144. Logic, Argument Structure, and Syllogistic Form

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Lecture Notes

Main Topics #

Two Ways an Argument Can Be Defective #

Form Error (Invalid Reasoning)

• The logical form is bad, meaning the conclusion does not follow necessarily from the premises
• Example: “If you’re the man who robbed the bank, then you are a man. You are a man. Therefore, you’re the man who robbed the bank.”
• This violates the rule against affirming the consequent
• The premises may be true, but the conclusion is not warranted

Matter Error (False Premises)

• The premises are false or one premise is false
• The logical form may be correct, but the argument rests on faulty information
• Example: Stating there are three students in one group and three in another, then multiplying to get 60 instead of recognizing a different number

Both Defects Can Occur Together

• An argument can have both invalid form and false premises
• Only one defect is needed to render an argument bad
• Both correct form AND true premises are necessary for a good argument

The Two Valid Forms of Conditional Syllogisms #

Modus Ponens (Affirmation of the Antecedent)

• “If A is so, then B is so. A is so. Therefore, B is so.”
• This form is obvious and directly valid
• When the antecedent is affirmed, the consequent must be affirmed

Modus Tollens (Denial of the Consequent)

• “If A is so, then B is so. B is not so. Therefore, A is not so.”
• This form is less obvious but can be proven through the first form
• Works by showing that affirming A while denying B would contradict the conditional statement
• The three statements (conditional, A, and not-B) are incompatible

The Two Invalid Forms #

Denying the Antecedent

• “If A is so, then B is so. A is not so. Therefore, B is not so.”
• Invalid because B could result from other causes besides A
• Example: “If Rosie is a cat, then Rosie is an animal. Rosie is not a cat. Therefore, Rosie is not an animal.”
• Can be disproven by example

Affirming the Consequent

• “If A is so, then B is so. B is so. Therefore, A is so.”
• Invalid because B may have multiple possible causes
• Example: “If the sun goes around the Earth, then there will be day and night. There is day and night. Therefore, the sun goes around the Earth.”
• The Earth rotating on its axis is an alternative explanation

Multiple Causes and Invalid Inference #

• When an effect B can result from multiple causes (A, C, D, etc.), one cannot infer which cause produced B
• Example: “If Burkwist is visiting grandchildren, he’ll be absent. If Burkwist is sick, he’ll be absent. Burkwist is absent. Therefore, he’s sick.”
• This commits the fallacy of denying the antecedent or affirming the consequent
• Berquist warns students this is “wishful thinking, not logical thinking”

Proof by Example vs. Disproof by Example #

• One example can disprove a universal claim (show it’s not always true, hence not necessarily true)
  • Example: One black man disproves “man is always white”
  • One counterexample to a logical form shows it is not necessarily valid
• Infinite examples cannot prove a universal claim
  • Countless odd numbers do not prove “every number is odd”
  • Many white men do not prove “man is always white”
• This asymmetry is crucial in logic: invalidity can be demonstrated by counterexample, but validity cannot

Aristotle’s Two Analytics #

Prior Analytics

• Examines whether the conclusion follows necessarily from the premises
• Focuses on logical form and valid inference

Posterior Analytics

• Examines whether the premises are necessarily true
• Focuses on the truth content and status of premises

Demonstration (Aristotelian Syllogism)

• A syllogism whose premises are necessarily true
• The conclusion therefore becomes necessarily true
• Requires both valid form (Prior Analytics) and true premises (Posterior Analytics)

Dialectical Syllogisms vs. Demonstrations #

• Demonstration: Premises are necessarily true, conclusion is necessarily true
• Dialectical syllogism: Premises are only probable opinions, conclusion is only probable
• Induction and Examples: Conclusion does not follow necessarily, even if premises are true
  • Example: “The last Honda I bought lasted ten years. The next Honda will last ten years.”
  • The conclusion may be reasonable but does not follow necessarily

Application to Argument Analysis #

• In evaluating any argument, check two independent factors:
  1. Is the logical form valid? (Can the conclusion fail to follow even if premises were true?)
  2. Are the premises true? (Is the information content accurate?)
• Deficiency in either factor makes the argument unsound
• Like a checkbook error: the balance can be wrong either from arithmetic error or from writing incorrect amounts

Key Arguments #

The Conditional Form Analysis #

• Starting point: “If A is so, then B is so” establishes that B follows necessarily from A
• Modus Ponens: Affirm the antecedent → must affirm the consequent (obvious)
• Modus Tollens: Deny the consequent → must deny the antecedent (proven through Modus Ponens)
  • If A were true, then B would be true (by the conditional)
  • But B is not true (given)
  • Therefore A cannot be true (otherwise B would have to be true)

Why Other Forms Fail #

• Affirming the Consequent: B does not determine which of multiple possible causes produced it
• Denying the Antecedent: A may not be the only cause of B; B’s negation doesn’t negate all possible causes

The Necessity Connection #

• Valid form ensures that if the premises are true, the conclusion must be true
• But this conditional relationship is not enough; the premises themselves must be true
• Truth of premises + valid form = certain knowledge

Important Definitions #

Syllogism: An argument in which some statements are laid down (premises) and another follows necessarily from those laid down (conclusion)

Demonstration (Demonstratio): A syllogism whose premises are necessarily true, yielding necessarily true conclusions

Dialectical Syllogism: A syllogism whose premises are only probable opinions, yielding only probable conclusions

Modus Ponens: The valid form of affirming the antecedent in a conditional (if-then) argument

Modus Tollens: The valid form of denying the consequent in a conditional argument

Induction/Enumeratio/Example: Types of argument where the conclusion does not follow necessarily, even when premises are true

Examples & Illustrations #

The Bank Robber Fallacy #

• “If you’re the man who robbed the bank, then you are a man.”
• “You are a man.”
• “Therefore, you’re the man who robbed the bank.”
• Shows affirming the consequent: many people are men; being a man doesn’t identify the robber

The Checkbook Analogy #

• A checkbook balance can be wrong in two independent ways:
  1. Arithmetic error: adding/multiplying/subtracting incorrectly
  2. Wrong numbers: writing the wrong amount
• One error makes the result untrustworthy; both errors certainly do
• Parallel to logical errors: bad form, bad matter, or both

The Book Purchase Error #

• “There are four people here. A book costs $20. Four times twenty is $100.”
• This is arithmetically correct but the conclusion is false because the number of people or the price was wrong
• Contrast: “There are three students here, three students there. A book is $20. Three times 20 is 60.”
• Here the arithmetic is correct, but 60 is the wrong answer because the premise “four students” was wrong

The Absence of Burkwist #

• Multiple causes could explain absence: visiting grandchildren, being sick, forgetting, or even dying
• Observing his absence does not determine which cause operated
• Inferring sickness from absence commits the fallacy of multiple causes

The Celestial Motion Example #

• “If the sun goes around the Earth, then there will be day and night.”
• “There is day and night.”
• “Therefore, the sun goes around the Earth.”
• This is invalid because the Earth’s rotation on its axis also produces day and night
• The ancients committed this fallacy; the effect (day and night) has multiple causes

The Political Voting Example #

• “If I’m opposed to Wesley Clark on abortion, I’ll vote against him.”
• One may oppose a candidate for multiple reasons (other issues, character, etc.)
• Observing opposition to the candidate doesn’t prove which specific reason caused the opposition

Rosie the Dog #

• “If Rosie is a cat, then Rosie is an animal. Rosie is not a cat. Therefore, Rosie is not an animal.”
• Invalid: Rosie could be an animal (a dog) without being a cat
• Demonstrates denying the antecedent fallacy
• “If Rosie is a dog, then Rosie is an animal. Rosie is an animal. Therefore, Rosie is a dog.”
• Invalid: Rosie is an animal but may be a cat or horse, not necessarily a dog
• Demonstrates affirming the consequent fallacy

Notable Quotes #

“If he reasons this way, his matter is good, his premises are true, but the conclusion doesn’t follow… If he reasons this way, the conclusion follows, but the premises are…not both true.”

“For an argument to be good, both the premises have to be true, and the conclusion has to follow necessarily.”

“You can’t reason from the affirmation of the consequent to the affirmation of the antecedent. You can reason from the affirmation of the antecedent to the affirmation of the consequent.”

“That’s wishful thinking. It’s not logical thinking.”

“You can’t prove a form by examples. You can disprove it by examples.”

“If it’s necessarily so, then it’s always so. It’s not always so; therefore, it’s not necessarily so.”

“One example is enough to show that something is not always so. Sure. So if I say man is always white, how many black men do you need to disprove me? Just one, right? But how many white men would prove that man is always white? The world went on forever…”

Questions Addressed #

How can we distinguish valid from invalid argument forms? #

• Through understanding the two valid conditional forms (Modus Ponens and Modus Tollens) and recognizing when other forms are used
• Invalid forms fail because effects can have multiple causes or consequences can follow from multiple sources

Why is one example sufficient to disprove a form but not to prove it? #

• A universal claim (“always so”) is falsified by even one counterexample
• But proving a universal claim requires exhaustive verification, which is impossible
• Therefore, one counterexample suffices for disproof; infinite confirmations cannot constitute proof

How do the Prior and Posterior Analytics relate? #

• Prior Analytics ensures form is valid (conclusion follows from premises)
• Posterior Analytics ensures premises are true (premises are necessarily true)
• Both must be satisfied for a demonstration

What makes Modus Tollens less obvious than Modus Ponens? #

• Modus Ponens is direct: affirm the cause, affirm the effect
• Modus Tollens requires understanding that three statements (the conditional, the antecedent, and the negation of the consequent) are incompatible
• It must be proven by reducing it to Modus Ponens