37. The Four Forms of Hypothetical Syllogism and Scientific Confirmation

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Lecture Notes

Main Topics #

The Four Forms of If-Then (Hypothetical) Syllogism #

All four forms share the structure: If A is so, then B is so (the hypothetical premise), followed by a statement about either the antecedent or consequent:

1. Affirming the Antecedent: A is so; therefore B is so. ✓ VALID
2. Denying the Consequent: B is not so; therefore A is not so. ✓ VALID
3. Denying the Antecedent: A is not so; therefore B is not so. ✗ INVALID
4. Affirming the Consequent: B is so; therefore A is so. ✗ INVALID

The Problem of False Imagination #

The human mind naturally assumes that invalid forms are valid, particularly because Forms 1 and 4 appear structurally parallel, as do Forms 2 and 3. This “false imagination” is the fundamental cause of logical error. Berquist notes from teaching experience that even when students understand the distinctions intellectually, they frequently make mistakes on exams because the mind is prone to this deception.

Why Only Two Forms Are Syllogisms #

Form 1 (Affirming the Antecedent): If A is so, then B is so. A is so. Therefore B is so.

• This is self-evident and obvious
• It directly follows from the logical structure of the conditional

Form 2 (Denying the Consequent): If A is so, then B is so. B is not so. Therefore A is not so.

• Not immediately obvious but can be proven through Form 1
• Proof: Either A is so or A is not so. If A were so, then B would have to be so (by Form 1). But B is not so. Therefore A cannot be so.
• Uses either-or reasoning to establish its validity

Form 3 (Denying the Antecedent): If A is so, then B is so. A is not so. Therefore B is not so.

• INVALID: A being false tells us nothing about whether B is true or false
• A true antecedent is sufficient for B, but a false antecedent does not determine B’s truth value

Form 4 (Affirming the Consequent): If A is so, then B is so. B is so. Therefore A is so.

• INVALID: B being true does not prove A caused it
• B could follow from some other source entirely

Key Arguments #

The Berkowitz Example (Form 4 - Invalid) #

“If Berkowitz dropped dead last night, he would be absent from class today. He is absent from class today, therefore he dropped dead.”

This is wishful thinking, not logical thinking. His absence could result from:

• His car breaking down
• Illness
• Any number of other causes

The same effect (absence) can follow from multiple different antecedents. Therefore, the fact that the consequence occurs does not prove which antecedent caused it.

Scientific Hypothesis Confirmation and Form 4 #

Einstein recognized a critical problem: scientific method typically confirms hypotheses using Form 4 (affirming the consequent), which is NOT a valid syllogism.

How scientific confirmation works:

• “If my hypothesis is correct, then such-and-such will occur” (conditional premise)
• “Such-and-such does occur” (consequent is true)
• Therefore, “My hypothesis is correct” (affirmed antecedent)

The logical weakness:

• The same predicted effect could follow from a different hypothesis
• Example: Newtonian physics predicted planetary motions with remarkable success, yet Einstein showed a different theory (relativity) could predict the same things plus additional phenomena
• This demonstrates that confirmed predictions do not prove the hypothesis is correct

Confirmation vs. Rejection:

• Confirmation (Form 4): Weakly supports the hypothesis but does not prove it
• Rejection (Form 2): Rigorously disproves the hypothesis
  • “If hypothesis H is true, then observation O will occur. Observation O does not occur. Therefore H is not true.”
  • This is a valid syllogism with necessary force

Why Probability ≠ Necessity #

The more predictions that come true, the greater the probability that a hypothesis is correct. However:

• Probability is not the same as necessity
• No matter how many times a hypothesis is tested successfully, it can always be tested again
• Induction (more frogs with three-chambered hearts = more probable that all do) never proves necessity
• One counterexample is sufficient to disprove necessity, but infinity of confirmations cannot prove it

Rejection as More Rigorous #

Form 2 (Denying the Consequent) provides logical necessity:

• If hypothesis predicts an eclipse at 10 a.m. and no eclipse occurs, the hypothesis is definitively false
• This is the form: “If H then P. Not-P. Therefore not-H.”
• It has the force of a syllogism with necessary conclusion

Form 4 (Affirming the Consequent) provides only probability:

• If hypothesis predicts phenomenon X and X occurs, the hypothesis might be correct
• But X might result from some other cause
• This can never establish certainty

Important Definitions #

Antecedent (ἡγούμενος / antecedens) #

The first part of a conditional statement (the “if” clause). It is what comes before logically and serves as the sufficient condition.

Consequent (λῆγον / consequens) #

The second part of a conditional statement (the “then” clause). It is what follows logically from the antecedent as a necessary result.

False Imagination (falsae imaginationes) #

The mind’s natural tendency to imagine logical connections that do not hold. Specifically, the tendency to assume invalid forms of reasoning are valid based on superficial structural similarity.

Necessity (ἀνάγκη / necessitas) #

Something that must always be true. A claim is necessary if it cannot be otherwise. To show something is not necessary requires only one counterexample; to show something is necessary requires proof that it is always true.

Examples & Illustrations #

The Dog and Animal (Form 3 - Denying Antecedent) #

“If I’m a dog, I’m an animal. I’m not a dog, therefore I’m not an animal.”

This is invalid. Being a man, I am still an animal even though I am not a dog. The false antecedent does not determine the truth value of the consequent.

Euclid’s Sixth Theorem (Either-Or with If-Then Support) #

Euclid proves: If the angles at the base of a triangle are equal, then the sides are equal.

The proof structure:

• Either the sides are equal OR they are unequal (exhaustive division)
• If they are unequal, then by constructing appropriate triangles, a part would equal the whole (impossible)
• Therefore, they cannot be unequal
• Therefore, they must be equal

This uses both either-or reasoning (to establish the complete division) and if-then reasoning (to show one possibility is impossible).

Faith, Hope, and Charity (Either-Or Syllogism from Thomas) #

Thomas Aquinas: Every virtue is either a moral virtue or an intellectual virtue. Faith, hope, and charity are not among the moral virtues. Faith, hope, and charity are not among the intellectual virtues. Therefore, faith, hope, and charity are not virtues (in the natural sense).

Thomas then resolves this by distinguishing natural virtues from theological virtues.

Questions Addressed #

Why do students confuse valid and invalid forms? #

The mind is naturally prone to false imagination. Forms 1 and 4 have similar structure (affirmative in both), as do Forms 2 and 3 (negative in both). Without explicit logical training, the mind assumes parallel structure indicates parallel validity. Aristotle noted in the De Anima that philosophers had not adequately explained how deception occurs; they explained knowing but not false imagining.

How can we prove a form is invalid? #

By providing a counterexample where the premises are true but the conclusion is false. Since necessity means always true, a single counterexample suffices to show something is not necessary. (However, one true instance cannot prove necessity; all instances must be examined or necessity must be proven through logical structure.)

Why is scientific hypothesis never known to be “necessarily true”? #

Because scientific confirmation uses Form 4 (affirming the consequent), which is not a valid syllogism. No matter how many times predictions come true, the form does not guarantee that the hypothesis is the true cause of those predictions. Einstein maintained that scientific hypotheses are “freely imagined” like novels—they have no justification until tested, and even after testing and confirmation, they remain provisional guesses subject to future disproof.

What is the relationship between the either-or and if-then syllogisms? #

Both are “syllogisms from hypothesis” (hypothetical syllogisms). Both differ from categorical syllogisms:

• Categorical syllogism: Two premises are simple statements; middle term connects them; conclusion is simple statement
• Either-or syllogism: First premise is disjunctive (either A or B); second premise eliminates possibilities; conclusion is simple statement
• If-then syllogism: First premise is conditional (if A then B); second premise affirms or denies antecedent/consequent; conclusion is simple statement

The if-then syllogism is more important because it appears more frequently in reasoning, though the either-or is conceptually simpler.