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2.6 Arguments and Rules of Inference

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• Page ID 23240

• Harris Kwong
• State University of New York at Fredonia via OpenSUNY

In this section we will look at how to test if an argument is valid.  This is a test for the structure of the argument.  A valid argument does not always mean you have a true conclusion; rather, the conclusion of a valid argument must be true if all the premises are true. We will also look at common valid arguments, known as Rules of Inference as well as common invalid arguments, known as Fallacies.

An  argument is a set of initial statements, called premises , followed by a conclusion.

An argument is  valid if and only if in every case where all the premises are true, the conclusion is true.  Otherwise, the argument is invalid .

Here is an example:

If I read my text, I will understand how to do my homework.

I understand how to do my homework.

Therefore, I read my text.

Our first premise: is If I read my text, then I understand how to do my homework . 

Our second premise is: I understand how to do my homework .

Our conclusion is  I read my text.

Let's use t means I read my text and u means I understand how to do my homework .

Symbolically, our argument is:

\(t \rightarrow u\)

\(\therefore t\)

Testing the validity of an argument by truth table.

We represent this argument by working out its premises and conclusion on a truth table:

Notice we repeat the column for \(u\) and the column for \(t\) because one is a premise and one is a conclusion.

Since a valid argument must have a true conclusion in all cases where the premises are true, we need to examine the rows where all premises are true.

Given a truth table representing an argument, the rows where all the premises are true are called the  critical rows.

We test an argument by considering all the critical rows.  If the conclusion is true in all critical rows, then the argument is valid.  This is another way of saying the conclusion of a valid argument must be true in every case where all the premises are true. 

Look for rows where all premises are true.

                                                                                                                 premise 1                                          premise 2                                  conclusion

We see that the 1st and 3rd rows are critical rows.  In the 1st row, the conclusion is true.  However, in the 3rd row, a critical row, the conclusion is false.

Thus this argument is ________________. 

Example \(\PageIndex{1}\)

Consider this argument.

If Pat goes to the store, Pat will buy $1,000,000 worth of food.

Pat goes to the store.

Therefore, Pat buys $1,000,000 worth of food.

This is a valid argument (you can test it on a truth table). 

However, even though Pat goes to the store, Pat does not buy $1,000,000 worth of food.  The conclusion is false.

How can the conclusion of a valid argument be false?

The validity of an argument refers to its structure.  Given a valid argument, the conclusion must be true if the premises are true. In this case the first premise is NOT true, and thus the conclusion does not need to be true.  The conclusion of a valid argument can be false if one or more of the premises is false.

Rules of Inference

A number of valid arguments are very common and are given names.  Know these four:

Modus Ponens

\(p \rightarrow q\)

\(\therefore q\)

Modus Tollens

\(\therefore \mbox{~}p\)

Elimination

\(p \vee q\)

Transitivity

\(q \rightarrow r\)

\(\therefore p \rightarrow r\)

As you think about the rules of inference above, they should make sense to you.  Furthermore, each one can be proved by a truth table.

If you see an argument in the form of a rule of inference, you know it's valid.

Example \(\PageIndex{2}\)

Explain why this argument is valid:

If I go to the movies, I will not do my homework.

I do my homework.

Therefore, I did not go to the movies.

This is valid by Modus Tollens.

Fallacies 

Fallacies are invalid arguments.  Know the names of these two common fallacies.

Converse Error

\(\therefore p\)

Inverse Error

\(\therefore \mbox{~}q\)

If you think about the converse and inverse (and that they do not have the same meaning as the original implication) you can see why these fallacies have these names. You can use a truth table to show these fallacies are arguments that are_________________.

Example \(\PageIndex{3}\)

Explain why this argument is valid or invalid:

I did not go to the movies.

Therefore, I did do my homework.

This is invalid; it is an inverse error.

Exercises \(\PageIndex{}\)  

Exercise \(\PageIndex{1}\)

True or False?

(a) Given a valid argument with true premises, the conclusion must be true.

(b) Given a valid argument with false premises, the conclusion must be false.

(c) Given an invalid argument, the conclusion must be false.

(a) true  (b) false  (c) false

Exercise \(\PageIndex{3}\)

Decide if the following arguments are valid or invalid.  State the Rule of Inference of fallacy used.

If it snows, then school is closed.

School is open.

Therefore it is not snowing.

My pet is a cat or my pet is a dog.

My pet is not a dog.

Therefore my pet is a cat.

If the movie is long, I will fall asleep.

I do fall asleep.

Therefore the movie was long.

(a) VALID, Modus Tollens  (b) VALID, Elimination (c) INVALID, Converse Error

Exercise \(\PageIndex{5}\)

Use a truth table to determine if this argument is valid or invalid. Include a clear explanation.

\(\overline{p} \vee (q \rightarrow r)\)

\(\overline{r}\)

\(\therefore \overline{p \wedge q} \)

As seen below, there are three critical rows, namely the 4th, 6th and 8th rows. We can see that in every case where all the premises are true, the conclusion is also true. Thus, this is a valid argument.

                                                                                                                                           premise 1              premise 2                                            conclusion

Exercise \(\PageIndex{7}\)

Use a truth table and an explanation to prove Modus Ponens is a valid form of an argument.

As seen below, the only critical row is the first row. We can see that in the one case that all the premises are true, the conclusion is also true. Thus, Modus Ponens has the form of a valid argument.

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What is the inverse statement to "if I do my homework, then it will snow"?10 POINTSS!!!!!

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What is the inverse of the statement if the television is on you will not do homework?

If I do not do my homework, then it will not snow.

Add your answer:

What is the inverse to the statement If I do my homework then it will snow?

I honestly mathematily don't really know

The second statement is the what of the first.?

Contrapositive.

What is the contrapositive of the statement If he does his homework then he gets good grades?

If he does not get good grades, he does not do his homework.

What is the inverse of the statement 'If I like math then I like science'?

The inverse of 'If I like math then I like science' is 'If I do not like math then I do not like science'.

What does inverse fraction mean?

The inverse of a fraction is simple the result of flipping it's denominator with its numerator. It is equivalent to the statement (x/y)^-1 = y/x

State the converse the contrapositive and the inverse what conditional statement If the television is on you will not do homework?

Which of the following is the inverse of the statement if you do your homework then it will snow.

If you do not do your homework then it will not snow.If I do not do my homework, then it will not snow.

Is the inverse of the statement If I do my homework then it will snow A. If I do not do my homework then it will snow. B. If I do not do my homework then it will not snow. C. If?

You have to do this assignment. We can't do it for you. You need to develop your critical thinking skills and how well you understood the lesson.

What is the inverse of this statement?

What isn't the inverse of this statement(?)

What is the inverse of the statement if you do your homework then it would snow?

There is no inverse as such. All the statement gives is the possibility of snow IF homework is done. It says nothing about if there being snow homework has, or has not, been done; nor does it say anything about the possibility of snow if homework is not done. This is the "implies" logical operator, the truth table for which is: A B → F F T F T T T F F T T T It is equivalent to: (not A) or B.

What is Inverse statement?

It is what you get in an inference, after negating both sides. That is, if you have a statement such as: if a then b the inverse of this statement is: if not a then not b Note that the inverse is NOT equivalent to the original statement.

The statement formed by negating both the hypothesis and conclusion of a conditional statement?

What is an inverse statement of if a triangle is an equilateral triangle.

"if a triangle is an equilateral triangle" is a conditional clause, it is not a statement. There cannot be an inverse statement.

What is an inverse statement?

Given a conditional statement of the form:If "hypothesis" then "conclusion",the inverse is:If "not hypothesis" then "not conclusion".

What is the inverse of the original statement?

if A then B (original) if not A then not B (inverse)

What is logically equivalent to the inverse of a conditional statement?

The conditional statement "If A then B" is equivalent to "Not B or A" So, the inverse of "If A then B" is the inverse of "Not B or A" which is "Not not B and not A", that is "B and not A",

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Watch CBS News

Why is looking at a solar eclipse dangerous without special glasses? Eye doctors explain.

By Sara Moniuszko

Edited By Allison Elyse Gualtieri

Updated on: April 8, 2024 / 8:54 AM EDT / CBS News

The solar eclipse will be visible for millions of Americans on April 8, 2024, making many excited to see it — but how you watch it matters, since it can be dangerous for your eyes. 

A  solar eclipse occurs when the moon passes between the sun and Earth, blocking the sun's light . When the moon blocks some of the sun, it's a partial solar eclipse, but when moon lines up with the sun, blocking all of its light, a total solar eclipse occurs,  NASA explains . Either way, you need eye protection when viewing.

"The solar eclipse will be beautiful, so I hope that everyone experiences it — but they need to experience it in the right way," said Dr. Jason P. Brinton, an ophthalmologist and medical director at Brinton Vision in St. Louis.

Here's what to know to stay safe.

Why is looking at a solar eclipse dangerous?

Looking at the sun — even when it's partially covered like during an eclipse — can cause eye damage.

There is no safe dose of solar ultraviolet rays or infrared radiation, said  Dr. Yehia Hashad , an ophthalmologist, retinal specialist and the chief medical officer at eye health company Bausch + Lomb.

"A very small dose could cause harm to some people," he said. "That's why we say the partial eclipse could also be damaging. And that's why we protect our eyes with the partial as well as with the full sun."

Some say that during a total eclipse, it's safe to view the brief period time when the moon completely blocks the sun without eye protection. But experts warn against it. 

"Totality of the eclipse lasts only about 1 to 3 minutes based on geographic location, and bright sunlight suddenly can appear as the moon continues to move," notes an eclipse viewing guide published in JAMA , adding, "even a few seconds of viewing the sun during an eclipse" can temporarily or permanently damage your vision. 

Do I need special glasses for eclipse viewing?

Yes.  Eclipse glasses are needed to protect your eyes if you want to look at the eclipse.

Regular sunglasses aren't protective enough for eclipse viewing — even if you stack more than one. 

"There's no amount of sunglasses that people can put on that will make up for the filtering that the ISO standard filters and the eclipse glasses provide," Brinton said.

You also shouldn't look at the eclipse through a camera lens, phone, binoculars or telescope, according to NASA, even while wearing eclipse glasses. The solar rays can burn through the lens and cause serious eye injury.

Eclipse glasses must comply with the  ISO 12312-2 international safety standard , according to NASA, and should have an "ISO" label printed on them to show they comply. The American Astronomical Society  has a list  of approved solar viewers.

Can't find these, or they're sold out near you? You can also  make homemade viewers ,   which allow you to observe the eclipse indirectly — just don't accidentally look at the sun while using one.

How to keep kids safe during the solar eclipse

Since this eclipse is expected to occur around the time of dismissal for many schools across the country, it may be tempting for students to view it without the proper safety precautions while getting to and from their buses. That's why some school districts are  canceling classes early so kids can enjoy the event safely with their families.

Dr. Avnish Deobhakta, vitreoretinal surgeon at New York Eye and Ear Infirmary at Mount Sinai, said parents should also be careful because it can be difficult for children to listen or keep solar eclipse glasses on. 

"You want to actually, in my opinion, kind of avoid them even looking at the eclipse, if possible," he said. "Never look directly at the sun, always wear the right eclipse sunglasses if you are going to look at the sun and make sure that those are coming from a reliable source."

Brinton recommends everyone starts their eclipse "viewing" early, by looking at professional photos and videos of an eclipse online or visiting a local planetarium. 

That way, you "have an idea of what to expect," he said. 

He also recommends the foundation  Prevent Blindness , which has resources for families about eclipse safety.

What happens if you look at a solar eclipse without eclipse glasses?

While your eyes likely won't hurt in the moment if you look at the eclipse without protection, due to lowered brightness and where damage occurs in the eye, beware: The rays can still cause damage .

The harm may not be apparent immediately. Sometimes trouble starts to appear one to a few days following the event. It could affect just one or both eyes.

And while some will regain normal visual function, sometimes the damage is permanent. 

"Often there will be some recovery of the vision in the first few months after it, but sometimes there is no recovery and sometimes there's a degree to which it is permanent," Brinton said. 

How long do you have to look at the eclipse to damage your eyes?

Any amount of time looking at the eclipse without protection is too long, experts say. 

"If someone briefly looks at the eclipse, if it's extremely brief, in some cases there won't be damage. But damage can happen even within a fraction of a second in some cases," Brinton said. He said he's had patients who have suffered from solar retinopathy, the official name for the condition.

Deobhakta treated a patient who watched the 2017 solar eclipse for 20 seconds without proper eye protection. She now has permanent damage in the shape of a crescent that interferes with her vision. 

"The crescent that is burned into the retina, the patient sees as black in her visual field," he said. "The visual deficit that she has will never go away."

How to know if you've damaged your eyes from looking at the eclipse

Signs and symptoms of eye damage following an eclipse viewing include headaches, blurred vision, dark spots, changes to how you see color, lines and shapes. 

Unfortunately, there isn't a treatment for solar retinopathy.

"Seeing an eye care professional to solidify the diagnosis and for education I think is reasonable," Brinton said, but added, "right now there is nothing that we do for this. Just wait and give it time and the body does tend to heal up a measure of it."

Sara Moniuszko is a health and lifestyle reporter at CBSNews.com. Previously, she wrote for USA Today, where she was selected to help launch the newspaper's wellness vertical. She now covers breaking and trending news for CBS News' HealthWatch.

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