36. The Four Forms of Hypothetical Syllogisms

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Lecture Notes

Main Topics #

The Four Forms of If-Then Syllogisms #

Berquist presents four distinct forms of hypothetical syllogisms using the structure “If A is so, then B is so”:

1. Affirming the Antecedent (Valid): If A is so, then B is so. A is so. Therefore, B is so.
2. Denying the Consequent (Valid): If A is so, then B is so. B is not so. Therefore, A is not so.
3. Denying the Antecedent (Invalid): If A is so, then B is so. A is not so. Therefore, B is not so.
4. Affirming the Consequent (Invalid): If A is so, then B is so. B is so. Therefore, A is so.

The Problem of False Imagination #

Berquist emphasizes that forms 3 and 4 appear to follow necessarily but do not. This false imagination is the primary cause of deception in reasoning. Forms 3 and 4 structurally resemble the valid forms, causing the mind to mistakenly judge them as valid. The invalid forms seem as obvious as the valid ones when considered superficially.

Necessity vs. Probability #

A critical distinction: something following necessarily means it must always be true. If a conclusion sometimes holds and sometimes does not, it is not necessary and therefore cannot be a syllogism. Necessity and always-ness are connected: if something is necessarily true, it is always true, and if it is always true, it is necessarily true. The reverse holds: if something is not always true, it is not necessarily true.

Key Arguments #

Why Forms 1 and 2 Are Valid #

Form 1 (Affirming the Antecedent):

• If the antecedent (A) is true and the if-then statement is true, the consequent (B) must follow necessarily
• This form is the most obvious and rarely causes confusion

Form 2 (Denying the Consequent):

• If the consequent is false and the if-then statement is true, the antecedent cannot be true
• Berquist explains through contradiction: if A were true, then B would have to be true (by the if-then statement), but B is given as false, so the three statements (if A then B, A is true, B is false) are incompatible; therefore A must be false
• This form is less immediately obvious than Form 1 but can be demonstrated through logical contradiction

Why Forms 3 and 4 Are Invalid #

Form 3 (Denying the Antecedent):

• Counterexample: “If this number is 2, then this number is half of 4. This number is not 2. Therefore, it is not half of 4.”
• The form itself does not guarantee the conclusion, even though materially (2 and half of 4 are convertible) the conclusion happens to be true
• Another example: “If Tabitha is a dog, then Tabitha is an animal. Tabitha is not a dog. Therefore, Tabitha is not an animal.”
• Tabitha could be a cat or other animal and still be an animal; denying the antecedent does not necessarily deny the consequent

Form 4 (Affirming the Consequent):

• Counterexample: “If Socrates is a mother, then Socrates is a woman. Socrates is a woman. Therefore, Socrates is a mother.”
• Many things can be women without being mothers; the consequent is more general than the antecedent
• Aristotle notes this is the form “Homer taught the other poets how to tell a good lie”
• This form is the most deceptive because it seems plausible and probable
• Another example: “If he is losing the argument, then he will get angry. He got angry. Therefore, he is losing the argument.” But he could be angry for many other reasons

Testing Validity with Examples #

One cannot show a form is valid by examples alone (since one cannot examine all cases). However, one can show a form is invalid by providing counterexamples. For a form to be invalid, one must find cases where the premises are true but the conclusion is false—or where the conclusion is sometimes true and sometimes false.

Looking Before and After: Establishing vs. Overthrowing #

Berquist connects this to Shakespeare’s phrase “reason looks before and after”:

• To establish a statement B (to prove it true), look before it for a statement from which B follows (use Form 1: affirm the antecedent)
• To overthrow a statement B (to prove it false), look after it for a statement that follows from B, then show that statement is false (use Form 2: deny the consequent)
• In the structure of if-then statements, the antecedent is before and the consequent is after; the premises are before and the conclusion is after in any syllogism

Important Definitions #

Antecedent: The simple statement in the “if” part of a conditional; it is logically before and upon which something follows.

Consequent: The simple statement in the “then” part of a conditional; it is what follows from the antecedent.

Hypothetical Syllogism: An argument with one compound (if-then) premise and one simple premise, yielding a simple conclusion. Only two of the four forms are true syllogisms.

Necessity: A conclusion follows necessarily when it must always be true given true premises. Necessity is distinguished from mere probability or occasional truth. This is fundamental to the definition of syllogism itself.

False Imagination: The mind’s tendency to imagine that invalid forms are valid because they structurally resemble valid forms. This is the main cause of deception in knowing.

Examples & Illustrations #

Mathematical Examples #

• “If this number is 2, then this number is half of 4. This number is half of 4. Therefore, this number is 2.” (Affirming the consequent—seems to work because 2 and half of 4 are convertible, but the form itself is invalid)
• “If this number is 2, then this number is half of 4. This number is not 2. Therefore, it is not half of 4.” (Denying the antecedent—invalid form, though the conclusion is true materially)
• Demonstrating necessity: “2 is necessarily half of 4” means 2 is always half of 4. If one produces any 2 that is not half of 4, the necessity claim is false.

Geometric Example (Euclid, First Proposition) #

• Constructing an equilateral triangle: to prove two line segments are equal, Socrates notes they are radii of the same circle, and “all radii of the same circle are equal.”
• This uses Form 1 (affirming the antecedent): If something is a radius of the same circle, then it equals all other radii of that circle. AB and AC are radii of the same circle. Therefore, AB and AC are equal.

Dialectical Examples from Plato’s Meno #

• Socrates reasons both that virtue can be taught and that it cannot be taught, using hypothetical syllogisms on both sides
• On one side: “If virtue is knowledge, then it can be taught” (antecedent affirmable); evidence suggests virtue is knowledge; therefore, virtue can be taught (Form 1)
• On the other side: “If virtue can be taught, then there are teachers of virtue. There are no teachers of virtue. Therefore, virtue cannot be taught” (Form 2)
• The reasoning is dialectical (proceeding through probable opinions and even to contradictory conclusions) because the nature of virtue itself is unknown

Everyday Examples #

• “If he is losing the argument, then he will get angry. He got angry. Therefore, he is losing the argument.” (Invalid—affirming the consequent; he could be angry for other reasons like the other person being stubborn or denying the obvious)
• Teaching and knowledge: “If virtue could be taught, the great men of Athens would have taught it to their sons. But the great men’s sons are not virtuous. Therefore, virtue cannot be taught.” (Valid—denying the consequent)
• Hiring piano teachers but not teachers of justice: “Is it more important for my children to play piano than to be just? Obviously not. Yet I hired a piano teacher but no teacher of justice. This suggests there are no teachers of justice.”

Notable Quotes #

“False imagination is the main cause of deception on the side of the knowing powers.”

“If something followed necessarily, it would have to be always so… if something is sometimes so and sometimes not so, then nothing is necessarily so, and therefore you’re going to have no syllogism.”

“Aristotle says, this is the way that Homer taught the other poets how to tell a good lie.” (Regarding affirming the consequent)

“You have to go back to something obvious, because you can’t prove everything. If you prove everything, you can prove nothing.”

“If I say that man is necessarily white, and some men are white and some are not, forget about saying that man is necessarily white.”

Questions Addressed #

Why does denying the antecedent seem to work? #

The invalid form “If A is so, then B is so. A is not so. Therefore, B is not so” appears valid because it mirrors the structure of the valid form “If A is so, then B is so. B is not so. Therefore, A is not so.” However, the consequent is often more general than the antecedent. Denying a species does not deny the genus; denying that Socrates is a mother does not deny that he is an animal or a human.

Why is affirming the consequent so deceptive? #

This form is most deceptive because it deals with probability. When a probable connection exists between antecedent and consequent (e.g., losing an argument makes anger probable), and the consequent is observed to be true, the mind naturally imagines the antecedent must be true. However, multiple causes can produce the same effect.

How does one show a form is invalid? #

One must find concrete counterexamples where the premises are true but the conclusion is false (or sometimes true and sometimes false). This proves the form does not work necessarily. However, examples alone cannot show a form is valid, only that it is always valid in those cases examined.

How does this apply to the study of Euclid’s Elements? #

Euclid uses all three types of syllogisms (categorical, if-then, and either-or) even in simple proofs. The first proposition uses only one type of syllogism (categorical), but this knowledge is necessary foundation for understanding more complex geometric proofs that employ multiple forms.