Validity

In general, there are well defined rules that define when premises support conclusions.  The strongest form of support is known as deductive validity.   Some reasons or premises can be said to support their conclusion, while others do not offer good support.  The notion of "support" here refers to the logical connection between the reasons and the conclusion.  But "support" is vague.  Some arguments, in virtue of their logical structure, have the feature of being deductively valid.  

Consider these arguments:  

1.  All aliens are green.  

2. Gronk is an alien.  

_______________________

3.  Therefore, Gronk is green.  

1.  If you are a billionaire, then you are rich.

2. McCormick is a billionaire.

________________________

3.  Therefore, McCormick is rich.  

1.  No mammals are reptiles.

2. All snakes are reptiles.

_______________________

3. Therefore, no snakes are mammals.

It should be intuitively clear that all of these arguments have an important property of support between the premises and the conclusion.  The conclusion follows from the premises in a significant way.  By contrast, these arguments do not have the same feature:

1. If you win the lottery, then you will be rich.  

2. Elon Musk is rich.

________________________

3.  Therefore, Elon Musk won the lottery.  

1. No cats are invertebrates.  

2. No mammals are invertebrates. 

_________________________

3.  Therefore, no cats are mammals.  

What is the feature that the arguments in the first group have that the arguments in the second group do not?  Is it that they all have true premises and true conclusions?  No.  McCormick is not a billionaire.  

Is it that the premises support the conclusions?  Yes, but what does support mean here?  What exactly is the relationship between the premises and the conclusion?  

Validity defined:  An argument is valid when it has this property:  if the premises were true, then the conclusion would be guaranteed to be true.  In all of the arguments in the first group, it is impossible for the premises to be true while the conclusion is false.  The support relationship between them means that if the premises are true, then they would guarantee the truth of the conclusion.  That is the definition of validity. 

A valid argument is one where the premises, if they were true, would guarantee the truth of the conclusion. 

Put otherwise, and in terms of what we learned about logical consistency and logical impossibility in the last chapter, a valid argument is one where the premises’ being true and the conclusion being false would generate a logical contradiction.  

A Test for Validity

Another way to apply this definition is to think of it as a test.  Consider this argument:

1. All of the cars in the lot have CA plates.  

2. Vaughn's car does not have CA plates.

      ________________________________

3. Therefore, Vaughn's car is not in the lot.  

Test it for validity:

A) Imagine or pretend that the premises are true.  

B) Ask this question: is it possible for the conclusion to be false in some situation while we are imagining that the premises are true? 

C)  If we can imagine the premises being true while the conclusion is false, then it is invalid.  

D)  If we cannot imagine the premises being true while the conclusion is false, then it is valid.  

If you can imagine a situation where the premises are true but the conclusion is false, then it is invalid.  If you cannot imagine a situation where the premises are true and the conclusion is false, then the argument is valid.  So in the case above, we assume or imagine that according to premise 1, every single car that is in the lot has CA plates.  There are no non-CA plates on any cars in the lot.  Then we imagine that Vaughn's car does not have CA plates.  Now we ask the question, is it possible that Vaughn's car could be in the lot?  It cannot if all of the cars in the lot have CA plates.  So we cannot imagine a situation where the premises are true and the conclusion is false.  So this argument is valid.  The premises, if true, guarantee the truth of the conclusion.  

Invalid arguments are ones where there could be circumstances where the premises are true but the conclusion is false.  We can imagine situations where the premises are true but the conclusion is false.  Or, invalid arguments are ones where the truth of the premises does not guarantee the truth of the conclusion.  

In this case:

1. If you win the lottery, then you will be rich.  

2. Elon Musk is rich.

     ________________________

3.  Therefore, Elon Musk won the lottery.  

The premises are true–if you win the lottery then you will be rich, and Elon Musk is rich.  But the conclusion is false:  Elon Musk became rich from owning tech companies, not by winning the lottery.  

Validity concerns logical form, not content

We should be able to recognize some patterns across different arguments that have very different subjects.  These arguments are both valid:

1.  If you win the lottery, then you will be rich.  

2. McCormick won the lottery.

_____________________________

3.  Therefore, McCormick is rich.  

1.  If your headlights are out, then your car is not legal to drive.  

2. McCormick’s headlights are out.

_____________________________________

3.  Therefore, McCormick’s car is not legal to drive.  

They both have the same logical pattern or structure, even though they are about very different topics.  Formal sentential logic lets us represent this valid argument pattern, and it has a name:

1. P → Q

2. P

________________

3. ∴   Q Modus Ponens (MP)

We will use “P” for “You win the lottery,” and Q for “You will be rich.”  And an arrow → to represent the “if … then” logical connective.   And we will use ∴ to stand for “therefore.”  And we can recognize generally that this inference pattern, whenever it occurs and no matter what the content of the argument is about, is always valid.  A conditional is a sentence of the form “If P, then Q,” where P is the antecedent, and Q is the consequent.  There are four basic conditional inferences, two of them valid and two invalid, and they all have names.  Here they are, with clear English examples, for quick reference (assume for the sake of these examples that there is only one city named Sacramento).  We will use “~” for “not” when we are writing these out in logical form:  

These English examples should make it clear why in the last two cases these argument patterns are invalid.  In the Affirming the Consequent example, it is possible that 1.  If you are in Sacramento, then you are in CA, and that 2.  McCormick is in CA are both true, but the conclusion, 3. McCormick is in Sacramento is false.  It could be true that McCormick is in CA because he is in Los Angeles, so the premises would not guarantee the truth of the conclusion.  Likewise in the Denying the Antecedent example, 2. Could be true because McCormick is in Los Angeles, so the premises would not guarantee that he is not in CA.  In both of these cases, we could imagine that the premises could be true and the conclusion could be false, thus they fail the test for validity.  Notice that these two argument patterns are invalid even when the premises do not make it so obvious as they do in the geography examples.  

1.  If you are against capital punishment, then you are against abortion.  

2. McCormick is against abortion. 

      _____________________________

3.  Therefore, McCormick is against abortion.  

This argument is another invalid example of Affirming the Consequent.  Premise one says that if you are against capital punishment, then you are against abortion, but it doesn’t tell us about other grounds or reasons that might lead to being against abortion.  It does not tell us that if you are against abortion then you are against capital punishment.  And it does not assert that all and only people who are against capital punishment are against abortion.  So merely knowing that McCormick is against abortion does not imply that he is against capital punishment.  He might be, but this argument does not guarantee that conclusion, so it is invalid.  As it is in the Elon Musk example above.  Every argument of the logical form:

1. P → Q

2. Q

      _________

3. ∴  P 

Is invalid.  Going forward, we will need to recognize these four patterns and know their names.  

Here are some more valid and invalid argument patterns that we will need to recognize: 

Advanced Validity, Longer Arguments

There are several other basic logical inferences that we will encounter.  Consider this argument:

1. Mitch either has a cold or he has Covid.  

2. Mitch doesn’t have Covid.

____________________________

3.  Therefore, Mitch has a cold.    

This is an Argument by Elimination or Disjunctive Syllogism (DS)  We employ and encounter this sort of reasoning frequently.  We should also notice that this sort of reasoning can be extended and that arguments need not be confined to three lines:

1. Either McCormick’s battery is dead, his alternator is broken, or his starter is malfunctioning.  

2. His alternator is not broken.  

3. His starter is not malfunctioning.  

     _________________________________

4.  Therefore, McCormick’s batter is dead.  

We should recognize the formal and abstract logical form of these arguments.  We will use “v” for “and” when we write out the logical form:  

1.  A v B

2. ~A

     _____________

3. ∴ ~B DS

Here’s another valid inference rule that we should be familiar with and that we will encounter frequently.  Hypothetical Syllogism is an argument that conjoins two or more conditional premises; we can think of the “If ….then” logical inference as being transitive:

1. If McCormick is in Hollywood, then he is in California.  

2. If he is in California, then he is in the United States.  

     ______________________________________________

3.  Therefore, if McCormick is in Hollywood, then he is in the United States.  

Notice that this argument is hypothetical.  No premise actually asserts that McCormick is in Hollywood or in California.  The premises are about what would be true if the antecedent was true.  The logical form of this argument is:

1.  P→ Q

2. Q → R

     ____________

3. ∴  P → R

And like DS, these inferences can be continued for any number of conditionals.  We can think of HS as transitive reasoning like dominoes falling:

1. If I stay up late tonight, then I’ll be tired tomorrow.

2. If I’m tired tomorrow, then I’ll skip the gym.

3. If I skip the gym, then I’ll feel worse by the end of the week.

     ________________________________________________________

4. Therefore, if I stay up late tonight, I’ll feel worse by the end of the week.  

Longer arguments can be deductively valid and they can contain multiple kinds of inferences.  Consider this argument:

1. McCormick drove home on Stockton, 65th, or Folsom.  

2. If he drove home on Stockton, then he was late.  

3. If he drove home on Folsom, then he stopped at Trader Joe’s.  

4. He wasn’t late.  

5. He didn’t stop at Trader Joe’s.  

6. If he drove home on 65th, then he suffered through a lot of traffic.  

     ____________________________________________

7.  Therefore, he suffered through a lot of traffic.  

Although it might be harder to see immediately, this argument is deductively valid.  That is, if all of the premises were true, then the conclusion would be guaranteed to be true.  It has the logical form of:

1.  S v 65th v F

2. S → L

3. F → TJ

4. ~L

5. ~ TJ

6. 65th → T

     ____________

7. ∴  T

These versions of the argument leave some of the valid inferences implicit.  Let’s make them explicit by expanding all of the reasoning and identifying the different valid inference rules being employed.  Also notice that we are adding lines for the implicit references and we are using sub-arguments with subconclusions in service of the overall argument and conclusion.  

1. McCormick drove home on Stockton, 65th, or Folsom.  

2. If he drove home on Stockton, then he was late.  

3. He wasn’t late.

     ____________________

4. Therefore, he did not drive home on Stockton.  (this follows validly by 2, 3, and modus tollens.)

8. If he drove home on Folsom, then he stopped at Trader Joe’s.  

9. He didn’t stop at Trader Joe’s.

     _______________________________

10.  Therefore, he didn’t drive home on Folsom.     (this follows by 8, 9, and modus tollens.)

11.  If he drove home on 65th, then he suffered through a lot of traffic.  

12.  He drove home on 65th     (this follows validly by 1, 4, 10 and disjunctive syllogism)

     ____________________________________________

13.  Therefore, he suffered through a lot of traffic.    (this follows validly by 11, 12, and modus ponens)

You might have been able to see intuitively that the first 7 line argument in English was valid, but now with some formal reconstruction and by bringing out all of the inferences and sub-arguments it is much clearer.  For now, there are several points to remember here:

1. Arguments can have more than one inference and many lines.  

2. Arguments that are made up of multiple inferences where each of them is valid are also valid overall.  

3. Validity concerns the abstract form of reasoning and doesn’t address the actual truth of the premises.  This argument is valid whether McCormick actually stopped at Trader Joe’s or he didn’t.  A valid argument is one where the premises, if they were true, would guarantee the truth of the conclusion.  

Invalid Arguments

Let’s consider some more examples of invalid reasoning to get clearer on the concept.  Consider:  

1. If it rains overnight, then the ground will be wet in the morning.

2. If the ground is wet in the morning, then the soccer game will be canceled.

3. If the sprinklers run overnight, then the ground will be wet in the morning.

4. The soccer game was canceled.

     ______________________________________

5.  Therefore, the sprinklers ran overnight.

You might be able to see intuitively that this argument is invalid.  In a valid argument, the premises, if they were true, would guarantee the truth of the conclusion.  If 1-4 were all true, would they guarantee that the sprinklers ran overnight?  Would it be logically impossible that the sprinklers did not run overnight?  Imagine that the premises were true; picture the world where rain makes the ground wet, and sprinklers could make the ground wet overnight.  And if the ground is wet, then the soccer game is cancelled.  So you got news that the soccer game was cancelled, could you infer with certainty from what you know that it must have been the sprinklers that caused the cancellation?  No, you can’t know that for certain, so this argument is invalid.  There are two ways for the ground to get wet, resulting in the cancelled game, so if the game is cancelled, you can’t be certain from that alone what made the ground wet.  An important lesson about validity to draw here is that without a guarantee, there can be no validity.  

Consider this case:

1. If the smoke alarm is broken, then it will not beep.  

2. The smoke alarm did not beep.  

     _______________________________

3. Therefore, the smoke alarm is broken.

We can see in this case that the conclusion is not guaranteed by the premises.  A broken smoke alarm does have the outcome of no sound.  But not hearing a sound from the smoke alarm doesn’t necessarily imply or guarantee that it’s broken.  We’ve seen this fallacy before: Affirming the Consequent.  And the lesson to take away more formally is that  P→ Q does not imply Q → P.  “If you won the lottery, then you are rich” does not imply logically that “If you are rich, then you won the lottery.”  If your phone battery is dead, the phone won’t turn on ≠ If the phone won’t turn on, the battery is dead.  If it rained last night, then the grass will be wet ≠ if the grass is wet, then it rained last night.  We are all prone to conflate P→ Q with Q → P by association.  

We are also prone to object to or criticize P→ Q claims with examples of some other antecedent bringing about the consequent Q.  Smith says, “If you get a college degree, then you’ll have more job opportunities.”  And Jones objects by saying, “That’s not true, my cousin went to trade school and he’s got lots of job offers.”  The objection is misguided; it misinterprets the original claim as “The only way to get job opportunities is with a college degree.”  In the language of our last chapter, a good objection would point out a logically inconsistency or contradiction.  But “If you get a college degree, then you’ll have more job opportunities,” and a cousin who went to trade school and got a lot of job opportunities are logically consistent because  P→ Q does not mean or imply Q → P. 

In general, being careful with our claims, clarifying what we mean and what we don’t mean, and sharpening up the definitions of our concepts helps us be much better critical thinkers.  In formal logic, we have a different locution for a stronger claim:  P ↔ Q means “P if and only if Q, as in:  

A figure is a triangle if and only if it has exactly three straight sides.

A person is a bachelor if and only if he is an unmarried, adult, human male.  

A student passes this course if and only if they earn at least a 70%

Keeping conditional reasoning clear and separate from biconditional reasoning, avoiding the affirming the consequent fallacy, and not confusing P→ Q with Q → P are crucial for being a successful critical thinker.  

There are countless examples of invalid arguments and reasoning that become more apparent as we fully grasp the concept of validity.  Consider:  

1. If it’s a holiday, then the office will be closed.

2. It’s not a holiday.

     ___________________________

3. Therefore, the office is open.

Clearly, the conclusion is not guaranteed by the premises.  Even if they were true, the conclusion could be false; the office could be closed because of a power outage, or some other reason.  This mistake is Denying the Antecedent from our chart above, it’s invalid, and we should be on the watch for it going forward.  We will revisit the full list of the biases, fallacies, and mistakes and reasoning for the course in chapter 11, but we will begin using that chapter for reference, for definitions, and for examples now:  11 Biases, Fallacies, and Mistakes in Reasoning

ChatGPT on Validity

In order to succeed in this course, you will need to master the concept of validity as it has been presented here, and you’ll need to demonstrate that mastery in class assessments.  Beyond reading, studying, and taking careful notes on this chapter, asking questions in class, studying the YouTube lecture, and studying our class discussion slides, the way to improve your understanding of validity is to go through quiz questions with the custom ChatGPT agent for the class.  Go here:  

 McCormick Critical Thinking AI  and give it prompts like these:

“Quiz me, one at a time, with examples of valid and invalid arguments”

“Test me on valid and invalid arguments until I can reliable (80+%) identify them.” 

“Prompt me to give examples of invalid arguments that commit the fallacies outlined in our text.”  

“Quiz me with longer valid and invalid arguments”

And so on.  Ask the AI for clarifications, explanations, and examples.  Have it explain things you’re not clear on.  

Valid Argument Patterns

There are infinitely many valid argument patterns.  Philosophers and logicians have been studying, formalizing, and identifying them for 2,500 years.  For the purposes of this class, we adopt this relatively short list of valid argument patterns:

Summary  

What is deductive reasoning?  Deductive arguments are valid arguments.  They are arguments where the logical structure seeks to guarantee the conclusion or make it certain.  By contrast, inductive reasoning is typically probabilistic; it concerns the class of arguments where the evidence or the reasons given cannot guarantee the conclusion but they could make the conclusion probably true.  We will define deductive reasoning as valid reasoning.  

Valid arguments are arguments in which the premises, if they were true, would guarantee the truth of the conclusion.  

We’ve also introduced a number of important concepts in this chapter that you will be responsible for mastering in the course.

• Logical consistency 

• Epistemic Support

• Arguments: premises, conclusions, standard form

• Testing for Validity

• Invalid arguments

• Validity concerns logical form, not content

• Modus ponens (MP)

• Modus tollens (MT)

• Fallacy:  Affirming the Consequent

• Fallacy:  Denying the Antecedent

• Universal Syllogism-Affirmative (UA)

• UN Universal Syllogism-Negation (UN)

• Disjunctive Syllogism (DS) 

• Hypothetical Syllogism (HS)