychology by introducing a systematic approach that involved setting up controlled experiments to observe and record behaviors in a scientific manner. By using rigorous experimental designs, such as operant conditioning chambers, Skinner demonstrated how behavior could be shaped through reinforcement and punishment, providing empirical support for his theories. This empirical methodology allowed behaviorism to develop as a verifiable and testable framework within psychology.

What Are the Limitations of Falsifiability in Psychology?

While essential, falsifiability in psychology has limitations including an overemphasis on quantitative data, challenges in measuring complex psychological phenomena, and the potential for bias and misinterpretation in empirical studies.

Quantitative data, although valuable, may not always capture the depth and nuances of human behavior and emotions. In the realm of psychology, where intricacies abound, relying solely on numbers can restrict the understanding of subjective experiences. Complex concepts such as self-esteem or motivation defy simple quantification, posing a challenge for researchers aiming to validate or invalidate hypotheses. The inherent subjectivity in psychological assessments introduces the risk of confirmation bias, where researchers might unconsciously seek data that supports their preconceived notions, jeopardizing the objectivity of their findings.

Overemphasis on Quantitative Data

One limitation of falsifiability in psychology is the potential overemphasis on quantitative data, which may prioritize statistical analysis over qualitative insights, potentially diminishing the pursuit of scientific truth.

While quantitative data plays a crucial role in evidence-based research, solely relying on numerical figures can sometimes overshadow the rich qualitative nuances that characterize human behavior and experiences. Psychology, as a field deeply rooted in understanding the complexities of the mind, requires a balanced integration of both quantitative and qualitative approaches to capture the multidimensional nature of psychological phenomena.

Difficulty in Measuring Complex Psychological Phenomena

Measuring complex psychological phenomena poses a significant challenge to falsifiability in psychology, as intricate concepts, subjective experiences, and multifaceted behaviors may defy straightforward empirical assessment.

When attempting to quantify attributes such as emotions, consciousness, or personality traits, researchers encounter methodological hurdles that necessitate innovative approaches. The very nature of these phenomena often eludes simplistic measurement techniques, requiring a nuanced understanding of cognitive processes and human behavior.

Employing standardized assessments for subjective constructs risks oversimplification and undermines the rich tapestry of individual differences present within human psychology. This highlights the importance of utilizing a diverse array of methodologies, from qualitative interviews to experimental paradigms, to capture the multidimensionality of psychological phenomena.

Potential for Bias and Misinterpretation

The potential for bias and misinterpretation in empirical tests poses a limitation to falsifiability in psychology, as subjective influences, preconceptions, or methodological errors may distort the interpretation of research findings.

These biases can arise from various sources, such as the researchers’ personal beliefs, societal norms, or even the funding sources behind the study.

Methodological flaws like small sample sizes, lack of control groups, or inadequate statistical analyses can further exacerbate the risk of misinterpretation.

Misinterpretations can occur when researchers draw unwarranted conclusions from the data or overlook alternative explanations that could provide a more accurate understanding of the phenomena under investigation.

How Can Falsifiability Be Improved in Psychological Research?

Enhancing falsifiability in psychological research involves integrating qualitative data alongside quantitative measures, utilizing diverse methods of data collection, and promoting open science practices to augment the empirical foundation of psychological inquiry.

Integrating qualitative data enriches research outcomes by providing depth and context to statistical findings. This approach allows researchers to capture nuances, individual experiences, and subjective perspectives that quantitative measures alone may miss. Moreover, diverse data collection methodologies such as surveys, interviews, and observational studies offer a comprehensive view of the phenomenon under investigation, reducing potential biases inherent in singular data sources.

Adopting transparent and reproducible scientific practices is paramount in fostering credibility and trust within the scientific community. It involves sharing raw data, analysis scripts, and methodological details to ensure research findings can be verified by others. By advocating for open science practices , researchers contribute to a culture of accountability and rigor, enhancing the quality and reliability of psychological research outcomes.

Incorporating Qualitative Data

Enhancing falsifiability in psychology can be achieved by incorporating qualitative data, which enriches scientific discovery, supports empirical methods, and broadens the scope of psychological inquiry beyond quantitative measures.

Qualitative data in psychology serves as a valuable tool in enriching research methodologies by providing a deeper understanding of human behavior and experiences. By capturing the nuances, complexities, and unique perspectives that quantitative data may overlook, qualitative insights enable researchers to delve into the intricacies of the human mind and emotions. This not only enhances the credibility and rigor of psychological studies but also fosters a more holistic approach to investigating psychological phenomena.

Utilizing Multiple Methods of Data Collection

Expanding the repertoire of data collection methods is instrumental in enhancing falsifiability in psychological research, as employing diverse approaches enables comprehensive empirical testing and strengthens the scientific foundation of psychological inquiry.

By incorporating various data collection techniques such as surveys, experiments, observations, and interviews, researchers can cross-validate findings across multiple methodologies , thus reinforcing the validity and reliability of their results. The integration of quantitative and qualitative data gathering methods not only enriches the depth of analysis but also fosters a more nuanced understanding of complex psychological phenomena. The use of mixed-method approaches can help researchers triangulate findings, enhancing the overall credibility and robustness of their research outcomes.

Promoting Open Science Practices

Promoting open science practices is crucial for advancing falsifiability in psychological research, as transparency, replicability, and collaborative scrutiny foster scientific discovery, validate empirical experiments, and uphold the principles of rigorous inquiry.

Falsifiability in psychological research refers to the capability of a hypothesis or theory to be empirically disproven through observation or experimentation. By embracing open science practices, researchers can enhance the credibility and reliability of their findings. Transparency ensures that all aspects of a study are openly shared, allowing for scrutiny and replication by other researchers. Reproducibility is key in demonstrating the robustness of results and findings, enabling the scientific community to build upon existing knowledge collectively.

Open science encourages collaborative engagement among researchers, facilitating peer review processes that help weed out errors and biases while also promoting a culture of constructive criticism and continuous improvement. This collaborative scrutiny not only enhances the quality of psychological research but also contributes towards building a stronger foundation for future scientific inquiries.

Frequently Asked Questions

What is the concept of falsifiability in psychology and why is it important.

Falsifiability refers to the idea that scientific theories and hypotheses should be able to be proven false through empirical evidence. In psychology, this means that theories and claims must be testable and open to being disproven. This is important because it ensures that research and findings are based on solid evidence rather than speculation or personal bias.

How does falsifiability impact the validity of psychological research?

Falsifiability plays a crucial role in determining the validity of psychological research. If a theory or hypothesis cannot be disproven, it cannot be considered a valid scientific claim. This means that the conclusions drawn from the research may not be reliable or accurate.

Can you provide an example of falsifiability in psychology?

One example of falsifiability in psychology is the theory of cognitive dissonance, which suggests that individuals experience discomfort when they hold contradictory beliefs. This theory is falsifiable because it can be tested and potentially proven false through empirical research.

How does the concept of falsifiability impact the practice of psychology?

The concept of falsifiability is integral to the scientific method, which is the foundation of psychological research. It ensures that psychologists use rigorous and empirical methods to test their theories and claims, leading to more reliable and valid findings. Without falsifiability, psychology would not be considered a legitimate science.

In what ways is falsifiability applied in psychological research?

Falsifiability is applied in psychological research through the use of experiments, surveys, and other empirical methods. These methods allow researchers to test their theories and hypotheses in a controlled environment, making it possible to either support or disprove them.

How does the concept of falsifiability differ from the concept of verification?

Falsifiability and verification are two opposing concepts in the field of science. While falsifiability focuses on testing theories and hypotheses to prove them false, verification is the process of confirming them. In psychology, the emphasis is on falsifiability as it promotes a critical and skeptical approach to research rather than simply seeking evidence to support existing beliefs.

Vanessa Patel is an expert in positive psychology, dedicated to studying happiness, resilience, and the factors that contribute to a fulfilling life. Her writing explores techniques for enhancing well-being, overcoming adversity, and building positive relationships and communities. Vanessa’s articles are a resource for anyone looking to find more joy and meaning in their daily lives, backed by the latest research in the field.

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Being Scientific: Falsifiability, Verifiability, Empirical Tests, and Reproducibility

If you ask a scientist what makes a good experiment, you’ll get very specific answers about reproducibility and controls and methods of teasing out causal relationships between variables and observables. If human observations are involved, you may get detailed descriptions of blind and double-blind experimental designs. In contrast, if you ask the very same scientists what makes a theory or explanation scientific, you’ll often get a vague statement about falsifiability . Scientists are usually very good at designing experiments to test theories. We invent theoretical entities and explanations all the time, but very rarely are they stated in ways that are falsifiable. It is also quite rare for anything in science to be stated in the form of a deductive argument. Experiments often aren’t done to falsify theories, but to provide the weight of repeated and varied observations in support of those same theories. Sometimes we’ll even use the words verify or confirm when talking about the results of an experiment. What’s going on? Is falsifiability the standard? Or something else?

The difference between falsifiability and verifiability in science deserves a bit of elaboration. It is not always obvious (even to scientists) what principles they are using to evaluate scientific theories, 1 so we’ll start a discussion of this difference by thinking about Popper’s asymmetry. 2 Consider a scientific theory ( T ) that predicts an observation ( O ). There are two ways we could approach adding the weight of experiment to a particular theory. We could attempt to falsify or verify the observation. Only one of these approaches (falsification) is deductively valid:

Popper concluded that it is impossible to know that a theory is true based on observations ( O ); science can tell us only that the theory is false (or that it has yet to be refuted). He concluded that meaningful scientific statements are falsifiable.

Scientific theories may not be this simple. We often base our theories on a set of auxiliary assumptions which we take as postulates for our theories. For example, a theory for liquid dynamics might depend on the whole of classical mechanics being taken as a postulate, or a theory of viral genetics might depend on the Hardy-Weinberg equilibrium. In these cases, classical mechanics (or the Hardy-Wienberg equilibrium) are the auxiliary assumptions for our specific theories.

These auxiliary assumptions can help show that science is often not a deductively valid exercise. The Quine-Duhem thesis 3 recovers the symmetry between falsification and verification when we take into account the role of the auxiliary assumptions ( AA ) of the theory ( T ):

That is, if the predicted observation ( O ) turns out to be false, we can deduce only that something is wrong with the conjunction, ( T and AA ); we cannot determine from the premises that it is T rather than AA that is false. In order to recover the asymmetry, we would need our assumptions ( AA ) to be independently verifiable:

Falsifying a theory requires that auxiliary assumption ( AA ) be demonstrably true. Auxiliary assumptions are often highly theoretical — remember, auxiliary assumptions might be statements like the entirety of classical mechanics is correct or the Hardy-Weinberg equilibrium is valid ! It is important to note, that if we can’t verify AA , we will not be able to falsify T by using the valid argument above. Contrary to Popper, there really is no asymmetry between falsification and verification. If we cannot verify theoretical statements, then we cannot falsify them either.

Since verifying a theoretical statement is nearly impossible, and falsification often requires verification of assumptions, where does that leave scientific theories? What is required of a statement to make it scientific?

Carl Hempel came up with one of the more useful statements about the properties of scientific theories: 4 “The statements constituting a scientific explanation must be capable of empirical test.” And this statement about what exactly it means to be scientific brings us right back to things that scientists are very good at: experimentation and experimental design. If I propose a scientific explanation for a phenomenon, it should be possible to subject that theory to an empirical test or experiment. We should also have a reasonable expectation of universality of empirical tests. That is multiple independent (skeptical) scientists should be able to subject these theories to similar tests in different locations, on different equipment, and at different times and get similar answers. Reproducibility of scientific experiments is therefore going to be required for universality.

So to answer some of the questions we might have about reproducibility:

• Reproducible by whom ? By independent (skeptical) scientists, working elsewhere, and on different equipment, not just by the original researcher.
• Reproducible to what degree ? This would depend on how closely that independent scientist can reproduce the controllable variables, but we should have a reasonable expectation of similar results under similar conditions.
• Wouldn’t the expense of a particular apparatus make reproducibility very difficult? Good scientific experiments must be reproducible in both a conceptual and an operational sense. 5 If a scientist publishes the results of an experiment, there should be enough of the methodology published with the results that a similarly-equipped, independent, and skeptical scientist could reproduce the results of the experiment in their own lab.

Computational science and reproducibility

If theory and experiment are the two traditional legs of science, simulation is fast becoming the “third leg”. Modern science has come to rely on computer simulations, computational models, and computational analysis of very large data sets. These methods for doing science are all reproducible in principle . For very simple systems, and small data sets this is nearly the same as reproducible in practice . As systems become more complex and the data sets become large, calculations that are reproducible in principle are no longer reproducible in practice without public access to the code (or data). If a scientist makes a claim that a skeptic can only reproduce by spending three decades writing and debugging a complex computer program that exactly replicates the workings of a commercial code, the original claim is really only reproducible in principle. If we really want to allow skeptics to test our claims, we must allow them to see the workings of the computer code that was used. It is therefore imperative for skeptical scientific inquiry that software for simulating complex systems be available in source-code form and that real access to raw data be made available to skeptics.

Our position on open source and open data in science was arrived at when an increasing number of papers began crossing our desks for review that could not be subjected to reproducibility tests in any meaningful way. Paper A might have used a commercial package that comes with a license that forbids people at university X from viewing the code ! 6

Paper 2 might use a code which requires parameter sets that are “trade secrets” and have never been published in the scientific literature . Our view is that it is not healthy for scientific papers to be supported by computations that cannot be reproduced except by a few employees at a commercial software developer. Should this kind of work even be considered Science? It may be research , and it may be important , but unless enough details of the experimental methodology are made available so that it can be subjected to true reproducibility tests by skeptics, it isn’t Science.

• This discussion closely follows a treatment of Popper’s asymmetry in: Sober, Elliot Philosophy of Biology (Boulder: Westview Press, 2000), pp. 50-51.
• Popper, Karl R. “The Logic of Scientific Discovery” 5th ed. (London: Hutchinson, 1959), pp. 40-41, 46.
• Gillies, Donald. “The Duhem Thesis and the Quine Thesis”, in Martin Curd and J.A. Cover ed. Philosophy of Science: The Central Issues, (New York: Norton, 1998), pp. 302-319.
• C. Hempel. Philosophy of Natural Science 49 (1966).
• Lett, James, Science, Reason and Anthropology, The Principles of Rational Inquiry (Oxford: Rowman & Littlefield, 1997), p. 47
• See, for example www.bannedbygaussian.org

5 Responses to Being Scientific: Falsifiability, Verifiability, Empirical Tests, and Reproducibility

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“If we cannot verify theoretical statements, then we cannot falsify them either.

Since verifying a theoretical statement is nearly impossible, and falsification often requires verification of assumptions…”

An invalid argument is invalid regardless of the truth of the premises. I would suggest that an hypothesis based on unverifiable assumptions could be ‘falsified’ the same way an argument with unverifiable premises could be shown to be invalid. Would you not agree?

“Falsifying a theory requires that auxiliary assumption (AA) be demonstrably true.”

No, it only requires them to be true.

In the falisificationist method, you can change the AA so long as that increases the theories testability. (the theory includes AA and the universal statement, btw) . In your second box you misrepresent the first derivation. in the conclusion it would be ¬(t and AA). after that you can either modify the AA (as long as it increase the theories falsifiability) or abandon the theory. Therefore you do not need the third box, it explains something that does not need explaining, or that could be explained more concisely and without error by reconstructing the process better. This process is always tentative and open to re-evaluation (that is the risky and critical nature of conjectures and refutations). Falsificationism does not pretend conclusiveness, it abandoned that to the scrap heap along with the hopelessly defective interpretation of science called inductivism.

“Contrary to Popper, there really is no asymmetry between falsification and verification. If we cannot verify theoretical statements, then we cannot falsify them either.” There is an asymmetry. You cannot refute the asymmetry by showing that falsification is not conclusive. Because the asymmetry is a logical relationship between statements. What you would have shown, if your argument was valid or accurate, would be that falsification is not possible in practice. Not that the asymmetry is false.

Popper wanted to replace induction and verification with deduction and falsification.

He held that a theory that was once accepted but which, thanks to a novel experiment or observation, turns out to be false, confronts us with a new problem, to which new solutions are needed. In his view, this process is the hallmark of scientific progress.

Surprisingly, Popper failed to note that, despite his efforts to present it as deductive, this process is at bottom inductive, since it assumes that a theory falsified today will remain falsified tomorrow.

Accepting that swans are either white or black because a black one has been spotted rests on the assumption that there are other black swans around and that the newly discovered black one will not become white at a later stage. It is obvious but also inductive thinking in the sense that they project the past into the future, that is, extrapolate particulars into a universal.

In other words, induction, the process that Popper was determined to avoid, lies at the heart of his philosophy of science as he defined it.

Despite positivism’s limitations, science is positive or it is not science : positive science’s theories are maybe incapable of demonstration (as Hume wrote of causation), but there are not others available.

If it is impossible to demonstrate that fire burns, putting one’s hand in it is just too painful.

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The Falsification Principle – Towards Empirical Testing

The Falsification Principle is a pivotal concept in the philosophy of science . It was proposed by philosopher Karl Popper in the mid-20th century. This principle serves as a criterion for demarcating scientific theories from non-scientific ones. It also emphasizes the importance of empirical testing and the potential for falsification in scientific inquiry.

This principle challenges the traditional notion of verification. It argues that scientific theories should be formulated in a way that allows for the possibility of being proven false through observation and experimentation.

Here, we will delve into the theoretical background of the Falsification Principle, and explore its key concepts and assumptions. Also, we examine the criticisms it has faced and analyze its applications and implications in various fields.

Further, we compare it with other rival principles, discuss its contemporary relevance, and contemplate its future directions. By exploring the Falsification Principle, we aim to gain a deeper understanding of the nature and methodology of scientific inquiry.

1. Introduction to the Falsification Principle

Definition of the falsification principle.

Imagine you have a theory that claims something is true. According to the Falsification Principle, for that theory to be considered scientific, it must be able to be proven false. In other words, a scientific theory should make specific predictions. If these are found to be false through empirical observation or experimentation, would lead us to reject the theory.

The Falsification Principle, proposed by philosopher Karl Popper, revolutionized the way we understand scientific knowledge. It challenged the prevailing idea that science could only confirm theories through empirical evidence. This principle instead emphasized the importance of potential falsifiability.

Historical Context and Origins

The Falsification Principle emerged in the mid-20th century as a response to the Logical Positivism movement. That movement aimed to define science solely by its empirical verifiability. Popper, dissatisfied with the idea that science could only confirm theories, sought to establish a demarcation criterion that would separate science from non-science.

Popper was inspired by the work of philosopher David Hume. He argued that induction, which forms the foundation of empirical verification, was logically flawed. Instead, he proposed that science should focus on deductive reasoning and the testing of hypotheses through attempts to falsify them.

2. Theoretical Background of the Falsification Principle

Influential thinkers and philosophical movements.

Popper’s Falsification Principle drew inspiration from philosophers such as Hume and Immanuel Kant. Hume’s skepticism towards induction highlighted the problem of confirming theories solely based on past observations. Kant’s emphasis on deductive reasoning and the existence of synthetic a priori knowledge influenced Popper’s thoughts on scientific methodology.

In terms of philosophical movements, the Falsification Principle emerged as a critique. It is an alternative to Logical Positivism , which dominated the Vienna Circle during the early 20th century. Popper’s ideas gained momentum and went on to shape the philosophy of science .

Development of the Falsification Principle

Popper’s development of the Falsification Principle took place in his groundbreaking book, “The Logic of Scientific Discovery,” published in 1934. Throughout subsequent revisions and editions, Popper refined and expanded upon his ideas. He solidified the Falsification Principle as a key component of his overall philosophy of science .

Popper’s discussions on falsifiability and the demarcation between science and pseudo-science attracted both praise and criticism from the scientific community. Nevertheless, his insights have had a lasting impact, influencing scientific methodology and shaping debates surrounding the nature of scientific knowledge.

3. Key Concepts and Assumptions of the Falsification Principle

Empirical observation and testability.

At the core of the Falsification Principle is the belief that scientific theories must be empirically testable. This means that theories should make predictions that can be subjected to observation or experimentation. The ability to test and potentially falsify theories distinguishes them from unfalsifiable claims that fall outside the realm of science .

Falsifiability and Refutability

Falsifiability refers to the potential for a theory to be proven false through empirical evidence. According to the Falsification Principle, a theory gains scientific legitimacy when it puts itself at risk of being refuted by specific observations or experiments. The more a theory withstands attempts to falsify it, the more confidence we can have in its validity.

It is crucial to note that the absence of falsification does not inherently prove a theory to be true or certain. The principle merely highlights the importance of subjecting theories to rigorous testing. It then finds specific evidence that could potentially refute them.

Language and Meaning in Science

The Falsification Principle also delves into the role of language and meaning in scientific discourse. Popper argued that scientific statements should be formulated in precise and testable terms. These should allow clear criteria to determine their falsifiability. He emphasized the importance of avoiding vague or ambiguous statements that could hinder empirical scrutiny.

By emphasizing the need for precise language and clear meaning, Popper aimed to create a framework for scientific discourse that promotes objective evaluation and critical scrutiny.

4. Critiques and Challenges to the Falsification Principle

Objections from popper’s contemporaries.

While the Falsification Principle made remarkable contributions to the philosophy of science , it faced notable criticisms from Popper’s contemporaries. Some argued that the principle was overly simplistic and failed to account for the complex nature of scientific practice. Others contended that it was difficult to draw a clear line between falsifiability and confirmation. This fact makes the demarcation between science and non-science less straightforward.

Revisions and Modifications to the Principle

Over time, various philosophers and scientists have proposed revisions and modifications to the Falsification Principle to address its limitations. These adjustments often aim to strike a balance between the importance of empirical testing and the recognition that scientific knowledge is open to revision and refinement.

Some argue for a more nuanced understanding of falsifiability, considering degrees of confidence rather than binary outcomes. Others advocate for the incorporation of Bayesian inference, which allows for the updating of probabilities based on new evidence.

The Falsification Principle may not provide a definitive answer to the nature of scientific knowledge. It however has its lasting impact and influence on scientific methodology cannot be denied. It continues to shape discussions and debates, reminding us of the importance of critical thinking. This theory is an inspiration for rigorous testing in the pursuit of scientific understanding.

5. Applications and Implications of the Falsification Principle

Scientific methodology and research design.

The falsification principle has had a profound impact on scientific methodology and research design. By emphasizing the importance of testing hypotheses through empirical observation, it has helped scientists develop rigorous and reliable methods for conducting experiments. Researchers now strive to design experiments that can potentially falsify their hypotheses, as this strengthens the validity of their findings.

Furthermore, the falsification principle has led to the development of stringent criteria for evaluating scientific theories. Instead of simply accepting theories based on supporting evidence, scientists now actively seek to challenge and disprove them. This approach ensures that scientific knowledge is constantly refined and updated. The theories that withstand rigorous testing are more likely to accurately explain the phenomena they seek to describe.

Impact on Philosophy of Science

The falsification principle has had a significant impact on the philosophy of science . It has challenged previously held notions that scientific theories can be verified or proven true. Instead, it is asserting that they can only be disproven or falsified. This shift in perspective has led to a more refined understanding of the nature of scientific inquiry.

Philosophers of science now recognize that the falsifiability criterion plays a crucial role in distinguishing scientific theories from non-scientific claims. By requiring empirical evidence that could potentially disprove a theory, the falsification principle helps maintain the integrity and credibility of scientific knowledge.

6. Comparison and Contrast with other Philosophical Principles

Verificationism vs. falsificationism.

Verificationism, an opposing philosophical principle, asserts that a statement is meaningful only if it can be empirically verified. In contrast, the falsification principle argues that a statement must be potentially falsifiable to be meaningful. While verificationism seeks to confirm statements, falsificationism focuses on refutation.

The key distinction between these principles lies in their approach to knowledge. Verificationism aims to establish truth through empirical verification, whereas falsificationism emphasizes the importance of actively seeking evidence that could disprove a theory. Falsificationism’s inclination towards skepticism and critical evaluation of theories sets it apart from the more affirmative stance of verificationism.

Inductivism vs. Falsificationism

Inductivism, another philosophical principle, suggests that knowledge can be gained by accumulating observations and generalizing from them. Falsificationism, on the other hand, argues that knowledge progresses through the process of hypothesis testing and potential falsification.

Inductivism relies on the accumulation of evidence to support theories. The falsificationism, on the other hand, places greater emphasis on the rigorous testing of hypotheses. Falsificationists contend that even a vast amount of supporting evidence cannot definitively prove a theory, as there may always be the possibility of contradictory evidence arising in the future. This critical stance distinguishes falsificationism from the more inductive approach of accumulating evidence and generalizing conclusions.

7. Contemporary Relevance and Influence of the Falsification Principle

Contributions to the philosophy of science.

The falsification principle continues to make significant contributions to the philosophy of science . Its emphasis on empirical testing and critical evaluation has fostered a culture of rigorous scientific inquiry. It helped maintain the integrity of scientific knowledge. The falsification principle has influenced the development of various scientific methodologies and research practices. The theory has ensured that theories are subject to thorough scrutiny and falsifiability.

Moreover, the falsification principle has sparked debates and discussions among philosophers, scientists, and scholars alike. Its challenges to traditional notions of verification and induction have fueled new avenues of inquiry and inspired novel approaches to understanding the nature of scientific knowledge.

Influence on Scientific Practices and Methodology

The impact of the falsification principle extends beyond the realm of philosophy . It has significantly influenced scientific practices and methodology, particularly in fields where empirical testing and hypothesis-driven research are essential.

Scientists now place greater importance on designing experiments that can potentially falsify their hypotheses. This approach ensures that their findings are robust and reliable. This approach has led to more rigorous research practices, improved experimental designs, and increased transparency in reporting scientific results.

Additionally, the falsification principle has encouraged scientists to critically evaluate existing theories and actively seek to disprove them. This rigorous approach to scientific inquiry has advanced knowledge across various disciplines. It is also leading to the refinement, revision, and even rejection of previously accepted theories.

Final Thoughts

The falsification principle has revolutionized scientific methodology, challenged traditional philosophical frameworks, and shaped our understanding of how knowledge is gained and refined. Its emphasis on empirical testing, critical evaluation, and potential falsifiability has paved the way for more rigorous scientific inquiry and the continuous advancement of knowledge.

In the future, the influence of the falsification principle is likely to persist, as scientists and philosophers continue to grapple with questions surrounding the nature of scientific knowledge and the boundaries of empirical inquiry. By encouraging skepticism, curiosity, and rigorous testing, the falsification principle will undoubtedly contribute to further breakthroughs and discoveries in the realms of science and philosophy . So, let’s keep questioning, testing, and falsifying to unlock the mysteries of the universe !

The Falsification Principle has had a profound impact on the philosophy of science , challenging traditional notions of verification and highlighting the importance of empirical testing and falsifiability in scientific inquiry. While the principle has faced criticisms and undergone revisions, its core ideas continue to shape scientific methodology and contribute to our understanding of the nature of scientific theories.

As we look to the future, further exploration and refinement of the Falsification Principle can help us refine our scientific practices, deepen our knowledge, and push the boundaries of human understanding. By embracing the principles of falsifiability, we can continue to strive for progress and uncover new insights in the fascinating realm of scientific research.

1. Why is the Falsification Principle important in the philosophy of science?

The Falsification Principle is important in the philosophy of science as it provides a criterion for distinguishing scientific theories from non-scientific ones. It emphasizes the significance of empirical testing and the potential for falsifying theories through observation and experimentation. By promoting falsifiability, the principle encourages the formulation of testable hypotheses and the advancement of scientific knowledge.

2. What are some criticisms of the Falsification Principle?

The Falsification Principle has faced criticisms from various philosophers and scientists. Some argue that it sets an unrealistic standard for scientific theories, as it is often challenging to completely falsify complex theories. Others contend that the principle neglects the importance of confirmation and that scientific progress relies on a combination of refutation and verification. Additionally, there have been debates surrounding the demarcation problem, questioning whether the principle can reliably distinguish between science and pseudoscience.

3. How does the Falsification Principle impact scientific methodology?

The Falsification Principle has had a significant impact on scientific methodology. It has influenced the way scientific hypotheses are formulated, emphasizing the need for clear predictions that can be tested and potentially falsified. This principle encourages scientists to design rigorous experiments and observations that can challenge and refine existing theories. It also promotes an open and critical approach to scientific inquiry, encouraging researchers to actively seek evidence that could refute their hypotheses.

4. What is the relationship between the Falsification Principle and other philosophical principles?

The Falsification Principle is often contrasted with other philosophical principles, most notably verificationism. Whereas verificationism suggests that scientific statements should be verifiable to be meaningful, falsificationism argues that falsifiability is a more appropriate criterion for scientific theories. Additionally, the principle has been compared to inductivism, which emphasizes the role of inductive reasoning in scientific inquiry. Exploring these relationships helps to understand the nuances and debates within the philosophy of science and the various approaches to evaluating scientific theories.

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Karl Popper

Karl Popper is generally regarded as one of the greatest philosophers of science of the twentieth century. He was also a social and political philosopher of considerable stature, a self-professed critical-rationalist, a dedicated opponent of all forms of scepticism and relativism in science and in human affairs generally and a committed advocate and staunch defender of the “Open Society”. One of the many remarkable features of Popper’s thought is the scope of his intellectual influence: he was lauded by Bertrand Russell, taught Imre Lakatos, Paul Feyerabend and philanthropist George Soros at the London School of Economics, numbered David Miller, Joseph Agassi, Alan Musgrave and Jeremy Shearmur amongst his research assistants, was counted by Thomas Szasz as “among my foremost teachers” and had close ties with the economist Friedrich Hayek and the art historian Ernst Gombrich. Additionally, Peter Medawar, John Eccles and Hermann Bondi are amongst the distinguished scientists who have acknowledged their intellectual indebtedness to his work, the latter declaring that “There is no more to science than its method, and there is no more to its method than Popper has said”.

2. Backdrop to Popper’s Thought

3. the problem of demarcation, 4. basic statements, falsifiability and convention, 5. the growth of human knowledge, 6. probability, knowledge and verisimilitude, 7. objective knowledge and the three worlds ontology, 8. social and political thought—the critique of historicism and holism, 9. scientific knowledge, history, and prediction, 10. immutable laws and contingent trends, 11. critical evaluation, primary literature: works by popper, secondary literature/other sources, other internet resources, related entries.

Karl Raimund Popper was born on 28 July 1902 in Vienna. His parents, who were of Jewish origin, brought him up in an atmosphere which he was later to describe as “decidedly bookish”. His father was a lawyer by profession, but he also took a keen interest in the classics and in philosophy, and communicated to his son an interest in social and political issues. His mother inculcated in him such a passion for music that for a time he contemplated taking it up as a career; he initially chose the history of music as a second subject for his Ph.D. examination. Subsequently, his love for music became one of the inspirational forces in the development of his thought, and manifested itself in his highly original interpretation of the relationship between dogmatic and critical thinking, in his account of the distinction between objectivity and subjectivity, and, most importantly, in the growth of his hostility towards all forms of historicism, including historicist ideas about the nature of the “progressive” in music. The young Karl attended the local Realgymnasium , where he was unhappy with the standards of the teaching, and, after an illness he left to attend the University of Vienna in 1918, matriculating four years later. In 1919 he became heavily involved in left-wing politics and became for a time a Marxist. However, he was quickly disillusioned with the doctrinaire character of the latter, and soon abandoned it entirely. He also discovered the psychoanalytic theories of Freud and Adler (he served briefly as a voluntary social worker with deprived children in one of the latter’s clinics in the 1920s), and heard Einstein lecture on relativity theory. The dominance of the critical spirit in Einstein, and its total absence in Marx, Freud and Adler, struck Popper as being of fundamental importance: the pioneers of psychoanalysis, he came to think, couched their theories in terms which made them amenable only to confirmation, while Einstein’s theory, crucially, had testable implications which, if false, would have falsified the theory itself.

Popper took some time to settle on a career; he trained as a cabinetmaker, obtained a primary school teaching diploma in 1925 and qualified to teach mathematics and physics in secondary school in 1929. He undertook a doctoral programme with the department of psychology at the University of Vienna, then under the supervision of Karl Bühler, one of the founder members of the Würzburg school of experimental psychology. Popper’s project was initially designed as a psychological investigation of human memory, on which he conducted initial research. However, the subject matter of a planned introductory chapter on methodology assumed a position of increasing pre-eminence and this resonated with Bühler, who, as a Kant scholar (a professor of philosophy and psychology), had famously addressed the issue of the contemporary “crisis in psychology”. This “crisis”, for Bühler, related to the question of the unity of psychology and had been generated by the proliferation of then competing paradigms within the discipline which had undermined the hitherto dominant associationist one and problematized the question of method. Accordingly, under Bühler’s direction, Popper switched his topic to the methodological problem of cognitive psychology and received his doctorate in 1928 for his dissertation “Zur Methodenfrage der Denkpsychologie”. In extending Bühler’s Kantian approach to the crisis in the dissertation, Popper critiqued Moritz Schlick’s neutral monist programme to make psychology scientific by transforming it into a science of brain processes. This latter ideal, Popper argued, was misconceived, but the issues raised by it ultimately had the effect of refocusing Popper’s attention away from Bühler’s question of the unity of psychology to that of its scientificity. This philosophical focus on questions of method, objectivity and claims to scientific status was to become a principal life-long concern, bringing the orientation of Popper’s thought into line with that of such contemporary “analytic” philosophers as Frege and Russell as well as that of many members of the Vienna Circle.

Popper married Josephine Anna Henninger (“Hennie”) in 1930, and she also served as his amanuensis until her death in 1985. At an early stage of their marriage they decided that they would never have children. In 1937 he took up a position teaching philosophy at the University of Canterbury in New Zealand, where he was to remain for the duration of the Second World War.

The annexation of Austria in 1938 became the catalyst which prompted Popper to refocus his writings on social and political philosophy. He published The Open Society and Its Enemies , his critique of totalitarianism, in 1945. In 1946 he moved to England to teach at the London School of Economics, and became professor of logic and scientific method at the University of London in 1949. From this point on his reputation and stature as a philosopher of science and social thinker grew, and he continued to write prolifically—a number of his works, particularly The Logic of Scientific Discovery (1959), are now widely seen as pioneering classics in the field. However, he combined a combative personality with a zeal for self-aggrandisement that did little to endear him to professional colleagues. He was ill-at-ease in the philosophical milieu of post-war Britain which was, as he saw it, fixated with trivial linguistic concerns dictated by Wittgenstein, whom he considered his nemesis. Popper’s commitment to the primacy of rational criticism was counterpointed by hostility towards anything that amounted to less than total acceptance of his own thought, and in Britain—as had been the case in Vienna—he increasingly became an isolated figure, though his ideas continued to inspire admiration.

In later years Popper came under philosophical criticism for his prescriptive approach to science and his emphasis on the logic of falsification. This was superseded in the eyes of many by the socio-historical approach taken by Thomas Kuhn in The Structure of Scientific Revolutions (1962). In that work, Kuhn, who argued for the incommensurability of rival scientific paradigms, denied that science grows linearly through the accumulation of truths.

Popper was knighted in 1965, and retired from the University of London in 1969, remaining active as a writer, broadcaster and lecturer until his death in 1994. (For more detail on Popper’s life, see his Unended Quest [1976]).

A number of biographical features may be identified as having a particular influence upon Popper’s thought. His teenage flirtation with Marxism left him thoroughly familiar with the Marxian dialectical view of economics, class-war, and history. But he was appalled by the failure of the democratic parties to stem the rising tide of fascism in Austria in the 1920s and 1930s, and the effective welcome extended to it by the Marxists, who regarded fascism as a necessary dialectical step towards the implosion of capitalism and the ultimate victory of communism. The Poverty of Historicism (1944; 1957) and The Open Society and Its Enemies (1945), Popper’s most impassioned and influential social works, are powerful defences of democratic liberalism, and strident critiques of philosophical presuppositions underpinning all forms of totalitarianism.

Popper was also profoundly impressed by the differences between the allegedly “scientific” theories of Freud and Adler and the revolution effected by Einstein’s theory of Relativity in physics in the first two decades of the twentieth century. The main difference between them, as Popper saw it, was that while Einstein’s theory was highly “risky”, in the sense that it was possible to deduce consequences from it which were, in the light of the then dominant Newtonian physics, highly improbable (e.g., that light is deflected towards solid bodies—confirmed by Eddington’s experiments in 1919), and which would, if they turned out to be false, falsify the whole theory, nothing could, even in principle , falsify psychoanalytic theories. They were, Popper argues, “simply non-testable, irrefutable. There was no conceivable human behaviour which could contradict them” (1963: 37). As such, they have more in common with myths than with genuine science; “They contain most interesting psychological suggestions, but not in a testable form” (1963: 38). What is apparently the chief source of strength of psychoanalysis, he concluded, viz. its capability to accommodate and explain every possible form of human behaviour, is in fact a critical weakness, for it entails that it is not, and could not be, genuinely predictive. To those who would respond that psychoanalytic theory is supported by clinical observations, Popper points out that

… real support can be obtained only from observations undertaken as tests (by ‘attempted refutations’); and for this purpose criteria of refutation have to be laid down beforehand: it must be agree which observable situations, if actually observed, mean that the theory is refuted. (1963: 38, footnote 3)

Popper also considers that contemporary Marxism also lacks scientific status. Unlike psychoanalysis, he argues, Marxism had been initially scientific, in that it was genuinely predictive. However, when these predictions were not in fact borne out, the theory was saved from falsification by the addition of ad hoc hypotheses which made it compatible with the facts. By this means, Popper asserts, a theory which was initially genuinely scientific degenerated into pseudo-scientific dogma. As he sees it, the Hegelian dialectic was adopted by Marxists not to oppose dogmatism but to accommodate it to their cause by eliminating the possibility of contradictory evidence. It has thus become what Popper terms “reinforced dogmatism” (1963: 334).

These factors combined to make Popper take falsifiability as his criterion for demarcating science from non-science: if a theory is incompatible with possible empirical observations it is scientific; conversely, a theory which is compatible with all such observations, either because, as in the case of Marxism, it has been modified solely to accommodate such observations, or because, as in the case of psychoanalytic theories, it is consistent with all possible observations, is unscientific. However, Popper is not a positivist and acknowledges that unscientific theories may be enlightening and that even purely mythogenic explanations have performed a valuable function in the past in expediting our understanding of the nature of reality.

For Popper the central problem in the philosophy of science is that of demarcation, i.e., of distinguishing between science and what he terms “non-science” (e.g., logic, metaphysics, psychoanalysis, and Adler’s individual psychology). Popper is unusual amongst contemporary philosophers in that he accepts the validity of the Humean critique of induction, and indeed, goes beyond it in arguing that induction is never actually used in science. However, he does not concede that this entails scepticism and argues that the Baconian/Newtonian insistence on the primacy of “pure” observation, as the initial step in the formation of theories, is completely misguided: all observation is selective and theory-laden and there are no pure or theory-free observations. In this way he destabilises the traditional view that science can be distinguished from non-science on the basis of its inductive methodology. In contradistinction to this, Popper holds that there is no unique methodology specific to science; rather, science, like virtually every other organic activity, consists largely of problem-solving.

Popper accordingly rejects the view that induction is the characteristic method of scientific investigation and inference, substituting falsifiability in its place. It is easy, he argues, to obtain evidence in favour of virtually any theory, and he consequently holds that such “corroboration”, as he terms it, should count scientifically only if it is the positive result of a genuinely “risky” prediction, which might conceivably have been false. In a critical sense, Popper’s theory of demarcation is based upon his perception of the asymmetry which, at the level of logic, holds between verification and falsification: it is logically impossible to verify a universal proposition by reference to experience (as Hume saw clearly), but a single genuine counter-instance falsifies the corresponding universal law. In a word, an exception, far from “proving” a rule, conclusively refutes it.

Every genuine scientific theory then, in Popper’s view, is prohibitive , because the theories of natural science take the form of universal statements. “All A s are X ” is equivalent to “No A is not- X ” which is falsified if “Some A is not- X ” turns out to be true. For example, the law of the conservation of energy can be expressed as “There is no perpetual motion machine”.

However, the universality of such laws, he argues, does rule out the possibility of their verification. Thus, a theory that has withstood rigorous testing should be deemed to have received a high measure of corroboration. and may be retained provisionally as the best available theory until it is finally falsified and/or is superseded by a better theory.

Popper stresses in particular that there is no unique way, no single method such as induction, which functions as the route to scientific theory, and approvingly cites Einstein on that point:

There is no logical path leading to [the highly universal laws of science]. They can only be reached by intuition, based upon something like an intellectual love of the objects of experience. (2002: 8–9)

Science, in Popper’s view, starts with problems rather than with observations—it is, indeed, precisely in the context of grappling with a problem that the scientist makes observations in the first instance: his observations are selectively designed to test the extent to which a given theory functions as a satisfactory solution to a given problem.

On this criterion of demarcation physics, chemistry, and (non-introspective) psychology, amongst others, are classified as sciences, psychoanalysis is a pre-science and astrology and phrenology are pseudo-sciences.

Popper draws a clear distinction between the logic of falsifiability and its applied methodology . The logic of his theory is utterly simple: a universal statement is falsified by a single genuine counter-instance. Methodologically, however, the situation is complex: decisions about whether to accept an apparently falsifying observation as an actual falsification can be problematic, as observational bias and measurement error, for example, can yield results which are only apparently incompatible with the theory under scrutiny.

Thus, while advocating falsifiability as the criterion of demarcation for science, Popper explicitly allows for the fact that in practice a single conflicting or counter-instance is never sufficient methodologically for falsification, and that scientific theories are often retained even though much of the available evidence conflicts with them, or is anomalous with respect to them.

In this connection, in the Logic of Scientific Discovery Popper introduces the technical concept of a “basic statement” or “basic proposition”, which he defines as a statement which can serve as a premise in an empirical falsification and which takes the singular existential form “There is an X at Y ”. Basic statements are important because they can formally contradict universal statements, and accordingly play the role of potential falsifiers. To take an example, the (putative) basic statement “In space-time region k there is an apparatus which is a perpetual motion machine” contradicts the law of the conservation of energy, and would, if true, falsify it (2002: 48). Accordingly, Popper holds that basic statements are objective and are governed by two requirements: (a) the formal, that they must be both singular and existential and (b) the material, that they must be intersubjectively testable.

In essence, basic statements are for Popper logical constructs which embrace and include ‘observation statements’, but for methodological reasons he seeks to avoid that terminology, as it suggests that they are derived directly from, and known by, experience (2002: 12, footnote 2), which would conflate them with the “protocol” statements of logical positivism and reintroduce the empiricist idea that certain kinds of experiential reports are incorrigible. The “objectivity” requirement in Popper’s account of basic statements, by contrast, amounts to a rejection of the view that the truth of scientific statements can ever be reduced to individual or collective human experience. (2002: 25).

Popper therefore argues that there are no statements in science which cannot be interrogated: basic statements, which are used to test the universal theories of science, must themselves be inter-subjectively testable and are therefore open to the possibility of refutation. He acknowledges that this seems to present a practical difficulty, in that it appears to suggest that testability must occur ad infinitum , which he acknowledges is an operational absurdity: sooner or later all testing must come to an end. Where testing ends, he argues, is in a convention-based decision to accept a basic statement or statements; it is at that point that convention and intersubjective human agreement play an indispensable role in science:

Every test of a theory, whether resulting in its corroboration or falsification, must stop at some basic statement or other which we decide to accept . If we do not come to any decision, and do not accept some basic statement or other, then the test will have led nowhere. (2002: 86)

However, Popper contends that while such a decision is usually causally related to perceptual experience, it is not and cannot be justified by such experience; basic statements are experientially underdetermined.

Experiences can motivate a decision, and hence an acceptance or a rejection of a statement, but a basic statement cannot be justified by them—no more than by thumping the table. (2002: 87–88)

Statements can be justified only by other statements, and therefore testing comes to an end, not in the establishment of a correlation between propositional content and observable reality, as empiricism would hold, but by means of the conventional, inter-subjective acceptance of the truth of certain basic statements by the research community.

The acceptance of basic statements is compared by Popper to trial by jury: the verdict of the jury will be an agreement in accordance with the prevailing legal code and on the basis of the evidence presented, and is analogous to the acceptance of a basic statement by the research community:

By its decision, the jury accepts, by agreement, a statement about a factual occurrence—a basic statement, as it were. (2002: 92)

The jury’s verdict is conventional in arising out of a procedure governed by clear rules, and is an application of the legal system as a whole as it applies to the case in question. The verdict is accordingly represented as a true statement of fact, but, as miscarriages of justice demonstrate all too clearly,

the statement need not be true merely because the jury has accepted it. This … is acknowledged in the rule allowing a verdict to be quashed or revised. (2002: 92)

This is comparable, he argues, to the case of basic statements: their acceptance-as-true is also by agreement and, as such, it also constitutes an application of a theoretical system, and

it is only this application which makes any further applications of the theoretical system possible. (2002: 93)

However, the agreed acceptance of basic statements, like that of judicial verdicts, remain perennially susceptible to the requirement for further interrogation. Popper terms this “the relativity of basic statements” (2002: 86), which is reflective of the provisional nature of the entire corpus of scientific knowledge itself. Science does not, he maintains, rest upon any foundational bedrock. Rather, the theoretical systems of science are akin to buildings in swampy ground constructed with the support of piles:

The piles are driven down from above into the swamp, but not down to any natural or “given” base; and if we stop driving the piles deeper, it is not because we have reached firm ground. We simply stop when we are satisfied that the piles are firm enough to carry the structure, at least for the time being. (2002: 94)

For Popper, the growth of human knowledge proceeds from our problems and from our attempts to solve them. These attempts involve the formulation of theories which must go beyond existing knowledge and therefore require a leap of the imagination. For this reason, he places special emphasis on the role played by the creative imagination in theory formulation. The priority of problems in Popper’s account of science is paramount, and it is this which leads him to characterise scientists as “problem-solvers”. Further, since the scientist begins with problems rather than with observations or “bare facts”, he argues that the only logical technique which is an integral part of scientific method is that of the deductive testing of theories which are not themselves the product of any logical operation. In this deductive procedure conclusions are inferred from a tentative hypothesis and are then compared with one another and with other relevant statements to determine whether they falsify or corroborate the hypothesis. Such conclusions are not directly compared with the facts, Popper stresses, simply because there are no “pure” facts available; all observation-statements are theory-laden, and are as much a function of purely subjective factors (interests, expectations, wishes, etc.) as they are a function of what is objectively real.

How then does the deductive procedure work? Popper specifies four steps (2002: 9):

• The first is formal , a testing of the internal consistency of the theoretical system to see if it involves any contradictions.
• The second step is semi-formal , “the investigation of the logical form of the theory, with the object of determining whether it has the character of an empirical or scientific theory, or whether it is, for example, tautological” (2002: 9).
• The third step is the comparing of the new theory with existing ones to determine whether it constitutes an advance upon them. If its explanatory success matches that of the existing theories, and it additionally explains some hitherto anomalous phenomenon or solves some hitherto unsolvable problems, it will be adopted as constituting an advance upon the existing theories. In that sense, science involves theoretical progress: on this account, a theory X is better than a “rival” theory Y if X has greater empirical content , and hence greater predictive power , than Y .
• The fourth and final step is the testing of a theory by the empirical application of the conclusions derived from it. If such conclusions are shown to be true, the theory is corroborated (but never verified). If the conclusion is shown to be false, then this is taken as a signal that the theory cannot be completely correct (logically the theory is falsified), and the scientist begins his quest for a better theory. He does not, however, abandon the present theory until such time as he has a better one to substitute for it.

The general picture of Popper’s philosophy of science, then is this: Hume’s philosophy demonstrates that there is a contradiction implicit in traditional empiricism, which holds that universal scientific laws are in some way finally confirmable by experience, despite the open-ended nature of the latter being acknowledged. Popper eliminates the contradiction by removing the demand for empirical verification in favour of empirical falsification or corroboration. Scientific theories, for him, are not inductively inferred from experience, nor is scientific experimentation carried out with a view to verifying or finally establishing the truth of theories; rather, all knowledge is provisional, conjectural, hypothetical —the universal theories of science can never be conclusively established. Hence Popper’s emphasis on the importance of the critical spirit to science—for him critical thinking is the very essence of rationality. For it is only by critical thought that we can eliminate false theories and determine which of the remaining theories is the best available one, in the sense of possessing the highest level of explanatory force and predictive power.

In the view of many social scientists, the more probable a theory is, the better it is, and if we have to choose between two theories which differ only in that one is probable and the other is improbable, then we should choose the former. Popper rejects this. Science values theories with a high informative content, because they possess a high predictive power and are consequently highly testable. For that reason, the more improbable a theory is the better it is scientifically, because the probability and informative content of a theory vary inversely—the higher the informative content of a theory the lower will be its probability. Thus, the statements which are of special interest to science are those with a high informative content and (consequentially) a low probability, which nevertheless come close to the truth . Informative content, which is in inverse proportion to probability, is in direct proportion to testability. As a result, the severity of the test to which a theory can be subjected, and by means of which it is falsified or corroborated, is of fundamental importance.

Popper also argues that all scientific criticism must be piecemeal, i.e., he holds that it is not possible to question every aspect of a theory at once, as certain items of what he terms “background knowledge” must be taken for granted. But that is not knowledge in the sense of being conclusively established; it may be challenged at any time, especially if it is suspected that its uncritical acceptance may be responsible for difficulties which are subsequently encountered.

How then can one be certain that one is questioning the right thing? The Popperian answer is that we cannot have absolute certainty here, but repeated tests usually show where the trouble lies. As we saw, for Popper even observation statements are corrigible and open to review, and science in his view is not a quest for certain knowledge, but an evolutionary process in which hypotheses or conjectures are imaginatively proposed and tested in order to explain facts or to solve problems. For that reason, he emphasises both the importance of questioning the background knowledge when the need arises, and the significance of the fact that observation-statements are theory-laden and corrigible.

Popper was initially uneasy with the concept of truth, and in his earliest writings he avoided asserting that a theory which is corroborated is true—for clearly if every theory is an open-ended hypothesis, then ipso facto it has to be at least potentially false. However, he came to accept Tarski’s reformulation of the correspondence theory of truth, and in Conjectures and Refutations (1963) he integrates the concepts of truth and content to frame the metalogical concept of “truthlikeness” or “ verisimilitude ”. A “good” scientific theory, Popper argues in that work, has a higher level of verisimilitude than its rivals, and he explicates this concept by reference to the logical consequences of theories. A theory’s content is the totality of its logical consequences, which can be divided into two classes:

• the “ truth-content ”, which is the class of true propositions which may be derived from it, and
• the “ falsity-content ” of a theory, which is the class of the theory’s false consequences (which may be empty, and in the case of a theory which is true is necessarily empty).

Popper offers two accounts of how rival theories can be compared in terms of their levels of verisimilitude; these are the qualitative and quantitative definitions. On the qualitative account, verisimilitude is defined in terms of subclass relationships: a theory \(t_2\) has a higher level of verisimilitude than \(t_1\) if and only if their truth- and falsity-contents are comparable through subclass relationships, and either

• \(t_2\)’s truth-content includes \(t_1\)’s and \(t_2\)’s falsity-content, if it exists, is included in, or is the same as, \(t_1\)’s, or
• \(t_2\)’s truth-content includes or is the same as \(t_1\)’s and \(t_2\)’s falsity-content, if it exists, is included in \(t_1\)’s.

On the quantitative account, verisimilitude is defined by assigning quantities to contents, where the index of the content of a given theory is its logical improbability, given again that content and probability vary inversely (1963: 233–4). The utilisation of either method of computing verisimilitude shows, Popper argues, that even if a theory \(t_2\) with a higher content than a rival theory \(t_1\) is subsequently falsified, it can still legitimately be regarded as a better theory than \(t_1\), and “better” is here now understood to mean \(t_2\) is closer to the truth than \(t_1\) (1963: 235).

Thus, scientific progress involves, on this view, the abandonment of partially true, but falsified, theories, for theories with a higher level of verisimilitude, i.e., which approach more closely to the truth. In this way, verisimilitude allowed Popper to mitigate what some saw as the pessimism of an anti-inductivist philosophy of science. With the introduction of the concept, Popper was able to represent his account as an essentially realistic position in terms of which scientific progress can be seen as progress towards the truth, and experimental corroboration can be viewed as an indicator of verisimilitude.

However, subsequent research revealed defects in Popper’s formal definitions of verisimilitude. The concept is most important in his system because of its application to theories which are approximations (which are common in the social sciences) and thus known to be false . In this connection, Popper had written:

Ultimately, the idea of verisimilitude is most important in cases where we know that we have to work with theories which are at best approximations—that is to say, theories of which we know that they cannot be true… In these cases we can still speak of better or worse approximations to the truth (and we therefore do not need to interpret these cases in an instrumentalist sense). (1963: 235)

In 1974, David Miller and Pavel Tichý, working independently, demonstrated that the conditions specified by Popper in his accounts of both qualitative and quantitative verisimilitude for comparing the truth- and falsity-contents of theories can be satisfied only when the theories are true . In the crucially important case of false theories, Popper’s definitions are formally defective: for with regard to a false theory \(t_2\) which has excess content over a rival theory \(t_1\), both the truth-content and the falsity-content of \(t_2\) will exceed that of \(t_1\). With respect to theories which are false then, Popper’s conditions for comparing levels of verisimilitude, whether in quantitative and qualitative terms, can never be met.

Popper’s response was two-fold. In the first place, he acknowledges the deficiencies in his own formal account:

“…my main mistake was the failure to see at once that my ’A Theorem on Truth-Content’ [1966], could be extended to falsity content: if the content of a false statement a exceeds that of a statement b , then the truth content of a exceeds the truth content of b , and the same holds of their falsity contents.” (1979, 371)

But he argues that

“I do think that we should not conclude from the failure of my attempts to solve the problem that the problem cannot be solved. Perhaps it cannot be solved by purely logical means but only by a relativization to relevant problems or even by bringing in the historical problem situation” (1979, 372).

This suggestion was to precipitate a great deal of important technical research in the field. He additionally moves the task of formally defining the concept from centre-stage in his philosophy of science by protesting that he had never intended to imply

that degrees of verisimilitude … can ever be numerically determined, except in certain limiting cases. (1979: 59)

Instead, he argues, the chief value of the concept is heuristic, in which the absence of an adequate formal definition is not an insuperable impediment to its utilisation in the actual appraisal of theories relativised to problems in which we have an interest.

Many see the thrust of this latter strategy as genuinely reflecting the significance of the concept of verisimilitude in Popper’s system, but it has not satisfied all of his critics.

Popper’s hostility to psychologistic approaches to epistemology is evident from his earliest works. Questions relating to the origins of convictions, feelings of certainty and the like, he argues, are properly considered the province of psychology; their attempted use in epistemology, which has been characteristic in particular of some schools of empiricism, can lead only to confusion and ultimately to scepticism. Against it, he repeatedly insists on the objectivity of scientific knowledge and sees it as the principal task of epistemology to engage with the questions of justification and validity in that connection (2002: 7).

In some of his later works in the field, particularly his “Epistemology Without a Knowing Subject” (1967, republished in Objective Knowledge [1972]) and in his lecture “Three Worlds” in the Tanner Lectures on Human Values delivered in 1978 (published 1980), Popper develops the notion of objectivity further in a novel but controversial way by seeking to free it completely from all psychological constraints. What is central to epistemology, he reaffirms, is the concept of objectivity, which he seeks to show requires neither the notions of subjective mental states nor even that of a subject “possessing” knowledge: knowledge in its full objective sense, he argues, is knowledge “without a knowing subject”.

Popper’s angle of approach here is to situate the development of knowledge in the context of animal and human evolution. It is characteristic of evolutionary processes, he points out, that they come to take place in an environment which is itself in part fashioned by the species in question. Examples of this abound, such as the reefs built by corals, the hives built by bees, the dams built by beavers and the atmospheric effects yielded by plants. This, Popper contends, is also true of human beings: we too have created new kinds of products, “intellectual products”, which shape our environment. These are our myths, our ideas, our art works and our scientific theories about the world in which we live. When placed in an evolutionary context, he suggests, such products must be viewed instrumentally, as exosomatic artefacts. Chief amongst them is knowledge

in the objective or impersonal sense, in which it may be said to be contained in a book; or stored in a library; or taught in a university. (1979: 286)

On this view, termed “evolutionary epistemology”, the growth of human knowledge is an objective evolutionary process which involves the creation and promulgation of new problem-solving theories, which are then subjected to the challenge of criticism, modification, elimination and replacement. These latter activities are seen by Popper as growth-promoting in the evolution of knowledge, which he represents by means of a tetradic schema (1979: 287):

Here “\(P_1\)” stands for the “initial problem”; “ TT ” stands for “tentative theory” designed to solve it, “ EE ” stands for “attempts at error-elimination”, and “\(P_2\)” represents further problems that arise out of the critical process.

This kind of knowledge development, Popper argues, cannot be explained either by physicalism, which seeks to reduce all mental processes and states to material ones, or by dualism, which usually seeks to explicate knowledge by means of psychological categories such as thought, perception and belief. Consequently, he proposes a novel form of pluralistic realism, a “Three Worlds” ontology, which, while accommodating both the world of physical states and processes (world 1) and the mental world of psychological processes (world 2), represents knowledge in its objective sense as belonging to world 3, a third, objectively real ontological category. That world is the world

of the products of the human mind, such as languages; tales and stories and religious myths; scientific conjectures or theories, and mathematical constructions; songs and symphonies; paintings and sculptures. (1980: 144)

In short, world 3 is the world of human cultural artifacts, which are products of world 2 mental processes, usually instantiated in the physical world 1 environment.

Popper proceeds to explicate his distinction between the subjective and objective senses of knowledge by reference to this ontology. The subjective sense of knowledge relates to psychological processes and states, mental dispositions, beliefs and expectations, which may generically be termed “thought processes” and which belong to world 2. Knowledge in the objective sense, by contrast, consists not of thought processes but of thought contents , that is to say, the content of propositionalised theories: it is

the content which can be, at least approximately, translated from one language into another. The objective thought content is that which remains invariant in a reasonably good translation. (1980: 156)

And it is that thought content, when linguistically codified in texts, works of art, log tables, mathematical formulae, which constitutes world 3, to which objective knowledge relates.

For those who would suggest that such objects are mere abstractions from world 2 thought processes, Popper counters that world 3 objects are necessarily more than the thought processes which have led to their creation. Theories, for example, usually have logical implications beyond anything considered by their original author, as instanced in the case of Einstein’s Special Theory of Relativity. Moreover, what is most characteristic about such objects is that, unlike world 2 mental processes, they can stand in logical relationships to each other, such as equivalence, deducibility and compatibility, which makes them amenable to the kind of critical rational analysis and development that is one of the hallmarks of science. As he puts it,

Criticism of world 3 objects is of the greatest importance, both in art and especially in science. Science can be said to be largely the result of the critical examination and selection of conjectures, of thought contents. (1980: 160)

Popper takes Michelangelo’s sculpture The Dying Slave as an illustrative example of a world 3 object, embodied in a world 1 block of marble. Other examples given include memory engrams in the brain, the American Constitution, Shakespeare’s tragedies, Beethoven’s Fifth Symphony and Newton’s theory of gravitation. Each one of these, he contends, is a world 3 object that transcends both its physical, world 1 embodiments and its world 2 cognitive origins (1980: 145).

Popper was aware that he would be accused of hypostatising abstractions in asserting the reality and objectivity of world 3 objects. In response, he indicates strongly that he has no interest in what he regards as pointless terminological disputes over the meaning of the term “world” or “real”. He is therefore content, if required, to express his account of objective knowledge in more familiar and perhaps more mundane terms: world 3 objects are abstract objects while their physical embodiments are concrete objects. But that should not be allowed to disguise the fact that he saw the relationships between the three categories of his ontology as of enormous importance in understanding the role of science as an element of culture:

my thesis is that our world 3 theories and our world 3 plans causally influence the physical objects of world 1; that they have a causal action upon world 1. (1980: 164)

In the final analysis it is the causal interaction between the worlds that ultimately matters in Popper’s objectivist epistemology: it allows him to represent the growth of human knowledge as an evolutionary process of exosomatic adaptations, which is ultimately a function of the interplay of relations between the physical and mental worlds and the world of objective knowledge or thought content.

Given Popper’s personal history and background, it is hardly surprising that he developed a deep and abiding interest in social and political philosophy. He understood holism as the view that human social groupings are greater than the sum of their members, that they act on their human members and shape their destinies and that they are subject to their own independent laws of development. Historicism he identified as the belief that history develops inexorably and necessarily according to certain principles or rules towards a determinate end (as for example in the Marx’s dialectical materialism). The link between them is that holism holds that individuals are essentially formed by the social groupings to which they belong, while historicism suggests that we can understand such a social grouping only in terms of the internal principles which determine its development.

These lead to what Popper calls “The Historicist Doctrine of the Social Sciences”, the views (a) that the principal task of the social sciences is to make predictions about the social and political development of man, and (b) that the task of politics, once the key predictions have been made, is, in Marx’s words, to lessen the “birth pangs” of future social and political developments. Popper thinks that this view of the social sciences is both theoretically misconceived and socially dangerous, as it can give rise to totalitarianism and authoritarianism—to centralised governmental control of the individual and the attempted imposition of large-scale social planning. Against this, he advances the view that any human social grouping is no more (or less) than the sum of its individual members, that what happens in history is the (largely unforeseeable) result of the actions of such individuals, and that large scale social planning to an antecedently conceived blueprint is inherently misconceived—and inevitably disastrous—precisely because human actions have consequences which cannot be foreseen. Popper, then, is an historical indeterminist , insofar as he holds that history does not evolve in accordance with intrinsic laws or principles, that in the absence of such laws and principles unconditional prediction in the social sciences is an impossibility, and that there is no such thing as historical necessity.

The link between Popper’s theory of knowledge and his social philosophy is his fallibilism. We make theoretical progress in science by subjecting our theories to critical scrutiny, and abandoning those which have been falsified. So too in an open society the rights of the individual to criticise administrative policies will be safeguarded and upheld, undesirable policies will be eliminated in a manner analogous to the elimination of falsified scientific theories, and political differences will be resolved by critical discussion and argument rather than by coercion. The open society as thus conceived of by Popper may be defined as

an association of free individuals respecting each other’s rights within the framework of mutual protection supplied by the state, and achieving, through the making of responsible, rational decisions, a growing measure of humane and enlightened life. (R. B. Levinson 1953: 17)

Such as society is not a utopian ideal, Popper argues, but an empirically realised form of social organisation which is in every respect superior to its (real or potential) totalitarian rivals. His strategy, however, is not merely to engage in a moral defence of the ideology of liberalism, but rather to show that totalitarianism is typically based upon historicist and holist presuppositions, and of demonstrating that these presuppositions are fundamentally incoherent.

Historicism and holism, Popper argues, have their origins in what he terms

one of the oldest dreams of mankind—the dream of prophecy, the idea that we can know what the future has in store for us, and that we can profit from such knowledge by adjusting our policy to it. (1963: 338)

This dream was given impetus, he suggests, by the emergence of a genuine predictive capability regarding solar and lunar eclipses at an early stage in human civilisation, which became refined with the development of the natural sciences. Historicism derives a superficial plausibility from the suggestion that, just as the application of the laws of the natural sciences can lead to the prediction of such events as eclipses, knowledge of “the laws of history” as yielded by a social science or sciences can and should lead to the prediction of future social phenomena. Why should we not conceive of a social science which would function as the theoretical natural sciences function and yield precise unconditional predictions in the appropriate sphere of application? Popper seeks to show that this idea is based upon a series of misconceptions about the nature of science, and about the relationship between scientific laws and scientific prediction.

In relation to the critically important concept of prediction, Popper makes a distinction between what he terms “conditional scientific predictions”, which have the form “If X takes place, then Y will take place”, and “unconditional scientific prophecies”, which have the form “ Y will take place”. Contrary to popular belief, it is the former rather than the latter which are typical of the natural sciences, which means that typically prediction in natural science is conditional and limited in scope—it takes the form of hypothetical assertions stating that certain specified changes will come about if and only if particular specified events antecedently take place. This is not to deny that “unconditional scientific prophecies”, such as the prediction of eclipses, for example, do take place in science, and that the theoretical natural sciences make them possible. However, Popper argues that (a) these unconditional prophecies are not characteristic of the natural sciences, and (b) that the mechanism whereby they occur, in the very limited way in which they do, is not understood by the historicist.

What is the mechanism which makes “unconditional scientific prophecies” possible? Popper’s answer is that they are possible because they are derived from a combination of conditional predictions (themselves derived from scientific laws) and existential statements specifying that the conditions in relation to the system being investigated are fulfilled.

Given that this is the mechanism which generates unconditional scientific prophecies, Popper makes two related claims about historicism:

The first is that the historicist does not, as a matter of fact, derive his historical prophecies from conditional scientific predictions. The second … is that he cannot possibly do so because long term prophecies can be derived from scientific conditional predictions only if they apply to systems which can be described as well isolated, stationary, and recurrent. These systems are very rare in nature; and modern society is surely not one of them. (1963: 339)

Popper accordingly argues that it is a fundamental mistake for the historicist to take the unconditional scientific prophecies of eclipses as being typical and characteristic of the predictions of natural science; they are possible only because our solar system is a stationary and repetitive system which is isolated from other such systems by immense expanses of empty space. Human society and human history are not isolated systems and are continually undergoing rapid, non-repetitive development. In the most fundamental sense possible, every event in human history is discrete, novel, quite unique, and ontologically distinct from every other historical event. For this reason, it is impossible in principle that unconditional scientific prophecies could be made in relation to human history—the idea that the successful unconditional prediction of eclipses provides us with reasonable grounds for the hope of successful unconditional prediction regarding the evolution of human history turns out to be based upon a gross misconception. As Popper succinctly concludes, “The fact that we predict eclipses does not, therefore, provide a valid reason for expecting that we can predict revolutions” (1963: 340).

An additional mistake which Popper discerns in historicism is the failure of the historicist to distinguish between scientific laws and trends . This makes him think it possible to explain change by discovering trends running through past history, and to anticipate and predict future occurrences on the basis of such observations. Here Popper points out that there is a critical difference between a trend and a scientific law: the latter is universal in form, while a trend can be expressed only as a singular existential statement. This logical difference is crucial: neither conditional nor unconditional predictions can be based upon trends, because trends may change or be reversed with a change in the conditions which gave rise to them in the first instance. As Popper puts it, there can be no doubt that

the habit of confusing trends with laws, together with the intuitive observation of trends such as technical progress, inspired the central doctrines of … historicism. (1957: 106)

He does not, of course, dispute the existence of trends or deny that observing them can be of practical utility value. But the essential point is that a trend is something which itself ultimately stands in need of scientific explanation, and it cannot therefore function as the frame of reference in terms of which an unconditional prediction can be based.

A point which connects with this has to do with the role which the evolution of human knowledge has played in the historical development of human society. Human history has, Popper points out, been strongly influenced by the growth of human knowledge , and it is extremely likely that this will continue to be the case—all the empirical evidence suggests that the link between the two is progressively consolidating. However, this gives rise to a further problem for the historicist: no scientific predictor, human or otherwise, can possibly predict its own future results. From this it follows, he holds, that no society can predict, scientifically, its own future states of knowledge. Thus, while the future evolution of human history is extremely likely to be influenced by new developments in human knowledge, we cannot now scientifically determine what such knowledge will be.

From this it follows that if the future holds any new discoveries or any new developments in the growth of our knowledge, then it is impossible for us to predict them now, and it is therefore impossible for us to predict the future development of human history now, given that the latter will, at least in part, be determined by the future growth of our knowledge. Thus, once again historicism collapses—the dream of a theoretical, predictive science of history is unrealisable, because it is an impossible dream.

Popper’s argues against the propriety of large-scale planning of social structures on the basis of this demonstration of the logical shortcomings of historicism. It is, he argues, theoretically as well as practically misguided, because, again, part of what we are planning for is our future knowledge, and our future knowledge is not something which we can in principle now possess—we cannot adequately plan for unexpected advances in our future knowledge, or for the effects which such advances will have upon society as a whole. For him, this necessitates the acceptance of historical indeterminism as the only philosophy of history which is commensurate with a proper understanding of the provisional and incomplete nature of human knowledge.

This critique of historicism and holism is balanced, on the positive side, by Popper’s affirmation of the ideals of individualism and market economics and his strong defence of the open society—the view that a society is equivalent to the sum of its members, that the actions of the members of society serve to fashion and to shape it, and that the social consequences of intentional actions are very often, and very largely, unintentional. This part of his social philosophy was influenced by the economist Friedrich Hayek, who worked with him at the London School of Economics and who was a life-long friend. Popper advocates what he (rather unfortunately) terms “piecemeal social engineering” as the central mechanism for social planning: in utilising this mechanism, intentional actions are directed to the achievement of one specific goal at a time, which makes it possible to determine whether adverse unintended effects of intentional actions occur, in order to correct and readjust when this proves necessary. This, of course, parallels precisely the critical testing of theories in scientific investigation. This approach to social planning (which is explicitly based upon the premise that we do not, because we cannot, know what the future will be like) encourages attempts to put right what is problematic in society—generally-acknowledged social ills—rather than attempts to impose some preconceived idea of the “good” upon society as a whole. For Popper, in a genuinely open society piecemeal social engineering goes hand-in-hand with negative utilitarianism, the attempt to minimise the amount of suffering and misery, rather than, as with positive utilitarianism, the attempt to maximise the amount of happiness. The state, he holds, should concern itself with the task of progressively formulating and implementing policies designed to deal with the social problems which actually confront it, with the goal of mitigating human misery and suffering to the greatest possible degree. The positive task of increasing social and personal happiness, by contrast, can and should be left to individual citizens, who may, of course, act collectively to that end. “My thesis”, Popper states, is that

human misery is the most urgent problem of a rational public policy and that happiness is not such a problem. The attainment of happiness should be left to our private endeavours. (1963: 361)

Thus, for Popper, in the final analysis the activity of problem-solving is as definitive of our humanity at the level of social and political organisation as it is at the level of science, and it is this key insight which unifies and integrates the broad spectrum of his thought.

While it cannot be said that Popper was modest, he took criticism of his theories very seriously, and spent much of his time in his later years in addressing them. The following is a summary of some of the main ones which he had to address. (For Popper’s responses to critical commentary, see his “Replies to My Critics” (1974) and his Realism and the Aim of Science (1983).

First, Popper claims to be a realist and rejects conventionalist and instrumentalist accounts of science. But his account in the Logic of Scientific Discovery of the role played by basic statements in the methodology of falsification seems to sit uneasily with that. As we have seen, he follows Kant in rejecting the positivist/empiricist view that observation statements are incorrigible, and argues that they are descriptions of what is observed as interpreted by the observer with reference to a determinate conceptual and theoretical framework. He accordingly asserts that while basic statements may have a causal relationship to experience, they are neither determined nor justified by it.

However, this would seem to pose a difficulty: if a theory is to be genuinely testable, it must be possible to determine, at least in principle, whether the basic statements which are its potential falsifiers are actually true or false. But how can this be known, if basic statements cannot be justified by experience? As we have seen, Popper’s answer is that the acceptance or rejection of basic statements depends upon a convention-based decision on the part of the scientific community.

From a logical point of view, the testing of a theory depends upon basic statements whose acceptance or rejection, in its turn, depends upon our decisions. Thus it is decisions which settle the fate of theories. (2002: 91)

Traditional conventionalism, as exemplified in the work of Mach, Poincaré and Milhaud amongst others, holds that a “successful” science is one in which universal theories have assumed such explanatory critical mass that a convention emerges to pre-empt the possibility of their empirical refutation. This is strongly rejected by Popper, who differentiates his position from it by arguing that it is the acceptance of basic statements, rather than that universal theory, which is determined by convention and intersubjective agreement. For him, the acceptance or rejection of theory occurs only indirectly and at a higher investigative level, through critical tests made possible by the conventional acceptance of basic statements. As he puts it, “I differ from the conventionalist in holding that the statements decided by agreement are not universal but singular” (2002: 92). Simultaneously, however, he rejects any suggestion that basic statements are justifiable by direct experience:

I differ from the positivist in holding that basic statements are not justifiable by our immediate experiences, but are, from the logical point of view, accepted by an act, by a free decision. (2002: 92)

It is thus evident that Popper saw his account of basic statements as steering a course between the Scylla of orthodox conventionalism and the Charybdis of positivism/empiricism. However, while it is both coherent and consistent in that regard, there can be little doubt but that it constitutes a form of conventionalism in its own right. And it is not clear that it is compatible with scientific realism, understood as the view that scientific theories give true or approximately true descriptions of elements of a mind-independent world. As Lakatos puts it,

If a theory is falsified, it is proven false; if it is “falsified” [in Popper’s conventionalist sense], it may still be true. If we follow up this sort of “falsification” by the actual “elimination” of a theory, we may well end up by eliminating a true, and accepting a false, theory. (Lakatos 1978: 24)

Second, Popper’s theory of demarcation hinges quite fundamentally on the assumption that there are such things as critical tests, which either falsify a theory, or give it a strong measure of corroboration. Popper himself is fond of citing, as an example of such a critical test, the resolution, by Adams and Leverrier, of the problem which the anomalous orbit of Uranus posed for nineteenth century astronomers. They independently came to the conclusion that, assuming Newtonian mechanics to be precisely correct, the observed divergence in the elliptical orbit of Uranus could be explained if the existence of a seventh, as yet unobserved outer planet was posited. Further, they were able, again within the framework of Newtonian mechanics, to calculate the precise position of the “new” planet. Thus when subsequent research by Galle at the Berlin observatory revealed that such a planet (Neptune) did in fact exist, and was situated precisely where Adams and Leverrier had calculated, this was hailed as by all and sundry as a magnificent triumph for Newtonian physics: in Popperian terms, Newton’s theory had been subjected to a critical test, and had passed with flying colours.

Yet Lakatos flatly denies that there are critical tests, in the Popperian sense, in science, and argues the point convincingly by turning the above example of an alleged critical test on its head. What, he asks, would have happened if Galle had not found the planet Neptune? Would Newtonian physics have been abandoned, or would Newton’s theory have been falsified? The answer is clearly not, for Galle’s failure could have been attributed to any number of causes other than the falsity of Newtonian physics (e.g., the interference of the earth’s atmosphere with the telescope, the existence of an asteroid belt which hides the new planet from the earth, etc). The suggestion is that the “falsification/corroboration” disjunction offered by Popper is unjustifiable binary: non-corroboration is not necessarily falsification, and falsification of a high-level scientific theory is never brought about by an isolated observation or set of observations. Such theories are, it is now widely accepted, highly resistant to falsification; they are “tenaciously protected from refutation by a vast ‘protective belt’ of auxiliary hypotheses” (Lakatos 1978: 4) and so are falsified, if at all, not by Popperian critical tests, but rather within the elaborate context of the research programmes associated with them gradually grinding to a halt. Popper’s distinction between the logic of falsifiability and its applied methodology does not in the end do full justice to the fact that all high-level theories grow and live despite the existence of anomalies (i.e., events/phenomena which are incompatible with them). These, Lakatos suggests, are not usually taken by the working scientist as an indication that the theory in question is false. On the contrary, in the context of a progressive research programme he or she will necessarily assume that the auxiliary hypotheses which are associated with the theory can in time be modified to incorporate, and thereby explain, recalcitrant phenomena.

Third, Popper’s critique of Marxism has not, of course, gone unchallenged. The debate arising from, however, has in many cases tended to revolve around ideological rather than philosophical issues, which will be passed over here. However, there have also been some trenchant philosophical responses. Cornforth sees Marxism as a rational scientific discipline and Marxian thinking as “a rational and scientific approach to social problems” (Cornforth 1968: 6) of the kind which both Marx and Popper consider important. Consequently, he takes Popper to task for representing Marxism as a system of dogmas designed to close minds or pre-empt the operation of our rational faculties in addressing social issues. Against that view, he argues that it constitutes a way of thinking designed to open minds to the real possibilities of human life, and sees it as the philosophy best calculated to promote the ideals of the open society to which he, like Popper, subscribes. Hudelson (1980) argues that Marxian economics survives the Popperian critique of historicism and that, in any case, Marx did not hold many of the tenets of historicism identified by Popper. He also contends that Popper fails to show that there cannot be, and that we cannot know, laws of social development and that Marx did not in fact confuse trends and laws in the way that Popper suggests.

Fourth, In the case of Freudian psychoanalysis, the adequacy of Popper’s critique has been challenged on philosophical grounds by a number of commentators, particularly Adolf Grünbaum (1984). Grünbaum is highly critical, indeed scornful, of Popper’s dismissal of the claims of psychoanalysis to be scientific and argues that Freud showed “a keen appreciation of methodological pitfalls that are commonly laid at his door by critics” (Grünbaum 1984: 168) such as Popper. He argues that Freud was sensitive to the question of the logic of the confirmation and disconfirmation of psychoanalytic interpretations and cites Freud’s use of the concept of consilience, the convergence of evidence from disparate sources, as a serious and explicit attempt to meet the requirements of that logic. For Grünbaum, Popper’s critique of Freud amounts a veritable parody of the pioneering thinker who established the discipline of psychoanalysis, and he attributes that to Popper’s “obliviousness to Freud’s actual writings” (1984: 124). He points out, for example, that the case of the drowning child which Popper uses in Conjectures and Refutations (Popper 1963: 35), upon which he rests part of his case against psychoanalysis, is contrived and not in any way derived from Freud’s actual clinical texts.

Grünbaum contends that there are instances in Freud’s writings where he shows himself to be “a sophisticated scientific methodologist” (Grünbaum 1984: 128), keenly aware of the need for his theoretical system to meet the requirement of testability. One such occurs when Freud, in his assessment that anxiety neuroses are due to disturbances in sexual life, explicitly refers to the notion of falsifiability: “My theory can only be refuted when I have been shown phobias where sexual life is normal” (Freud 1895 [1962: 134]). Another occurs in Freud’s 1915 paper “A Case of Paranoia Running Counter to the Psycho-Analytic Theory of the Disease”, in which, as the title suggests, he saw the patient’s collection of symptoms as potentially falsifying the theory. Moreover, Freud’s entire account of paranoia as being due to an underlying repressed homosexuality is open to empirical refutation, Grünbaum argues, since it has the testable implication that a reduction in rates of paranoia should result from a removal or loosening of social sanctions against same-sex relationships (1984: 111).

Grünbaum accordingly holds that Freudian theory should be considered falsifiable and therefore genuinely scientific—albeit, in his view, ultimately unsuccessful, because the clinical evidence offered in its favour is contaminated by suggestion on the part of the analyst and cannot therefore support its conceptual weight. That is a verdict very different to what Grünbaum sees as the reductive dismissal of Freud offered by Popper: “the inability of certain philosophers of science to have discerned any testable consequences of Freud’s theory”, he concludes, “betokens their insufficient command or scrutiny of its logical content rather than a scientific liability of psychoanalysis” (1984: 113).

There can be little doubt of the seriousness of this challenge to Popper’s critique of the claims of psychoanalytic theory to scientific status. Further, the unparalleled cultural impact of Freudianism upon contemporary life can scarcely be denied, and even a cursory examination of the vast corpus of Freud’s works reveals a thinker of quite extraordinary theoretical power and imagination whose claims to scientific validity cannot be dismissed lightly.

However, while the detail of this psychoanalytic response to Popper is contestable, what is perhaps most striking and important about it is that, as articulated by Grünbaum, it is itself couched in broad terms of an acceptance of a Popperian account of science. That is to say, in rejecting the claim that psychoanalysis fails to meet the standard of falsifiability specified by Popper, Grünbaum—who also rejects the hermeneutic interpretation of Freud offered by thinkers such as Paul Ricoeur and Jurgen Habermas—nonetheless implicitly accepts that very standard, and with it the broad sweep of Popper’s theory of demarcation. For that reason alone, it seems clear that Popper’s work will continue to function as a critical reference point in the ongoing debate regarding the scientific status of Freudian thought.

Fifth, scientific laws are usually expressed by universal statements (i.e., they take the logical form “All A s are X ”, or some equivalent) which are therefore concealed conditionals—they have to be understood as hypothetical statements asserting what would be the case under certain ideal conditions. In themselves they are not existential in nature. Thus “All A s are X ” means “If anything is an A , then it is X ”. Since scientific laws are non-existential in nature, they logically cannot in themselves imply any basic statements, since the latter are explicitly existential. The question arises, then, as to how any basic statement can falsify a scientific law, given that basic statements are not deducible from scientific laws in themselves? Popper answers that scientific laws are always taken in conjunction with statements outlining the “initial conditions” of the system under investigation; these latter, which are singular existential statements, yield hard and fast implications when combined with the scientific law.

This reply is adequate only if it is true, as Popper assumes, that singular existential statements will always do the work of bridging the gap between a universal theory and a prediction. Hilary Putnam in particular has argued that this assumption is false, in that in some cases at least the statements required to bridge this gap (which he calls “auxiliary hypotheses”) are general rather than particular, and consequently that when the prediction turns out to be false we have no way of knowing whether this is due to the falsity of the scientific law or the falsity of the auxiliary hypotheses. The working scientist, Putnam argues (Putnam 1974; see also the 1991 reprinting with its retrospective note), always initially assumes that it is the latter, which shows not only that, but also why, scientific laws are, contra Popper, highly resistant to falsification, as Kuhn (1962) and Lakatos (1970, 1978) have also argued.

Popper’s final position is that he acknowledges that it is impossible to discriminate science from non-science on the basis of the falsifiability of the scientific statements alone; he recognises that scientific theories are predictive, and consequently prohibitive, only when taken in conjunction with auxiliary hypotheses, and he also recognises that readjustment or modification of the latter is an integral part of scientific practice. Hence his final concern is to outline conditions which indicate when such modification is genuinely scientific, and when it is merely ad hoc . This is itself clearly a major alteration in his position, and arguably represents a substantial retraction on his part: Marxism can no longer be dismissed as “unscientific” simply because its advocates preserved the theory from falsification by modifying it—for in general terms, such a procedure, it transpires, is perfectly respectable scientific practice. It is now condemned as unscientific by Popper because the only rationale for the modifications which were made to the original theory was to ensure that it evaded falsification, and so such modifications were ad hoc , rather than scientific. This contention—though not at all implausible—has, to hostile eyes, a somewhat contrived air about it, and is unlikely to worry the convinced Marxist. On the other hand, the shift in Popper’s own basic position is taken by some critics as an indicator that falsificationism, for all its apparent merits, fares no better in the final analysis than verificationism.

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Degrees of riskiness, falsifiability, and truthlikeness

A neo-Popperian account applicable to probabilistic theories

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• Published: 23 July 2021
• Volume 199 , pages 11729–11764, ( 2021 )

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In this paper, we take a fresh look at three Popperian concepts: riskiness, falsifiability, and truthlikeness (or verisimilitude) of scientific hypotheses or theories. First, we make explicit the dimensions that underlie the notion of riskiness. Secondly, we examine if and how degrees of falsifiability can be defined, and how they are related to various dimensions of the concept of riskiness as well as the experimental context. Thirdly, we consider the relation of riskiness to (expected degrees of) truthlikeness. Throughout, we pay special attention to probabilistic theories and we offer a tentative, quantitative account of verisimilitude for probabilistic theories.

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“Modern logic, as I hope is now evident, has the effect of enlarging our abstract imagination, and providing an infinite number of possible hypotheses to be applied in the analysis of any complex fact.”—Russell ( 1914 , p. 58).

A theory is falsifiable if it allows us to deduce predictions that can be compared to evidence, which according to our background knowledge could show the prediction—and hence the theory—to be false. In his defense of falsifiability as a demarcation criterion, Karl Popper has stressed the importance of bold, risky claims that can be tested, for they allow science to make progress. Moreover, in his work on verisimilitude, Popper emphasized that outright false theories can nevertheless be helpful on our way towards better theories. In this article, we try to analyze the relevant notion of riskiness, and take a fresh look at both falsifiability and verisimilitude of probabilistic theories. Our approach is inspired by the Bayesian literature as well as recent proposals for quantitative measures of verisimilitude and approximate truth (as reviewed in Sect.  3.1 ).

The paper is structured as follows. In Sect.  1 , we set the scene by discussing an example due to Jefferys ( 1990 ). In Sect.  2 , we make explicit the dimensions that underlie the notion of riskiness. We also examine if and how degrees of falsifiability can be defined, and how they are related to various dimensions of the concept of riskiness as well as the experimental context. Furthermore, we consider a crucial difference between deterministic and indeterministic theories in terms of their degrees of falsifiability. In Sect.  3 , we review consequence-based approaches to quantifying truthlikeness and, in Sect.  4 , we propose alternative definitions of (expected degrees of) truthlikeness and approximate truth that are better suited for probabilistic theories. We summarize our findings in Sect.  5 .

1 Three pundits, one true probability distribution

To illustrate the themes we want to explore here, we discuss an informal example given by Jefferys ( 1990 ) (in which we changed one of the numbers):

A priori, our “surprise” when we observe a value close to a sharp prediction is much greater than it would be if the theory made only a vague prediction. For example, consider a wholly imaginary world where stock market pundits provide standard deviations along with their predictions of market indexes. Suppose a pundit makes a prediction of the value of an index a year hence, and quotes a standard deviation of [2]% for his prediction. We would probably be quite surprised if the actual value turned out to be within several percent of the prediction, and if this happened we might want to investigate the pundit more closely. By making a precise prediction, this pundit takes a great risk of being proven wrong (and losing our business). By the same token, when his prediction turns out even approximately correct, we are surprised, and the likelihood that we will follow his advice in the future may be increased. We would probably be less interested in a second pundit, who predicted the same value for the index as did the first, but who quoted a standard deviation of 20%. We would doubtless have little interest at all in a third pundit who informed us only that “the market will fluctuate,” even though that prediction is virtually certain to be fulfilled!

We could reconstruct this scenario in terms of a unique, true value of the index at the specified time. However, since we want to address probabilistic theories (which make empirically verifiable, statistical hypotheses about the world), we assume a true probability function instead. This may either be an objective chance function or an epistemic probability assessment that is well-calibrated (say, from a group of experts), both of which can be modeled in a Bayesian context. Also the subjective probabilities expressed by the pundits can be represented as such. Throughout this paper, we will apply Bayesian ideas, so considering the evidence will typically lead to a revision of prior probabilities (assigned to theories or associated hypotheses) to posteriors, which can be computed via Bayes’ theorem.

One modeling choice could be to use Gaussian distributions and to choose the parameters such that the first prediction is strongly disconfirmed by the evidence. However, the results are even more striking when we opt for normalized boxcar functions (i.e., uniform distributions truncated on an interval), which allow for outright falsification of these probabilistic predictions. So, let’s assume both hypotheses of the first two pundits are normalized uniform distributions on intervals centred on the same value, \(\mu \) . The first distribution is nonzero on an interval that is ten times narrower than the second. Now assume that the interval where the true distribution takes on nonzero values and that includes the realized value turns out to fall in the second range and not in the first, but very close to the latter, as depicted in Fig.  1 . In this case, outright falsification of the first hypothesis occurs and the posterior probability of the second is unity (independently of the priors, as long as they start out as non-zero). Still, we could be surprised by how close the first pundit’s guess was and feel motivated to investigate it further, exactly as Jefferys ( 1990 ) described.

While the posterior for the first pundit’s hypothesis is zero, it was much bolder than the second’s and it was only a near miss. This may warrant looking into its neighborhood, rather than going on with the less precise, second hypothesis. If we only consider the posteriors, however, our surprise at finding a value relatively close to the precise, first prediction seems irrational, or at least: on their own, posteriors do not offer any justification for Jefferys’s move to inspect the precise, near-miss hypothesis closer.

This observation is related to another aspect that posteriors do not track: which of the competing hypotheses is more truthlike. The fact that the true hypothesis is twenty times as narrow as the second hypothesis and only half as narrow as the first or that the centre of the peak of the true hypothesis is close to that of both alternatives is not taken into account by posteriors at all. Doing this in general requires a method for measuring the similarity of the shape and position of hypothetical distributions compared to the true one.

Numerical example of normalized boxcar predictions by two pundits, one more precise than the other, as well as the true distribution. The area under each of the curves is unity (due to normalization of probability distributions)

Finally, consider the third pundit’s prediction that “the market will fluctuate”: this hypothesis is outright falsifiable (it is in principle possible that the market will turn out not to fluctuate and if this possibility is realized, this is observable), yet it is so extremely likely (on nearly any prior) that the market will fluctuate that this claim is not risky at all. Moreover, the prediction of the third pundit has a very different structure than the former two, which could be reconstructed as given by a single probability distribution (or perhaps a narrow family thereof). Instead, the third prediction excludes one possible realization (a constant path of the index through time), allowing all others, without assigning any probabilities to them at all. As such, this prediction is compatible with an infinite family of probability distributions. The negation of the third prediction corresponds to a singleton in the space of all possible paths for the index: “the market will not fluctuate” is equally falsifiable, but extremely risky, its prior being zero or at least extremely close to it (on nearly any probability distribution).

So, while the first two are in some sense very precise predictions, neither would be outright falsifiable or verifiable if we had chosen to reconstruct them as Gaussians (which would certainly be a defendable choice). But the third pundit’s prediction, which is a very general claim that strongly underdescribes the probabilities, is both falsifiable and verifiable.

To conclude this section, the example we borrowed from Jefferys ( 1990 ) suggests that Popper was right in emphasizing the relation between riskiness and falsifiability. Moreover, especially if we consider a Gaussian reconstruction, it seems to suggest that there are degrees of falsifiability, according to which the first pundit’s claim would be falsifiable to a higher degree than the second’s. The third pundit’s claim, however, shows it is possible for a prediction to be outright falsifiable yet not risky at all.

2 Riskiness and falsifiability

Popper ( 1959 ) identified falsifiability as a demarcation criterion for scientific theories. On his account, theories that make riskier predictions are assumed to be more easily falsifiable than others. Popper’s idea of falsifiability was inspired by an asymmetry from mathematics: if a conjecture is false, it can be falsified by providing a single counterexample, whereas, if the conjecture is true, finding a proof tends to require more work. In the empirical sciences, proving a universal hypothesis (if it is indeed true) requires verifying all instances, which is impossible if the domain is infinite, whereas falsification (if the universal hypothesis is false) seems to require only a single counterexample, just like in mathematics. In practice, even for deterministic hypotheses, the picture is more complicated, due to measurement error and the Duhem–Quine problem. Moreover, for probabilistic hypotheses, falsification seems to be unobtainable (but see Sect.  2.2 ).

In any case, Popper’s falsifiability, proposed as a demarcation criterion on hypotheses (or theories), is a condition on the form of the statements. Scientific hypotheses should be formulated such that empirically testable predictions can be derived from them and if the hypothesis is false, it should be possible to make an observation showing as much. This requires hypotheses to be formulated in an explicit and clear way, geared toward observational consequences. Although Popper’s falsifiability was a categorical notion (either it applies to a hypothesis or it does not), not all falsifiable hypotheses are created equal: Popper preferred those that made riskier predictions.

Our first task in this section is to disentangle the various dimensions of riskiness and to track their relation to falsifiability. We also aim to formulate clear desiderata for a formal account of degrees of falsifiability, albeit without fleshing out a full proposal that achieves them.

2.1 Two dimensions of riskiness

In the context of his falsificationism, Popper ( 1959 ) preferred bold or risky predictions: those that were rich in content, that were unlikely according to competing, widely accepted theories or that predicted entirely new phenomena.

On our analysis, this Popperian concept of riskiness (or boldness) consists of at least two aspects. Teasing these apart is essential for getting a good grip on the interplay between riskiness and falsifiability—as well as between riskiness and verisimilitude (to which we turn in Sect.  3 ). According to our analysis, the two main ingredients of riskiness are the following.

Informativeness For example, a hypothesis that gives a point prediction or a small range of possible measurement outcomes is more informative than one that gives a wide interval of possible values. That makes the former more risky. Of course, an informative prediction, which is rich in content and precise, may be wide off the mark; in other words, it may be very inaccurate, but that is precisely the point: the more informative a prediction is, the more opportunities it allows to detect discrepancies with the facts if it is false. Moreover, a substantive and precise prediction can be viewed as a conjunction of less substantive and precise hypotheses, so its prior cannot be higher than those of the latter: this is precisely what makes an informative prediction risky.

Conceptual novelty A hypothesis may predict phenomena that have no counterpart in any previous theory and that may not have been observed yet. This aspect is related to discussions in philosophy of science concerning radically new theories, language change, etc. (see, e.g., Masterton et al. 2017 and Steele and Stefansson 2020 ). Compared to the previous two aspects of riskiness, this one is more difficult to represent formally. For probabilistic theories, this issue is related to changes to the sample space, or at least the partition thereof, and it creates additional issues for how to set the priors (see, e.g., Wenmackers and Romeijn 2016 ).

While the notion of conceptual novelty is interesting in its own right, there is still plenty of formal work to do on the first dimension, informativeness, which will be our focus here. Informativeness is language-dependent: the granularity of the language depends on what can be expressed. Of the three pundits in Sect.  1 , the first one scored the highest and the third one the lowest on this dimension. See Appendix A for a formal development of this idea.

Informativeness does not equal improbability. To disentangle these concepts, we add a dimension of improbability that is not a dimension of Popperian riskiness:

Low probability despite equal informativeness Although probabilities are constrained by a partial ordering tracking informativeness, the latter is not sufficient to settle the numerical prior probability values in a unique way. Hence, subjective Bayesianism allows for individual variation across probability assignments by rational agents. These variations may be due to differences in the prior, differences in past learning or a combination thereof. As a result, when we compare hypotheses that are equally rich in content and equally precise, they may still have unequal probabilities prior to testing according to a particular rational agent (e.g., a scientist). Advancing a hypothesis with a lower prior may be regarded as a more “risky” choice, but in this case, it seems irrational to pursue it. However, if we tease apart the two reasons why a hypothesis may have a low prior—i.e., due to high informativeness and subjective variation across equally informative hypotheses—it becomes clear that only the former source of improbability is fruitful for advancing science.

2.2 Experimental context

From Sect.  2.1 we retain informativeness as an important gradable variable associated with riskiness, which makes it a promising ingredient of an analysis of falsifiability as a gradable notion. However, informativeness does not depend on the experimental context, which we analyze here.

It has become commonplace in philosophy of science and (Bayesian) epistemology to claim that probabilistic theories can only be disconfirmed to an arbitrarily high degree, but that they can never be outright falsified. After all, the argument goes, no finite length of observations of heads falsifies the hypothesis that the coin is fair. This example supposedly shows that there are hypotheses that are unfalsifiable, but highly disconfirmable. However, falsification is not unobtainable for all probabilistic hypotheses in all circumstances. In fact, we have already encountered an example of the possible falsification of a probabilistic hypothesis in Sect.  1 : assuming market indexes can only take on a finite number of discrete values, a distribution that is zero except for a narrow interval (such as the first pundit’s curve in Fig.  1 ) is outright falsified by any observation outside of that interval.

The next two examples illustrate in yet another way why we need to fix a reference class of experiments explicitly. They both show that even probabilistic hypotheses that do not rule out any part of the sample space in advance may still be falsifiable, given sufficient experimental resources.

( Emptying the bag ) Suppose one has an opaque bag with three marbles inside: either two black marbles and one white marble or vice versa. The only experiment allowed to gauge the color of the marbles in the bag is taking one marble out of the bag and placing it back before another draw can be made. We might have some prior credences with regard to drawing a black marble: this constitutes a probabilistic theory.

Without the restriction to draw one marble at a time, however, there is a very simple way to find out the truth: empty the bag and see what is in it.

( Superexperiments ) Consider a demigod who can do a certain experiment an infinite number of times in a finite time frame: we call this a “superexperiment”—an idea analogous to supertasks and hypercomputation. Some theories would remain falsifiable, while others become falsifiable in this context. Yet, a statement like “all ravens are black” does not become verifiable; the demigod can only test all ravens that exist at the time of the experiment, for instance.

Now, consider a jar containing a countable infinity of numbered marbles. We know that all marbles are marked by a unique natural number; we do not know, however, whether each natural number is printed on a marble. For instance, it could be the case that the jar only contains marbles with even numbers printed on them. Consider the statement h : “all numbers except for one unknown number, n , are printed on a marble in the jar.” This hypothesis is not falsifiable by usual methods but it is falsifiable by means of a superexperiment. Indeed, it is possible for the demigod to pick all naturals, thereby falsifying h .

The first example may strike the reader as trivial and the second as extravagant, but this is exactly the point. These examples (as well as more systematic studies along the lines of Kelly 1996 ) bring to light that we already had an implicit reference class of experiments in mind, which did not include emptying the whole bag at once or performing a superexperiment.

For a more realistic example, consider a physical theory that can only be tested by building a particle accelerator the size of the solar system. Clearly, resource constraints make falsifying such a theory (quasi-)impossible. This indicates that the binary distinction between possibility and impossibility of falsification does not tell the whole story and that practical constraints should be taken into account. This general observation (also made, for instance, by Carroll 2019 ) applies to probabilistic theories, too.

At this point, we hope that we have convinced the reader that the (degree of) falsifiability of a hypothesis co-depends on the severity of available experiments. Hence, we should include the experimental context in our formal models. Following Milne ( 1995 ), we formalize an experiment as a finite, mutually exclusive and jointly exhaustive set of propositions (equivalent to a partition of the sample space of the probability function, P , associated with the probabilistic hypothesis at hand). In a probabilistic context, the goal of learning from experiments is to reduce the uncertainty concerning a variable of interest; formally, this uncertainty can be modeled using an entropy measure (Crupi et al. 2018 ). Shannon entropy is one well-known measure, but there are other options.

2.3 Severity

The experimental context plays a key role in the notion of the severity of a test, for which quantifiable measures have been proposed in the literature. As discussed by Kuipers ( 2000 , pp. 60–62), Popper’s ( 1983 ) notion of severity refers to the “improbability of the prediction”. Possibly, then, the qualitative notion of severity can be viewed as giving us degrees of falsifiability. This will be our working hypothesis in this section.

Severity can be defined as expected disconfirmation (as proposed by Milne 1995 ) and can be related to entropy. We illustrate this in detail for an example in Appendix B. Severity can also be related to boldness as follows: relative to a reference set of possible experiments (which could represent, say, all experiments that are possible at a given point in time), we can define the boldness of a hypothesis as the maximum severity of tests of the hypothesis. We owe this suggestion to Wayne Myrvold (personal communication), who also observed that the confirmation measure can be relativized by working through the derivation in Appendix B from bottom to top: instead of considering the Kullback–Leibler divergence, one could start from the family of Rényi divergences, which can be related to a family of severity measures. On the one hand, a special case of Rényi entropy measures is the Hartley entropy, which is of interest to our Popperian project since the expected entropy reduction associated with this measure is positive just in case the test has a possible outcome that excludes at least one of the hypotheses under study (Crupi et al. 2018 ). This is in line with Popper’s ( 1959 ) falsificationism, which advises learners to seek to falsify hypotheses. On the other hand, if outright falsification is not taken to be the guiding notion, the definition of entropy could be generalized even beyond Rényi entropies, to a two-parameter family of entropy measures (Crupi et al. 2018 ).

Remark that, like Milne ( 1995 ), we assume that an ideal experiment induces a partition on the sample space. So, all possible experiments can be represented by considering the lattice of partitions. Typically, as science advances, experiments become more precise, resulting in a refinement of the experimental partition. A more radical change in experimental context occurs when qualitatively different phenomena become measurable: this is closely related to severe theory change and, like the corresponding aspect of riskiness, we will not consider it here.

Real-world experiments are not ideal, due to measurement errors. This, together with the Duhem–Quine problem and the issue of underdetermination of theories by empirical evidence, obviously complicates matters of outright falsification or verification. Here we will not go into these complications, but observe that measurement error can be represented by an additional probability distribution. (For an example of a Bayesian account, see, e.g., Jefferys and Berger, 1992 .)

Our discussion so far suggests a desideratum for an adequate formal representation of the notion of gradable falsifiability. Observe that, in the context of probabilistic theories, the language of the theory takes the shape of a partition on the sample space, as does the experimental context. This allows for a unified treatment. Hence, we require that degrees of falsifiability should depend on these two algebras:

an algebra related to the language of the theory, and

an algebra related to the experimental context.

In the next section, we turn to the question of how the falsiability of probabilistic theories compares to non-probabilistic ones.

2.4 Deterministic versus indeterministic theories

Thyssen and Wenmackers ( 2020 ) proposed a classification of theories in terms of how much freedom they allow. Although their classification was presented in a different context (the free-will debate), it is suggestive of an ordering in terms of falsifiability. Footnote 1 On their proposal, Class I theories are deterministic and Class II theories are probabilistic. However, the latter do not include all indeterministic theories. Probabilistic theories specify all possible outcomes and assign probabilities to all of them. This leaves room for two additional classes of theories: Class III theories allow for probability gaps and Class IV theories also allow for possibility gaps.

Class I theories are deterministic and complete. Note that a deterministic theory without free variables can be regarded as an extreme case: it assigns probability one to a single possible outcome; all other possibilities are assigned probability zero. Also note that this notion of determinism includes completeness, which is stronger than usual: incomplete theories such as “All F s are G s” are usually considered to be deterministic, although they remain silent on atomic possibilities; in this classification, they belong to Class III (see below).

Class II theories are probabilistic. Within this class, some theories may assign prior probability density zero to a subset of the possibility space: the larger this set of probability zero, the closer this theory is to Class I . Theories that do not assign prior probability zero to any subset of the possibility space can accommodate any possible empirical observation of frequencies, albeit not all with the same probability: the more spread out the probability assignments, the lower the degree of falsifiability. All else being equal, equiprobability leads to the lowest degree of falsifiability, although it does satisfy other theoretic virtues (such as simplicity of description and symmetry).

Class III theories have probability gaps: they specify all possible outcomes and may even specify relative probabilities for a subset of possible events, but they do not specify probabilities for all possible outcomes. This class includes theories that yield predictions with free variables or fudge factors that are not constrained by a probability measure. Like Class II theories, theories in this class can accommodate any possible empirical observation of frequencies, but they do not even assign probabilities to them. The third pundit’s prediction from Sect.  1 (“the market will fluctuate”) belongs to this class: it does not assign probabilities to the different ways in which the market might fluctuate.

Class IV theories have possibility gaps: they allow for radically new possibilities, not specified by the theory, to be realized. They may specify some possible outcomes, and even some relative probabilities of a subset of possible events, but at least under some circumstances they allow for radical openness regarding possible outcomes. The most extreme theory in this class takes the form: “anything can happen.” According to Popper’s demarcation criterion, that is the opposite of a scientific theory, because this statement cannot be falsified by any data. Its degree of falsifiability should equal the minimal value of the scale. One could even argue that they are not theories at all, but observe that, except for the extreme case, theories belonging to this class may offer a probabilistic or even a deterministic description of certain situations.

This classification mainly tracks informativeness (increasing from Class IV to Class I ), which we already know to correlate with riskiness and falsifiability, but it does not yet account for the experimental context. Again, this is important: when taking into account the empirical measurement errors in a probabilistic way, even a deterministic theory will lead to a strongly spiked probability distribution at best (as already mentioned at the end of Sect.  2.3 ). That is, even though Class I theories are maximally falsifiable in principle, measurement errors prevent even such theories to be perfectly falsifiable in practice and, as mentioned before, the Duhem–Quine problem complicates which conclusion should be drawn even in cases where outright falsifiability seems feasible.

Let us now briefly elaborate on the possibility of empirical equivalence of theories in the light of possible outright falsification. (Alternatively, one could define this for any degree of (dis-)confirmation, but our focus in this section is on decisive evidence.) Our approach is inspired by Sklar’s ( 1975 ) work on “transient underdetermination”, which refers to in-principle differences between theories that are not measurable on present empirical evidence. Likewise, we relativize the notion of empirical equivalence to a given empirical context. In terms of falsification, two theories, \(\tau _1\) and \(\tau _2\) , are empirically equivalent relative to an experimental context \({\mathcal {E}}\) , if every experiment \(e_i \in {\mathcal {E}}\) that can decide \(\tau _1\) can decide \(\tau _2\) , and vice versa. In particular, it may happen that two theories that belong to different classes are empirically equivalent relative to the current experimental context. For example, consider two different approaches to quantum mechanics: while Bohmian mechanics belongs to Class I , spontaneous collapse theories belong to Class II . They are empirically indistinguishable relative to the current experimental context, but this need not remain the case when experimental resolution improves in the future.

Instead of only quantifying over the elements in a given class of experiments, we can also quantify over the reference classes of experiments. For instance, if a theory \(\tau _1\) can be decided by any reference class of experiments that can decide a theory \(\tau _2\) , we can say that \(\tau _1\) is at least as decidable as \(\tau _2\) . If the converse holds as well, then \(\tau _1\) and \(\tau _2\) are empirically equivalent simpliciter (which need not imply that \(\tau _1\) and \(\tau _2\) are logically equivalent, of course).

At first sight, this analysis gives us no ground for preferring one among two (or more) theories that are empirically equivalent relative to the current experimental context. However, if we take into account the possibility of future refinements (or even radical changes) to the experimental context, it may still give grounds for preferring the theory that belongs to the lowest class number (and among those, the one that is the most precise)—provided, of course, the required future experimental improvement that distinguishes among the theories is achievable at all. While Milne’s severity measure does not take this into account, it could in principle be extended with an expectation ranging over changes of \({\mathcal {E}}\) .

This vindicates a preference for deterministic theories over indeterministic ones, and for fully probabilistic theories over underdescribed one. This is so, not due to any preconceptions about determinism or chance in the world, but simply because testing maximally falsifiable theories first allows us to weed out untenable options as fast as possible. Not because Class I theories are more likely to be true than theories of Class II and beyond, or even closer to the truth, but because they make maximally risky predictions and hence are maximally easy to refute when false. This seems a solution to Wheeler’s problem ( 1956 , p. 360; also quoted approvingly by Popper 1972 ):

Our whole problem is to make the mistakes as fast as possible [...]

A similar tension pertains to the other end of the spectrum. In the face of the fallibility of science, acknowledging that there may be, as of yet, unknown possibilities seems virtuous. This would suggest a preference for Class IV theories, rather than Class I . However, the parts of the theory that lead to it being Class IV have low to minimal falsifiability. Therefore, we argue that acknowledging the possibility of radical uncertainty should happen preferably only at the meta-theoretical level. This prevents a conflict with the virtue of falsifiability as well as most of the difficulties in formalizing radical uncertainty. Again, this is related to the second dimension of riskiness: conceptual novelty. Theories of Class IV should be considered as last resorts, if all else fails, and given the infinitude of alternatives to explore, temporary theories of this kind can always be superseded.

This concludes our discussion of degrees of falsifiability. In the next section, we turn our attention to a different question: how to define a measure of truthlikeness? Some of the notions that we explored in this section will reappear. For instance, it will turn out that the language of the theory is crucially important in both contexts.

3 Formal frameworks for truthlikeness and approximate truth

Various authors have contributed to improving our understanding of truthlikeness and approximate truth through formal work, such as Schurz and Weingartner ( 1987 , 2010 ), Gemes ( 2007 ), Niiniluoto ( 2011 ) and Cevolani and Schurz ( 2017 ). As far as we know, however, these ideas have not yet been applied explicitly to probabilistic theories.

One thing we appreciate about these studies is that they show how central concepts from the traditional debates in philosophy of science can be formalized and applied to other subjects in epistemology. An illustration of this is the elegant approach to the preface paradox, presented by Cevolani and Schurz ( 2017 ).

Drawing on these formal and quantitative accounts of Popperian ideas from the truthlikeness literature, we can take a fresh look at the Popperian theme from our previous section: falsifiability, which has so far eluded proper formalization and lacks a quantitative account. One of the dimensions of Popperian riskiness that we identified in Sect.  2 and that seems a crucial ingredient for any measure of falsifiability was informativeness. As we will see at the end of Sect.  3.1 , the content of a theory emerges as a natural measure of informativeness from the Schurz–Weingartner–Cevolani framework.

3.1 Review of the Schurz–Weingartner–Cevolani consequence approaches

The aforementioned formal literature on verisimilitude has developed some quantitative ways of comparing the verisimilitude of theories as well as their approximate truth. In particular, Cevolani and Schurz ( 2017 ) have proposed full definitions of verisimilitude and approximate truth, which we will restate below for ease of reference. Before we can do so, however, we have to sketch some of the background of the Schurz–Weingartner–Cevolani framework.

First, Schurz and Weingartner ( 1987 ) tackled verisimilitude by defining a notion of the relevant consequences of a theory . One can make sense of relevant consequences in both propositional languages and first-order languages. Any given theory, \(\tau \) , gives rise to a set of true relevant consequences, \(E_t(\tau )\) , and a set of false relevant consequences, \(E_f(\tau )\) . Theories are then compared by means of the sets of their relevant consequences.

Next, Schurz and Weingartner ( 2010 ) built further on this account to give a quantitative definition of truthlikeness for theories represented by relevant elements (those relevant consequences that are not logically equivalent to a conjunction of shorter relevant consequences) in a propositional language with a fixed number of variables, n . Definition 5 in Section 5 of their paper is important for our purposes, so we include a version of it here. This version takes into account two modifications introduced by Cevolani and Schurz ( 2017 ), who added the parameter \(\varphi >0\) to balance the relative weight of misses compared to matches, and normalized through division by n (cf. Definition 9 in their paper):

Definition 1

Cevolani–Schurz: quantitative truthlikeness for a theory \(\tau \) , represented by relevant elements in a propositional language with n variables.

where \(k_\alpha \) equals the number of \(\alpha \) ’s literals and \(v_\alpha \) equals the number of \(\alpha \) ’s true literals. The truthlikeness for Verum ( \(\top \) ) and Falsum ( \(\bot \) ) is set by the following convention: \({\textit{Tr}}_\varphi (\top ):=0\) and \({\textit{Tr}}_\varphi (\bot ):=-\varphi (n+1)/n\) .

On this proposal, evaluating the truthlikeness of a theory boils down to a lot of bookkeeping. The major upside of this account is its naturalness: the more true statements (of a certain kind) a theory makes, the better. Moreover, the less contaminated these true statements are with false statements, the higher the bonus.

In addition, Cevolani and Schurz ( 2017 ) introduced a separate notion of approximate truth, which expresses closeness to being true, irrespective of how many other things may be true outside the scope of the hypothesis under consideration (cf. Definition 10 in their article). In this context, we want to assess amounts of content, rather than bonuses versus penalties, so we suppress the subscript \(\varphi \) by setting its value to unity, as follows: \({\textit{Tr}}(\alpha ) := {\textit{Tr}}_{\varphi =1}(\alpha )\) .

Definition 2

Cevolani–Schurz: approximate truth.

On this account, it is natural to think of \(E_t(\tau )\) as the truth content of \(\tau \) and of \(E_f(\tau )\) as the falsity content of \(\tau \) . The corresponding quantitative measures of truth content ( TC ) and falsity content ( FC ) of a theory \(\tau \) could be defined as follows.

Definition 3

Quantitative content.

For true \(\tau _1\) and \(\tau _1 \models \tau _2\) , it holds that \({\textit{Content}}(\tau _1) \ge {\textit{Content}}(\tau _2)\) , as proven in Appendix C. This is intuitively compelling in light of the first dimension of riskiness (informativeness), which tracks the amount of content.

Using this notation, and now also suppressing \(\varphi \) in the formula for quantitative truthlikeness, we can bring the essential form of the Cevolani–Schurz definitions into clear focus:

The measure of truth content, \({\textit{TC}}\) , is always positive and acts as a reward in the definition of truthlikeness. The measure of falsity content, \({\textit{FC}}\) , is always negative and acts as a penalty in the definition of truthlikeness.

We can now apply these ideas to riskiness and falsifiability, as announced at the start of Sect.  3 . The first dimension of riskiness, informativeness, can now be understood as the \({\textit{Content}}\) of a theory. Ceteris paribus, as the \({\textit{Content}}\) of a theory increases, its falsifiability increases. Notice that this provides a way of measuring content that does not depend on relative logical strength alone. On the one hand, provided that \(\tau _1\) and \(\tau _2\) are true, \(\tau _1 \models \tau _2\) implies \({\textit{Tr}}_\varphi (\tau _1) \ge {\textit{Tr}}_\varphi (\tau _2)\) . (This follows from \({\textit{Content}}(\tau _1) \ge {\textit{Content}}(\tau _2)\) , as shown above; it was also shown in Schurz and Weingartner 1987 .) On the other hand, one can also consider theories \(\tau _1\) and \(\tau _2\) such that \({\textit{Content}}(\tau _1) < {\textit{Content}}(\tau _2)\) while \(\tau _2 \not \models \tau _1\) . However, when we will turn our attention away from propositional logic towards probabilistic theories, which are a special kind of quantitative theories, merely tracking the fraction of correct assignments without measuring the distance of wrong assignments to the correct values will become implausible.

3.2 Content of probabilistic theories

The main thrust of the approach reviewed in the previous section is that there are meaningful notions of truth content and falsity content to begin with. Before the advent of this approach, it seemed that content alone would never suffice to capture intuitively compelling desiderata about the structure of a truthlikeness ordering. To see this, consider a falsehood that constitutes a near miss: for instance, a long conjunction of many true atomic conjuncts and only one false conjunct. If we compare this to a tautology, which is the least informative kind of true statement, we would intuitively judge the former to be more truthlike than the latter. However, Tichý ( 1974 ) and Miller ( 1974 ) have both shown that, as long as we define the content of a theory as the set of sentences closed under the consequence relation, a tautology always ranks as more truthlike than any false theory. Cevolani and Schurz ( 2017 ) saves their notions of truth content and falsity content from the Tichý–Miller wrecking ball by restricting the notion of relevant consequence. The approach to truth content and falsity content proposed by Gemes ( 2007 ) proceeds along similar lines.

Our goal here goes beyond what was already achieved by these authors: we aim to consider the truthlikeness of probabilistic theories. An important hurdle is that there do not seem to be well-behaved counterparts to truth content and falsity content. We now give two examples that illustrate problems at the heart of all the issues we discuss below.

( Coarse-graining to a subalgebra ) Assume the following probabilistic theory is true. It is characterized by the probability space \(\langle \varOmega , {\mathcal {A}}, P \rangle \) determined by the sample space \(\varOmega := \{A, B, C\}\) , the \(\sigma \) -algebra \({\mathcal {A}} := {\mathcal {P}}(\varOmega )\) , and the probability function \(P(\{A\}) := 0.2\) ; \(P(\{B\}):=0.5\) ; \(P(\{C\}):= 0.3\) . In this probability space, the following holds: \(P(\{A, B\}) = 0.7\) and \(P(\{C\}) =0.3.\)

There are many other probability measures over the \(\sigma \) -algebra \({\mathcal {A}}\) that satisfy this requirement, too. This is related to the following observation: \(\{A, B\}\) and \(\{C\}\) uniquely determine a single subalgebra of \({\mathcal {A}}\) : \(\{\emptyset , \{A, B\}, \{C\}, \varOmega \}\) .

Now compare two theories probabilistic theories (which we will denote by T rather than \(\tau \) ) \(T_L\) and \(T_M\) that both get everything right on this subalgebra: \(T_L\) says nothing else, while \(T_M\) also includes probability assignments to \(\{A\}\) and \(\{B\}\) . We are now faced with a trade-off. Clearly \(T_M\) has more content, but if that content is very unlike the true theory, the truthlikeness of \(T_M\) might be lower than that of \(T_L\) . (This can also be nicely illustrated in terms of our own proposal, as we will see in Example  8 in Sect.  4.3 .)

Example  3 illustrates the importance of aligning the notion of content with the “ambient structure”, in particular, with the \(\sigma \) -algebra of the true probability space and coarse-grained versions of it. This is actually a quite natural requirement, in that it tracks our intuition regarding the relation between content and logical strength. The next example shows that a probabilistic theory can be wrong on all non-trivial events and still be very close to the truth.

( Wrong probabilities for all non-trivial events ) Building on the previous example, we can complicate matters further by considering two theories Q and R , such that \(Q(\{A\}) := 0.2; Q(\{B\}) := 0.1; Q(\{C\}):=0.7\) and \(R(\{A\}) := 0.18; R(\{B\}) := 0.51; R(\{C\}) := 0.31\) . Clearly, R only assigns the right values to the events \(\emptyset \) and \(\varOmega \) . Q , on the other hand, gets the event \(\{A\}\) right as well. Yet intuitively, R seems to lie closer to P than Q does.

Both examples illustrate that in tracking the truthlikeness of probabilistic theories, mere counting will no longer do. One reason is that the theories come with a native structure, which allows approximation by coarse-graining, so we cannot ignore this structure. Another reason is that the theories we are comparing are inherently quantitative, so we will have to switch from counting to measuring. This already applies for finite, discrete sample spaces—on which we will focus here—and is exacerbated for continuous distributions.

As a result, the orthodox consequence approach to verisimilitude no longer suffices in the probabilistic context, where verisimilitude does not depend on truth simpliciter. We need some content-specific measure to determine how close to the truth a theory really is. Notions like truth content and falsity content are therefore out of place. Issues like these just do not pop up that often when one works in propositional logic or predicate logic, so we feel that it is worthwhile to emphasize this nuance. Cevolani and Schurz ( 2017 ) seem to agree with us, at least tacitly, because they introduced a measure, \({\textit{App}}\) , to measure closeness in quantitative contexts, as we will see below.

3.2.1 Representing probabilistic theories

Schurz and Weingartner ( 2010 ) are certainly right in claiming that philosophers should take knowledge representation seriously. The difference between qualitative beliefs and quantitative beliefs is so large, however, that we cannot simply transplant ideas regarding verisimilitude and approximate truth from the qualitative setting to the quantitative setting. As mentioned above, one of the key issues is that we do not have natural notions of truth content and falsity content in this setting.

The proposal of Cevolani and Schurz ( 2017 , section 7.2) is the only one we are aware of that is applicable to quantitative theories. Still, they do not consider probabilistic theories explicitly. Hence, the goal of this section is to fit probabilistic theories into the general mold they proposed for quantitative theories.

Cevolani and Schurz ( 2017 ) started by assuming that there is an object, a , that has various magnitudes, including \(X_i\) . The true value of this particular magnitude is assumed to be \(X_i(a)=r_i^*\) . Then they suggest a quantitative theory \(\tau \) is a conjunction with conjuncts of the form:

Furthermore, they require a measure, \({\textit{App}}\) , ranging over these conjuncts and with the following properties:

\({\textit{App}}(\tau _i)\) is the degree to which \(r_i\) approximates the true value \(r_i^*\) of the magnitude \(X_i\) for object a .

\({\textit{App}}(\tau _i)\) ranges between \(-1\) and 1, where \({\textit{App}}(\tau _i) = 1\) means that \(r_i = r_i^*\) and \({\textit{App}}(\tau _i) = -1\) means that the distance between \(r_i\) and \(r_i^*\) is maximal.

To apply this to probabilistic theories, we consider the following choices for a , \(X_i\) and \(r_i\) :

a is a process or situation characterized by a probability space, including probability function P ;

i indexes an event, \(E_i\) , from the algebra of P (or an equivalent proposition);

and \(X_i\) is some evaluation function associated with said event, such that \(X_i(a)=P(E_i)=r_i\) , where \(r_i\) a real number in the unit interval.

Using \({\textit{App}}\) , Cevolani and Schurz ( 2017 ) proposed the following measure of verisimilitude for quantitative theories:

Definition 4

Cevolani–Schurz: verisimilitude for quantitative theories.

Let \({\textit{App}}\) be an adequate measure that measures how well \(r_i\) approximates \(r_i^*\) , the true value of \(X_i(a)\) for the relevant a . We define the verisimilitude for a conjunctive, quantitative theory \(\tau \) as:

The definition in Eq. ( 2 ) has the familiar form: like in Eq. ( 1 ), the first term rewards statements that approximate the truth well enough, while the second term penalizes statements that do not.

The driving force of the Schurz–Weingartner–Cevolani tradition consists of relevant consequences. Hence, we should assume that the theory \(\tau \) is written down in terms of its elementary consequences. Unfortunately, it is not exactly clear from a mathematical perspective what elementary consequences in a quantitative context comprehend. So, there remains some work to be done to determine what a probabilistic theory \(\tau \) written down in terms of its elementary consequences looks like. We assume that it includes all the probability assignments for the events in the sub- \(\sigma \) -algebra on which the theory is defined without the assignments of 1 to \(\varOmega \) and 0 to \(\emptyset \) . Footnote 2 , Footnote 3

To illustrate the proposal, let us now reconsider the theories \(T_L\) and \(T_M\) from Example  3 . In terms of its elementary consequences, we believe that \(T_L\) should be written down as Footnote 4

and that \(T_M\) should be written down as

From now on, we will employ an abuse of notation: if \(\alpha \) is a conjunct of a theory \(\tau \) presented in terms of its elementary consequences, we will denote this by \(\alpha \in \tau \) . For instance, we can write: \(M(\{ A,B \}) = r_{AB} \in T_M\) . Now let us consider \({\textit{Tr}}_1(T_L)\) and \({\textit{Tr}}_1(T_M)\) . We can see that

In Example  3 we assumed that M was way off the mark on A and B . Let us assume then that \(\frac{1}{n}\sum _{\tau _i \in (T_M - T_L)} {\textit{App}}(\tau _i)\) is negative. In this case

which is what we intuitively expect. If, on the other hand, M lies fairly close to P , then \(\frac{1}{n}\sum _{\tau _i \in (T_M - T_L)} {\textit{App}}(\tau _i)\) should be positive. In that case

which also makes sense. Let us quickly summarize: \(T_L\) and \(T_M\) will be assigned the same values for elementary statements that they have in common. The more fine-grained theory, \(T_M\) , also has the chance to get a bonus or a penalty for the statements it does not share with \(T_L\) .

In the preceding example, we have assumed that \({\textit{App}}\) takes on both positive and negative values (in agreement with the second requirement for \({\textit{App}}\) by Cevolani and Schurz 2017 ). In fact, there is no straightforward way to amend the theory such that \({\textit{App}}\) can only take negative or positive values. Footnote 5 Hence, we cannot simply define \({\textit{App}}(M(E) = r_E) := |r_E - p_E|\) , where \(p_E\) is the value that the true probability function assigns to event E . Moreover, the observation also rules out defining \({\textit{App}}\) as a statistical distance, which is a pity since it makes it harder to tie in this approach with existing work. (We return to this point in the next section.)

Again, without the full formal framework for thinking about relevant consequences in quantitative or probabilistic settings, the above account remains speculative. Further logical research is needed to assess whether the above account holds up as a concrete example of a Schurz–Weingartner–Cevolani-style theory of verisimilitude or as a variant. The ideas outlined here should make us hopeful for a well-behaved theory.

3.2.2 Candidates for \({\textit{App}}\)

If we try to apply Definition  4 to probabilistic theories, we need to make a choice for the measure of approximation, \({\textit{App}}\) . As shown above, the range of \({\textit{App}}\) cannot simply be a subset of either \({\mathbb {R}}^+\) or \({\mathbb {R}}^-\) , but has to take on both positive and negative values. This means that \({\textit{App}}\) needs to distinguish elementary consequences that perform ‘well enough’ qua likeness to truth (by assigning a positive value to them) from those that do not (by assigning a negative value). Given that it is usually better that the proposed probability of an event E lies close to the real probability of E , it is natural to work with a threshold: \({\textit{App}}\) assigns positive values to probabilities that differ by less than the threshold, negative values to those that differ more, and zero to those exactly at the threshold. How large the threshold ought to be might be context-dependent.

One way to meet the criteria for such a measure is as follows, where \(\epsilon \in \ ]0,1]\) is the threshold and each \(T_i\) is an element of theory T that assigns to event \(E_i\) the following probability, \(P(E_i) := r_i\) :

Definition 5

where \(r_i^*\) is the true value of \(P(E_i)\) .

Consider, for example, an event \(E_i\) with a true probability \(r_i^* = 0.7\) and consider a threshold \(\epsilon = 0.1\) , then the corresponding graph of \({\textit{App}}_\epsilon \) is depicted in Fig.  2 .

Numerical example of \({\textit{App}}_\epsilon \) as a function of \(r_i\) (the probability value that \(T_i\) assigns to the event of interest), for true probability \(r_i^* = 0.7\) and threshold \(\epsilon = 0.1\)

Alternatively, we could consider a definition in which all occurrences of \(|r_i^*-r_i|\) are replaced, mutatis mutandis, by \((r_i^*-r_i)^2\) for instance. So, while we do not find a definition for the measure of approximation that is uniquely well-motivated, at this point it might look like we have an almost complete account.

( \({\textit{App}}_\epsilon \) -based truthlikeness for P and Q ) Let us now illustrate this proposal for \({\textit{App}}\) by considering the probability functions P , Q , and R from Examples  3 and  4 . We stipulated that P was the true distribution and both Q and R are defined on the full algebra \({\mathcal {A}}\) . When we apply Definition  5 with \(\epsilon = 0.1\) , we obtain the numerical values of \({\textit{App}}\) listed below:

The truthlikeness for \(T_R\) and \(T_Q\) can now be calculated by plugging these values into Definition  4 :

We see that \({\textit{Tr}}(T_R) > {\textit{Tr}}(T_Q)\) , as we would intuitively expect.

In the next section, we discuss the strengths and weaknesses of \({\textit{App}}\) -based approaches in a more general setting.

3.3 Taking stock of consequence approaches for verisimilitude of probabilistic theories

Initially, we were rather sceptical towards the applicability of Schurz–Weingartner–Cevolani-style approaches to verisimilitude of probabilistic theories. Footnote 6 We have since warmed up considerably to this approach and related ideas. As mentioned above, we are hopeful that further research will yield a well-behaved theory. Nevertheless, we still believe that this approach might also be plagued by some issues. These issues all revolve around the same theme: the framework’s limited compatibility with important ideas from information theory.

Firstly, \({\textit{App}}\) forces us to compare distributions on all events that are in the set of elementary consequences. As such, \({\textit{App}}\) as envisioned by Cevolani and Schurz ( 2017 ) does not seem very flexible. Indeed, if one only wants to compare distributions on a couple of specific events, one might run into problems. Secondly, while the threshold-based construction for \({\textit{App}}\) that we considered in the previous section is adequate (in the sense of the criteria reviewed in Sect.  3.2.1 ), it remains artificial in the context of information theory. Usually, probability distributions are compared by (pre-)distances, but there is no such ready-made candidate for \({\textit{App}}\) , because it needs to take both positive and negative values (as explained in footnote 5).

Presumably, when Cevolani and Schurz ( 2017 ) developed their account for quantitative theories, they had in mind measures that are qualitatively different, such as the mass and the length of an object. As such, applying it to probabilistic theories is off-label use and we should not be surprised that we obtain heterodox proposals.

The approach we develop below is not a Schurz–Weingartner–Cevolani framework. Nevertheless, we are still indebted to their approach since we compare theories in terms of their ambient logical structure. The following example will allow us to quickly revisit some of the Popperian themes.

( High and low resolution ) Consider a theory, \(T_C\) , that is defined with respect to the Boolean algebra with semantic atoms \(A_1, A_2, \ldots , A_{100}\) and another theory, \(T_X\) , that is defined on an algebra with ‘atoms’ \(B_1, B_2, \ldots , B_{10}\) , where

( C and X refer to the Roman numerals for 100 and 10, respectively.)

Theory \(T_C\) consists of the following set of 100 statements:

Theory \(T_X\) consists of the following set of 10 statements:

In this case, \(T_X\) can be regarded as a low-resolution or coarse-grained version of \(T_C\) , since (i) the algebra of \(T_X\) only contains unions of elements of the algebra of \(T_C\) and (ii) the probabilities that \(T_X\) assigns to these unions are equal to the sum of probabilities that \(T_C\) assigns to the composing sets. In other words, where they are both defined, the probability assignments agree, and the cases where the probability is defined on \(T_X\) form a strict subset of the cases where the probability is defined on \(T_C\) . If we assume \(T_C\) is true, so is \(T_X\) , but \(T_C\) will be assigned a higher verisimilitude than \(T_X\) .

This example is analogous to the case of non-probabilistic theories differing in information content and precision (cf. Sect.  2 ). In practice, fine-graining can come into view due to theoretical refinement or due to increases in measurement resolution. Observe that there may also be theories that are more fine-grained in some aspects and more coarse-grained in others, so the order need not be total.

We acknowledge that the ambient logical structure and the relations it induces between theories are an essential part of a good understanding of verisimilitude and related Popperian notions. At the same time, we believe that we also need to look at the content of the theories, when considering the probabilistic case. A probabilistic theory might have little to no (non-trivial) true consequences at all, while still being a very good approximation (recall Example  3 and its subsequent discussion). The crux of Popperian verisimilitude consists in notions of true and false consequence. Indeed, Schurz and Weingartner ( 2010 ) as well as Oddie ( 2016 ) show that Popper’s account of verisimilitude can best be regarded as a consequence account. Even though we do not claim that more impure consequence accounts like Cevolani and Schurz’s ( 2017 ) \({\textit{App}}\) -based approach are doomed to fail, we do not aim to resolve these issues in our own account either. We side-step the issue because our proposal below is not a consequence account.

4 Towards an alternative definition of truthlikeness for probabilistic theories

At this point, we step away from the account that was based on propositional logic. Our goal here is to come up with an adequate alternative notion of verisimilitude that takes the structure of the underlying \(\sigma \) -algebra (which allows coarse-graining) into account. In Sect.  4.4 , we will consider the analogous question for approximate truth. Our main topic in this section, however, is the notion of truthlikeness in the context of probabilistic theories. Our goal is not to give a full, formal account, which would constitute a complete research program in itself, requiring technical work akin to that of Schurz and Weingartner ( 1987 ) and Gemes ( 2007 ). Instead, our more modest aim here is twofold. First, we want to show that we can apply some central Popperian ideas to probabilistic theories. Secondly, we want to suggest some pitfalls and desiderata for the development of a fully fledged formal account along these lines. We believe, in the spirit of Russell’s epigraph, that propositional logic or possible-world accounts cannot hope to capture actual scientific theories; neither can the alternative we wish to prepare for. Much like how scientists use the harmonic oscillator or other toy models and idealizations, we use formal methods to study specific aspects of science itself—without any pretense of being able to capture all aspects of actual examples of scientific theories. In other words, the point of our work is to expand the “abstract imagination” and thus to help us pinpoint fundamental issues.

4.1 Compatibility of probabilistic theories at a certain level of grain

We consider probabilistic theories that are fully specified by listing the probability assignments to the events that form the basis of a \(\sigma \) -algebra. From here on, we will refer to probabilistic theories by their probability function. In Example  6 , we have encountered a case of “compatibility” between probability functions that were defined on algebras with a different resolution. Let us now turn this idea into a general definition.

Definition 6

Compatibility of refinements.

Consider two probability functions, P defined on algebra \({\mathcal {A}}\) and \(P'\) defined on algebra \({\mathcal {A}}'\) that is a subalgebra of \({\mathcal {A}}\) . We say that probability distribution P is a compatible refinement of \(P'\) if \(P'(E) = P(E)\) for each event E in \({\mathcal {A}}'\) .

In this case, \(P'\) is called a compatible coarsening of P ; P and \(P'\) are said to be compatible with each other.

So, to determine whether two probabilistic theories are compatible, we compare their probability functions at the level of the coarsest algebra among the two. Clearly, this comparison only makes sense when the algebra of one probability function is a subalgebra of the other, as is indeed required by the definition. Another way of coarsening that could in principle be considered is related to rounding probability values. This is not what we are dealing with here, but it could help to restrict the set of all possible probability functions to a finite set.

If we apply Definition  6 to Example  6 , we see that \(P_C\) is a compatible refinement of \(P_X\) . Also observe that any probability function on any non-trivial algebra is a compatible refinement of the trivial (0, 1)-probability function on the minimal algebra (containing only the empty set and the sample space).

We denote the compatibility of a coarser probability distribution \(P'\) with P by \(P' \le P\) , which is read as: \(P'\) is a compatible coarsening of P . The symbol “ \(\le \) ” is fitting, since the compatible coarsening relation is reflexive ( \(P \le P\) for all P ), antisymmetric ( \(P' \le P\) and \(P \le P'\) implies \(P' = P\) ), and transitive ( \(P'' \le P'\) and \(P' \le P\) implies \(P'' \le P\) ); hence, it is a partial order.

Given an algebra \({\mathcal {A}}\) and a probability function \(P'\) on \({\mathcal {A}}'\) , which is a subalgebra of \({\mathcal {A}}\) , there may be many compatible refinements defined on an algebra \({\mathcal {B}}\) such that \({\mathcal {A}}' \subseteq {\mathcal {B}} \subseteq {\mathcal {A}}\) . We now fix a collection of compatible refinements to a given, coarser probability function in the following way. First, we fix a finite set of probability functions on subalgebras of \({\mathcal {A}}\) , calling it \(D_{\mathcal {A}}\) . Then, for a probability function \(P'\) defined on a subalgebra of \({\mathcal {A}}\) , we call \(D_{\mathcal {A}}(P')\) the subset of \(D_{\mathcal {A}}\) consisting of distributions compatible with \(P'\) .

Put differently,

which is a set of compatible refinements of \(P'\) . The idea behind this notion is sketched in Fig.  3 .

Schematic representation of a set of compatible refinements of a probability function on a coarser algebra

4.2 Quantifying verisimilitude of probabilistic theories

A natural way to define the likeness between two probabilistic theories would be to consider a statistical (pre-)distance (such as the Kullback–Leibler divergence or the Jensen–Shannon distance, to name but two examples) between their probability functions. However, these (pre-)distances are only defined if the two functions have the same domain. We want to define the truthlikeness of a probabilistic theory, so we need to compare a hypothesized probability function with the true probability function. Since these will typically have a different domain, we need to do a little more work first. To achieve this, we consider the set of compatible probability functions that are defined on the same domain as the true probability function.

This way, verisimilitude can be defined as follows:

Definition 7

Verisimilitude of a probabilistic theory.

Let \(P^*\) be the true probability function and \({\mathcal {A}}\) its algebra. Let \(D_{\mathcal {A}}\) be a finite set of probability functions on subalgebras of \({\mathcal {A}}\) , with \(P^* \in D_{\mathcal {A}}\) . Let \(P'\) be a probability function on a subalgebra of \({\mathcal {A}}\) . Given a statistical (pre-)distance m , the verisimilitude of \(P'\) relative to m is: Footnote 7

The idea here is simple. For every \(P'\) , we calculate the average “penalty” (hence the minus sign) that the distributions in its associated set of compatible refinements \(D_{\mathcal {A}}(P')\) accrue. There are good reasons to opt for the average penalty here. By proposing a theory with a more coarse-grained probability function \(P'\) , one casts a wider net: this way, one has a better chance to catch some of the theories that lie close to the truth ( \(P^*\) ), but one risks ensnaring some very inadequate theories as well.

The idea presented in Definition  7 is similar in flavor to the Tichý–Oddie average measure of verisimilitude, Vs (Oddie 1986 ). Their proposal was made in a possible-world context. The verisimilitude of the theory was then calculated as unity minus the average distance from the worlds \(w'\) covered by a given theory \(\tau \) to the actual world w . In other words:

Instead of possible worlds, we use the probabilistic theories P that are compatible with a given theory \(P'\) . Footnote 8 Similar ideas have recently been explored by Cevolani and Festa ( 2020 ).

Let us briefly elaborate on the fact that we have assumed the algebra \({\mathcal {A}}\) and the associated true probability function \(P^*\) to be the most fine-grained among all probabilistic theories under consideration. This is related to an important point raised by Kuipers ( 1982 ), who distinguished between descriptive verisimilitude and theoretical verisimilitude: whereas descriptive verisimilitude tracks the closeness of a descriptive statement to the true description of the actual world, theoretical verisimilitude tracks the closeness of a candidate theory to the theoretical truth (which is compatible with more than one physical possibility). Given that we assume the true probabilistic theory (formulated in terms of \({\mathcal {A}}\) and \(P^*\) ) to be maximally fine-grained, it may look as though we are dealing with the former. However, the very fact that we start from a probabilistic theory shows otherwise: by their very nature, probabilistic theories allow for multiple possibilities. To emphasize this, we can exclude theories that assign all prior probability mass to a singleton. Moreover, the structure of the algebra typically does not include the power set of the sample space. This shows that we aim to track theoretical verisimilitude rather than descriptive verisimilitude. In addition, we allow that scientists can usefully coarse-grain relative to \({\mathcal {A}}\) and \(P^*\) , so we do not presuppose that the true probabilistic theory is at the right level of grain for all purposes.

In light of the previous section, some readers might expect us to opt for an approach analogous to introducing an \({\textit{App}}\) function, with positive and negative contributions. While this would in principle be possible, we have chosen otherwise. Instead, we divide by \(|D_{\mathcal {A}}(P')|\) , in order to penalize for coarseness. This factor is also related to riskiness and falsifiability: a more coarse-grained probability function is compatible with a larger set of refinements. Hence it is less risky and has a lower degree of falsifiability.

( m -based truthlikeness of P and Q ) To illustrate Definition  7 , let us again consider the theories R and Q from Examples  3 and  4 . To measure the distance between two probability distributions, \(P_1\) and \(P_2\) defined on the same algebra \({\mathcal {A}}\) , we pick the Jensen–Shannon distance, \(m_{JS}\) . Footnote 9 Note that in this special case, where P and Q are specified on the same, full algebra, the reference probability sets are singletons \(D_{{\mathcal {A}}}(Q)=\{Q\}\) and \(D_{{\mathcal {A}}}(R)=\{R\}\) . Hence, comparing truthlikeness amounts to comparing the Jensen–Shannon distances between P and Q , and between P and R . In this case, we have that

Again, we find that R is more truthlike than Q , as we would intuitively expect.

From the perspective of machine learning and finance, our approach looks quite familiar, especially if we consider \(-{\textit{Tr}}_m(P')\) instead of \({\textit{Tr}}_m(P')\) : \(m(P^*,P)\) plays the role of a loss function , the average of which we want to minimize. Footnote 10 So, \(-{\textit{Tr}}_m(P')\) , which averages over the possible losses, can be considered to be a risk function that expresses the expected utility of \(P'\) .

Definition  7 does not specify which statistical (pre-)distance m one has to consider. Given that the literature on similarity between functions as well as the literature on distances between statistical objects has produced various measures, which capture different aspects of likeness, we currently do not see a uniquely well-motivated choice among them for truthlikeness either. A probabilistic theory may be very similar to the truth in some ways and less so in a different way. It may depend on the context, such as the goals of an inquiry, which of these aspects matters most. Hence, we have chosen to relativize the notion of truthlikeness to the likeness measure at hand.

Of course, there are additional variations on Definition  7 that we may consider. For instance, we may increase the penalties by applying monotone functions to the (pre-)distance. Hence, a possible alternative definition is given by:

4.3 Further relativization of verisimilitude of probabilistic theories

As already mentioned, more coarse-grained probability functions, \(P'\) , defined on a smaller subalgebra \({\mathcal {A}}\) , cast a wider net, which we represented by \(D_{\mathcal {A}}(P')\) . There is an alternative way to define the relevant net. We might, for instance, be interested in all probability distributions that lie close to a given (set of) distributions. To do this we, define the notion of \(\epsilon \) -compatibility.

Definition 8

\(\epsilon \) -compatibility of coarsenings.

Consider an algebra \({\mathcal {A}}\) and a non-empty but finite set, \(D_{\mathcal {A}}\) , of probability functions on subalgebras of \({\mathcal {A}}\) . Consider a probability function \(P \in D_{\mathcal {A}}\) defined on \({\mathcal {A}}\) and a probability function \(P'\) , defined on a subalgebra of \({\mathcal {A}}\) . Fix an \(\epsilon \in {\mathbb {R}}^+_0\) and fix a (pre-)distance, m , on the space of probability distributions. We say that \(P'\) is an \(\epsilon \) -compatible coarsening of P if there exists a probability distribution \(Q \in D_{\mathcal {A}}(P')\) such that \(m(P,Q) < \epsilon \) . In this case, we call P an \(\epsilon \) -compatible refinement of \(P'\) .

We denote the set of distributions that are \(\epsilon \) -compatible refinements of a given distribution \(P'\) by \(D_{{\mathcal {A}},m,\epsilon }(P')\) . This allows us to expand Definition  7 as follows.

Definition 9

\(\epsilon \) -verisimilitude of a probabilistic theory.

Let \(P^*\) be the true probability function and \({\mathcal {A}}\) its algebra. Let \(D_{\mathcal {A}}\) be some non-empty but finite set of probability functions defined on subalgebras of \({\mathcal {A}}\) . Let \(P'\) be a probability function on a subalgebra of \({\mathcal {A}}\) . Given a statistical (pre-)distance m and an \(\epsilon \in {\mathbb {R}}_0^+\) , the verisimilitude of \(P'\) is:

This account, like that of Cevolani and Schurz ( 2017 ), features a parameter. In our account, \(\epsilon \) plays a very different role than \(\varphi \) does in theirs: by changing \(\epsilon \) , we can change the scope of probability distributions that we would like to capture by our theories, as the following example shows.

Let us start again from Examples  3 and 4 and assume that P , Q , and R are distributions in some finite set of distributions that are under consideration. For ease of presentation (and especially calculation), we take the \(L_\infty \) metric (which measures the distance between two discrete probability distributions as the maximum of the distance between the probability assignments over sets in the algebra). Further assume that \(P'\) is defined on the algebra \(\{ \emptyset , \{A,B\}, \{C\}, \varOmega \}\) , with \(P'(\{A, B\}) := 0.7\) and \(P'(\{C\}) := 0.3\) . We will, again for ease of presentation, also assume that P , Q , and R are the only inhabitants of \(D_{{\mathcal {A}},L_\infty , 0.1}(P')\) . We can now compare the truthlikeness of the following theories: (1) \(D_{{\mathcal {A}},L_\infty , 0.1}(P')\) , (2) \(D_{{\mathcal {A}},L_\infty }(P')\) , (3) \(D_{{\mathcal {A}},L_\infty , 0.05}(P)\) , and (4) \(D_{{\mathcal {A}},L_\infty }(P)\) .

Since \(D_{{\mathcal {A}},L_\infty , 0.1}(P') = \{P, Q, R\}\) , Definition  9 leads us to:

Since \(D_{{\mathcal {A}},L_\infty }(P') = \{P, Q\}\) , applying Definition  7 yields:

Since \(D_{{\mathcal {A}},L_\infty ,0.05}(P) = \{P, R\}\) , with Definition  9 we find that:

Since \(D_{{\mathcal {A}},L_\infty }(P) = \{P\}\) , Definition  7 shows:

We see all sorts of interesting behavior here. First, \({\textit{Tr}}_{L_\infty }(P)\) , the truthlikeness of the theory containing only the true theory, is the highest of all theories. This makes perfect sense: this theory has maximal likeness qua truth and maximal content. Secondly, we can compare theories that do not imply each other. For instance, we can see that \({\textit{Tr}}_{L_\infty , 0.05}(P) > {\textit{Tr}}_{L_\infty }(P')\) . So, our proposal is not trivial in the sense that it only allows us to compare theories that stand in a certain logical relation to one another. Finally, let us have a look at how the truthlikeness of \(D_{{\mathcal {A}},L_\infty , 0.1}(P')\) relates to the truthlikeness of \(D_{{\mathcal {A}},L_\infty }(P')\) . We can see that \({\textit{Tr}}_{L_\infty , 0.1}(P') > {\textit{Tr}}_{L_\infty }(P')\) . This means the following: it is not necessarily the case that the truthlikeness increases as the content increases. The proposal also takes into account how close to the truth the stronger theory is. If the stronger theory has thrown out a lot of good parts of the weaker theory, it might get assigned a lower truthlikeness than the weaker theory. All of these are desirable properties for a theory of truthlikeness.

Moreover, Definition  9 can be extended to second-order probabilistic theories, which assign probabilities to elements of a set of (first-order) probabilistic theories, using a weighted average over \(\epsilon \) -verisimilitudes of the latter (using the higher-order probabilities as weight factors).

Nevertheless, there is a major downside to Definition  9 : if we coarsen \(P'\) to \(P''\) in such a way that \(D_{{\mathcal {A}},m,\epsilon }(P'') - D_{{\mathcal {A}},m,\epsilon }(P')\) only contains distributions that have the average distance from \(P^*\) to the elements of \(D_{{\mathcal {A}},m,\epsilon }(P')\) , then \({\textit{Tr}}_m(P')={\textit{Tr}}_m(P'')\) . This problem occurs because averaging “forgets” about the size of sets. In fact, this is a problem for all approaches to verisimilitude that use averaging.

We can make verisimilitude “remember” the size of sets by dividing the sum of \(m(P^*, P)\) by a factor smaller than \(|D_{{\mathcal {A}},m,\epsilon }(P')|\) . In order to achieve this, we consider functions \(f : {\mathbb {N}}_0 \rightarrow {\mathbb {R}}\) that satisfy the following three properties: (1) \(f(1) = 1\) , (2) f is increasing, and (3) \(n- f(n)\) is monotonically increasing on \({\mathbb {N}}_0\) . This entails that \(f(n) \le n\) for all \(n \in {\mathbb {N}}_0\) . We will call these functions well-behaved .

Definition 10

f -verisimilitude of a probabilistic theory.

Given a statistical distance m and an \(\epsilon \in {\mathbb {R}}^+_0\) . Now suppose that f is well-behaved. Then the f -truthlikeness of \(P'\) is defined as follows:

As compared to Definition  7 , this definition relativizes truthlikeness to two additional choices: a value for \(\epsilon \) and a particular function f (for example, a root function).

In fact, this ongoing relativization points us towards a crucial conceptual difficulty in quantifying versimilitude: how should one balance broadness of scope versus amount of truth? In the case of classical logics, this problem is “solved” by virtue of the naturalness of the Schurz–Weingartner–Cevolani framework. For probabilistic theories, however, there does not seem to be an equally natural choice. In particular, it is not clear what the trade-off between coarseness of the theory and losses incurred should be. This can be set by selecting a particular function f and, arguably, this is a subjective or at least context-dependent choice.

Although Definition  10 allows for various choices, it is still not fully general, since it only applies to probabilistic theories that specify probabilities on a subalgebra of the probability function associated with the true theory. As such, the possibility of being mistaken about the relevant sample space and the related matter of conceptual novelty have not been addressed here—a shortcoming shared with all extant proposals for verisimilitude.

4.4 Approximate truth of probabilistic theories

Some of the ideas in the previous section can also be applied to the notion of approximate truth of probabilistic theories. As before, we refer to such theories by their probability functions. Unlike verisimilitude, approximate truth is not affected by how much or how little a theory addresses, merely which fraction of the claims that the theory does make is true. In terms of probability functions, it does not matter how coarse the subalgebra is. The trivial algebra (empty set and sample space) and the trivial (0, 1)-probability assignment to it give rise to a maximal approximate truth, represented by a value of 1. This is the same approximate truth value obtained by a theory that stipulates a probability \(P'=P^*\) .

As long as we are dealing with finite algebras, we can define the approximate truth value as the fraction of correct probability assignments in the basis of the function’s algebra normalized by the number of elements of that basis. However, this would again raise the problem that no distinction is made between near misses and assignments that are far off the mark.

We propose to reuse our solution from Sect.  4.2 : introducing a statistical (pre-)distance between the hypothesized probability function and the compatible coarsening of the true probability function at the level of the hypothesized function. This idea is applied in the following definition.

Definition 11

Approximate truth of a probabilistic theory.

Let \(P^*\) be the true probability function on an algebra \({\mathcal {A}}\) and let \(P'\) be a theoretically proposed probability function defined on \({\mathcal {A}}' \subseteq {\mathcal {A}}\) . Let \(D_{{\mathcal {A}}'}\) be a finite set of probability functions defined on \({\mathcal {A}}'\) (contextually defined by which probabilistic theories are relevant). Given a statistical (pre-)distance m , the approximate truth of \(P'\) is:

4.5 Outlook

In future work, we plan to put the proposals forwarded here to the test: by applying the proposed definitions to concrete examples, we can check how various choices of measures and parameters influence the attainment of various goals, such as convergence to the truth in the limit or speed of approach to the truth. Such a study will also help us to clarify whether the proposals in Definitions  10 and  11 are robust in the following sense: different (pre-)distances m usually lead to different numerical results, but subsets of them could still lead to the same ordinal ordering. The methodology for investigating this hypothesis relies on numerical simulations, similar to the work of Douven and Wenmackers ( 2017 ), and requires a separate study.

Another matter that requires follow-up is more conceptual in nature: although probabilistic theories played a central role in our paper, we have not touched upon the thorny issue of the interpretation of probabilities. Yet, especially when we want to analyze the verisimilitude of probabilistic theories, it is highly relevant what the probabilities are taken to represent: are they transient features related to a lack of knowledge that may be improved upon, or are they permanent markers of irreducible stochasticity? This question can be related to a more general one in the verisimilitude literature about the goal of science: does science aim at a complete and true description of the actual world, which can be related to a particular realization of a probabilistic model? Or does it aim at finding true laws that pick out physical possibilities among all logical possibilities?

As mentioned in Sect.  4.2 , Kuipers ( 1982 ) clarified that these viewpoints lead to two different notions of verisimilitude, called descriptive verisimilitude and theoretical verisimilitude, respectively. In the context of probabilistic theories, this leads to an analogous question that can be phrased as follows: should “the truth” be represented as a degenerate probability distribution that assigns all probability mass to single possible outcome, which is equal to the unique physical realization that unfolds in time, or not? We have assumed that there is a most fine-grained algebra and a true probability function, \(P^*\) , defined on it. As such, our formalism does not presuppose that the goal of science is merely to describe the actual world, but instead to describe physical possibilities with their associated probabilities. However, if someone were to add the assumption that \(P^*\) is degenerate (in the sense of being equivalent to a Class I theory), the same formalism may perhaps be used to make sense of descriptive verisimilitude as well.

Finally, we think these discussions can be enriched by case studies that go beyond toy problems and that consider applications outside of philosophy of science proper, for instance in computer science.

5 Conclusions

Taking stock, in this paper we have made some progress in analyzing three Popperian concepts—riskiness, falsifiability, and truthlikeness—in a formal and unified context, in particular when considering probabilistic theories. In Sect.  2 , we have disentangled two dimensions of riskiness. The first one, informativeness, correlates positively with gradable falsifiability and can be modeled formally. We have also clarified that a formal account of degrees of falsifiability should capture the interplay between two algebras, that of the language of the theory and that of the experimental context. We have shown that this analysis applies to deterministic and indeterministic theories, allowing for a unified treatment. In Sect.  3 , we reviewed the extant proposals for a formal treatment of truthlikeness and approximate truth. Despite the indisputable virtues of the Schurz–Weingartner–Cevolani framework, we also found some shortcomings. One issue is that they involve a lot of bookkeeping that is, as of yet, incapable of capturing the structure of probabilistic theories. We believe that capturing this structure is essential for measuring the “likeness” among theories, and for estimating the likeness to the true theory in particular. After all, a stopped clock tells the time exactly twice in twenty-four hours, and by letting the hands turn rapidly (even counterclockwise!) a clock can be made to indicate the right time as often as one likes. Yet, only a clock with hands that rotate clockwise at about one or twelve hours per turn can hope to be like a true time teller. In response to some of these shortcomings, in Sect.  4 , we have given a general form for plausible definitions of both truthlikeness and approximate truth.

Let us now reflect on the relation between the three Popperian concepts in the title. We have seen that an important ingredient of Popperian riskiness is informativeness. Informativeness is related to both falsifiability and truthlikeness, albeit with different caveats. In the case of falsifiability, informativeness is an important ingredient. This can be seen, for instance, from Example  6 : there exists a lot of potentially corroborating evidence for the coarse-grained theory \(T_X\) that disconfirms or falsifies the more informative theory \(T_C\) . In the case of truthlikeness, riskiness increases with it provided that the theory is in fact true. So, whereas falsifiability is unrelated to truth or falsehood of a theory, truthlikeness does depend on truth. And whereas truthlikeness is experiment-independent, falsifiability is related to experimental severity.

Hence, informativeness does not tell the whole story: the severity of available experiments should be taken into account as well. Clearly, improbability alone is not going to cut it either: it correlates with informativeness, but also with low priors due to variation independent of logical strength and with past disconfirmation. This observation is related to an impossibility result: Sprenger ( 2018 ) has shown that no corroboration measure based on statistical relevance can simultaneously represent informativeness and the severity of tests that a theory has passed successfully. Whereas informativeness is related to improbability of priors (as far as they are correlated to logical strength), past predictive success leads to increased posterior probability: these are clearly two dimensions that cannot be meaningfully combined into one total order.

Of course, actual scientific theories and their historical development are more complicated than any formal model can hope to capture, but we think studies like ours should aim for the Russellian goal of “enlarging our abstract imagination”. As such, we hope our proposals will encourage further debate and development of a formal account that further unifies Popperian concepts and Bayesian or information-theoretic methods.

We thank Pieter Thyssen for an early discussion on this possible connection.

The reasoning for including all events, \(E_i\) , where the relevant probability measure is defined is thus: first, if \(P(E_1) = r_1; P(E_2) = r_2; P(E_1 \cap E_2) = 0\) , then \(P(E_1 \cup E_2) = r_1+r_2\) is a relevant consequence. Indeed, we cannot replace \(E_1\) or \(E_2\) by just any event salva validate . Secondly, it does not seem the case that \(P(E_1 \cup E_2) = r_1 + r_2\) is equivalent to conjunctions of elements of the form \(X_i(a) = r_i\) . For instance, \(P(E_1) = r_1; P(E_2) = r_2; P(E_1 \cap E_2) = 0\) is stronger than \(P(E_1 \cup E_2) = r_1 + r_2\) .

We have omitted the events \(\emptyset \) and \(\varOmega \) because otherwise the trivial theory, \(T_{{\textit{trivial}}}:=(P_{{\textit{trivial}}}(\varOmega )=1; P_{{\textit{trivial}}}(\emptyset ) = 0)\) , would always get assigned a high verisimilitude, which would be counterintuitive. Our choice allows us to set the default value for the trivial theories to 0. If one is not working in a set-theoretic context, one can just replace \(\emptyset \) and \(\varOmega \) by the \(\bot \) and \(\top \) elements of the algebra, respectively.

Note that consequences of the form \(L(\{A,C\}) \ge r_c\) are not elementary, since we could replace A by any element of \(\varOmega \) . We would like to thank an anonymous referee for pointing us in this direction.

Suppose that the range of \({\textit{App}}\) would be (a subset of) \({\mathbb {R}}^+\) . Then \({\textit{Tr}}(T_M) \ge {\textit{Tr}}(T_L)\) , no matter how accurate or inaccurate \(T_M\) is on \((T_M - T_L)\) . Indeed, on the assumption that the range of \({\textit{App}}\) is (a subset of) \({\mathbb {R}}^+\) , we have that \({\textit{App}}(\tau _i) \ge 0\) for all \(\tau _i \in (T_M - T_L)\) . In this case, the second term of Eq. ( 3 ):

In other words, if the range of \({\textit{App}}\) is (a subset of) \({\mathbb {R}}^+\) , any theory \(T_M\) that implies \(T_L\) has a higher truthlikeness than \(T_L\) , irrespective of its likeness qua truth on \((T_M- T_L)\) . Similarly, if the range of \({\textit{App}}\) was (a subset of) \({\mathbb {R}}^-\) , any theory \(T_M\) that implies \(T_L\) would be lower in truthlikeness than \(T_L\) , irrespective of its likeness qua truth on \((T_M - T_L)\) . Neither option captures how truthlikeness should behave. To avoid this, the range of \({\textit{App}}\) should include both positive and negative real values.

We are thankful to the referees for making us reconsider our assessment.

In principle, we should write \({\textit{Tr}}_{{\mathcal {A}},m}\) rather than \({\textit{Tr}}_m\) , but the reference algebra \({\mathcal {A}}\) is usually clear from the context.

We would like to thank an attentive referee for pointing this out.

The Jensen–Shannon distance is defined as follows:

where \(D_{KL}\) refers to the Kullback–Leibler divergence, in turn defined as:

with the base of the logarithm an arbitrary but fixed real \(>1\) . We have chosen the natural logarithm for our numerical results.

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Acknowledgements

We are grateful to the editors for inviting us to contribute to this topical collection, which prompted us to start this investigation. We thank Igor Douven, Wayne Myrvold, and Wouter Termont for very helpful feedback on our paper. We are grateful to two anonymous referees, whose detailed reports helped us to improve this paper. LV acknowledges his doctoral fellowship of the Research Foundation Flanders (Fonds Wetenschappelijk Onderzoek, FWO) through Grant Number 1139420N; SW acknowledges funding for part of this research project through FWO Grant Number G066918N.

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This article belongs to the topical collection “Approaching Probabilistic Truths”, edited by Theo Kuipers, Ilkka Niiniluoto, and Gustavo Cevolani.

Appendix A: Degrees of falsifiability and logical strength

The point of bold theories is to describe a true state of affairs with sentences of the language (say \({\mathcal {L}}\) ) that are as precise as possible. Any observation, \(\phi \) , that is inconsistent with a theory falsifies it.

Here we propose a sufficient condition for relative falsifiability: theory \(\tau _1\) is more falsifiable than theory \(\tau _2\) if every observation that falsifies \(\tau _2\) also falsifies \(\tau _1\) . Footnote 11 So, intuitively, theories become more falsifiable with logical strength. Indeed, if \(\tau _1\) is logically stronger than \(\tau _2\) , then we have the class of models \({\textit{Mod}}(\tau _1 \cup \{\phi \}) \subseteq {\textit{Mod}}(\tau _2 \cup \{\phi \})\) for every \(\phi \) .

Note that, even though we naturally prefer true theories over false ones, this does not matter for relative falsifiability: this is exactly as it should be. Secondly, even though we believe that the above is a sufficient condition for relative falsifiability, we do not believe it to be a necessary condition. For example, Popper ( 1963 ) argued that the theory of relativity is more falsifiable than psychoanalysis, even though neither seems to logically entail the other. Our sufficient condition also plays well with our ideas regarding prior credences, since probabilities decrease with logical strength.

Appendix B: Relation between severity and entropy

We illustrate that a severity measure corresponds to an entropy measure by means of an example. In particular, we show that the severity measure defended by Milne ( 1995 ) is related to the Kullback–Leibler divergence.

Milne ( 1995 ) has proposed to define the severity of a test as its expected disconfirmation. First, suppose we have a hypothesis h that we would like to test with an experiment \({\mathcal {E}} := \{e_1, e_2, \dots , e_N\}\) . Assume that \(P(h) \ne 0\) and \(P(e_i) \ne 0\) for all \(e_i \in {\mathcal {E}}\) . Then, Milne ( 1995 ) defined a measure of the degree of confirmation (first proposed by I. J. Good) as follows (with the base of the logarithm arbitrary but fixed \(>1\) ):

Finally, he identified the severity of a test of hypothesis h as the expected disconfirmation, that is:

Milne considered \(c_M(h, e)\) to be a uniquely well-chosen and natural measure of confirmation, which combines various aspects of Bayesianism and vindicates certain Popperian maxims. Nevertheless, Milne realized the superiority claim about \(c_M(h, e)\) as a measure confirmation might not convince everyone. Indeed, in the later comparison of various measures of confirmation relative to a list of four (a-)symmetry desiderata by Eells and Fitelson ( 2002 ), \(c_M(h, e)\) did not stand out as the best choice. However, Milne did maintain that \(c_M(h, e)\) was at least superior in another regard: as a measure of the decrease in informativeness and/or uncertainty. This claim stood the test of time better; see, e.g., Myrvold ( 2003 ).

So, can \(\langle d_{\mathcal {E}}(h) \rangle \) , based on this measure, also be interpreted as a—or perhaps the —measure of the degree of falsifiability? Unfortunately, the controversy over \(c_M(h, e)\) as an adequate measure of confirmation is not the only issue here. Observe that if there exists a possible experiment outcome, \(e_i \in {\mathcal {E}}\) , that would falsify h (in the classical sense), then \(P(h \mid e_i) = 0\) . In this case, however, the expected degree of confirmation accruing to h with respect to experiment \({\mathcal {E}}\) is undefined. Hence, the expected disconfirmation of h , \(\langle d_{\mathcal {E}}(h) \rangle \) , is undefined as well. For our purposes, it is especially unfortunate that cases of outright falsification remain undefined on this approach. As long as there is no outright falsification possible, the measure is sensitive enough to pick out differences among experiments that allow for different degrees of strong disconfirmation, but it cannot be denied that in scientific practice the possibility of outright falsification is assumed. Even if we extend the measure to the extended reals, thus formally allowing it to take on the values \(\pm \infty \) , this measure does not adequately distinguish between combinations of theories and experiments that are more likely to result in an outright falsification. (We discuss a qualitative classification of this kind in Sect.  2.4 .)

Nevertheless, there is something intuitively compelling about Milne’s proposal. First, consider a hypothesis, h , and imagine an experiment, \({\mathcal {E}}\) , that leaves the posterior upon any outcome, \(e_i\) , equal to the prior. Since such an experiment does not provide any confirmation or disconfirmation, it should be assigned a minimal severity score. We can now consider the severity of any other experiment of the same hypothesis as an assessment of how much the posterior can vary across the possible measurement outcomes as compared to the reference of such a completely insensitive experiment (and we are grateful to Wayne Myrvold for suggesting this). Milne’s proposal to use \(\langle d_{\mathcal {E}}(h) \rangle \) is of this form, which you can see by substituting the values for an insensitive experiment (i.e., \(P(h \mid e_i)=P(h)\) ) into the equation: this yields \(\langle d_{\mathcal {E}}(h) \rangle = 0\) , which is indeed the minimal value that this measure can attain. While this viewpoint does not favour a particular confirmation measure, it does help to understand why, once an adequate measure of confirmation has been chosen, positing expected disconfirmation is a sensible way of assigning severity scores.

It is also interesting to consider the relation between the severity of an experiment and the boldness of a hypothesis. A first step in the analysis is to elucidate the severity of a given experiment across two hypotheses and to relate this notion to relative entropy (in terms of the Kullback–Leibler divergence). Consider two hypotheses, \(h_1\) and \(h_2\) , relative to an experiment \({\mathcal {E}}\) , and assume all relevant terms are defined. Now suppose that the expected degree of confirmation that \(h_1\) will accrue relative to \({\mathcal {E}}\) is higher than that of \(h_2\) . Or, in terms of disconfirmation:

Applying the definitions in Eqs. ( 4 ) and ( 5 ) and Bayes’ theorem yields:

We apply some precalculus to obtain:

We can rewrite this in terms of the cross-entropy between two distributions, \(H(\mu , \nu )\) , which is only defined relative to a set of possible outcomes; here we use \({\mathcal {E}}\) . Effectively, this requires us to restrict the algebra on which the probability functions are defined to a coarse-grained partition of the sample space. (We draw attention to this here because the interplay between various algebras and coarse-graining is a recurrent theme in our paper; see in particular the end of Sects.  2.2 and 4 .) So, assuming we restrict P , \(P(\cdot \mid h_1)\) , and \(P(\cdot \mid h_2)\) to \({\mathcal {E}}\) , we obtain:

This can be linked to elementary information theory as follows. Subtracting the Shannon entropy of \(P_{\mathcal {E}}\) , \(H(P_{\mathcal {E}})\) , from both sides of the inequality, we obtain an inequality between relative entropies (given by the Kullback–Leibler divergence, \(D_{KL}\) ):

This means that, relative to the prior \(P_{\mathcal {E}}\) , \(P_{\mathcal {E}}(\cdot \mid h_1)\) is more conservative than \(P_{\mathcal {E}}(\cdot \mid h_2)\) . In other words, \(h_2\) is more surprising or bolder than \(h_1\) . Initially, we observed that Milne’s preferred measure of confirmation, \(c_M\) , has the drawback of being undefined for hypotheses that allow for outright falsification. It occurs when the above Kullback–Leibler divergence goes to infinity, which can indeed be regarded as a sign of bold theories. This derivation suggests that bolder statements are easier to falsify. Or rather, they are expected to accrue less confirmation relative to a given experiment \({\mathcal {E}}\) .

Setting aside experiments that allow outright falsification of a hypothesis for a moment, it is also interesting to observe that maximally bold hypotheses, as measured by their Kullback–Leibler divergence relative to a prior and relative to an experiment, are in a sense very precise distributions, maximally concentrated on a single atomic possibility. This shows that boldness measured in this way (which is relative to an experiment) nicely aligns with the first dimension of riskiness: informativeness.

Appendix C: Relation between logical strength and truthlikeness

Proposition 1.

Suppose that \(\tau _1\) is true and that \(\tau _1 \models \tau _2\) . Then it holds that \({\textit{Content}}(\tau _1) \ge {\textit{Content}}(\tau _2)\) .

Conceptually, we obtain this result by noting that (1) truthlikeness should increase with logical strength and (2) the content of a true theory is just its truth content. Let us now do it more formally.

Given that \(\tau _1\) is true, so is \(\tau _2\) . This means that \(\tau _1 \models E_{t}(\tau _1)\) and \(E_{t}(\tau _1) \models \tau _1\) , and analogously for \(\tau _2\) . Furthermore, \(E_f(\tau _1)\) is empty; likewise for \(E_f(\tau _2)\) . This means that \(\tau _1 \ge _{SW} \tau _{2}\) —for the definition of \(\ge _{SW}\) consult Definition 4 of Schurz and Weingartner ( 2010 ). Theorem 1 from Schurz and Weingartner ( 2010 ) now yields that \({\textit{Tr}}(\tau _1) \ge {\textit{Tr}}(\tau _2)\) . But since \(\tau _1\) and \(\tau _2\) are true, \({\textit{Tr}}(\tau _1)\) and \({\textit{Tr}}(\tau _2)\) reduce to \({\textit{Content}}(\tau _1)\) and \({\textit{Content}}(\tau _2)\) , respectively. This concludes the proof. \(\square \)

This works nicely with one of the spearhead results of Cevolani and Schurz ( 2017 ). Verisimilitude contains two parts: likeness (qua truth) and content. If the likeness is perfect (it only pertains to true theories in the proposition above), then the only thing that can differ is the content.

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Vignero, L., Wenmackers, S. Degrees of riskiness, falsifiability, and truthlikeness. Synthese 199 , 11729–11764 (2021). https://doi.org/10.1007/s11229-021-03310-5

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DOI : https://doi.org/10.1007/s11229-021-03310-5

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