Three Kinds of Syllogism: The Four Forms of If-Then Speech, the Meno and Euclid

Orientation #

He says at the start that they will “first be finishing the form and matter of the syllogism” and then go into the forms of the either-or and if-then syllogisms; the first handout passed out distinguishes the three kinds of syllogism, the second gives the form of the simple syllogism, to be read “the next time”. He says they will “come back” to Euclid after going through the syllogisms, and that the simple syllogism will require “48 combinations” to be considered later. Verified sources identify handout pages read in class (“Contradiction and the Growth of Reason” p. 22; “The First Definition of Reason” p. 103), though the reading itself is not audible in the transcript. The previous class defined syllogism against enthymeme, induction and example; the next continues with form and matter of the syllogism and the conditional forms.

The class, in order #

1. Three kinds of syllogism and the handouts (“distinguish the three kinds of syllogism”) #

The point: there are three kinds of syllogism to be learned, and Euclid uses all three.

• The three: what Aristotle calls “syllogism, period” (others call it categorical); the hypothetical or if-then; the either-or (disjunctive).
• Euclid I.1 (equilateral triangle on a line AB, two circles, lines to the cut C) uses only the simple syllogism, twice, in identical form: all radii of the same circle are equal; AB and AC are radii of the same circle; therefore they are equal — then the same for AC and BC. “You have to really declare everything.”
• Even a simple theorem like I.6 uses all three kinds (worked out in segment 11).
• Aside: he has taught logic a long time, interrupted by being chairman.

2. First form: affirming the antecedent (“you can’t prove everything”) #

The point: If A is so, then B is so; A is so; therefore B is so is a syllogism and obvious.

• Reason given: it is obvious; “you have to go back to something obvious because you can’t prove everything, right? If you had to prove everything, you could prove nothing.”
• [Transcript gap: the passage between Euclid I.1 and this statement is missing in the recording.]

3. Second form: denying the antecedent (“false imagination”) #

The point: If A, then B; A is not so; therefore B is not so is not a syllogism, and examples show it.

1. Students think it follows because to the imagination it looks “exactly the same” as the first form: this is false imagination, “the main cause of deception on the side of the knowing powers.”
2. How to show a form is not a syllogism: find examples where both premises are true and in one B is in fact so, in another B is not so.
3. Why that suffices: syllogism by definition means the conclusion follows necessarily; what is necessary is always so (“two is necessarily half of four” → always half of four). If man were necessarily white he would always be white; one black man shows man is not always white, hence not necessarily white.
4. Examples: “If Socrates is a dog, then Socrates is an animal; Socrates is not a dog” — but Socrates is an animal (B so). “If Socrates is a mother, then Socrates is a woman; Socrates is not a mother” — and not a woman (B not so). Sometimes so, sometimes not: no necessity.
5. Converse limit: you cannot show a form is a syllogism by examples, since examples do not show even “always” (2, 4, 6, 8… infinite examples do not show numbers are always even); but one counterexample suffices to show it is not.

• Why people are taken in: they know “Socrates is not a mother” and “Socrates is not a woman” are both true and related; but then the women students who are not mothers “would not be a woman, right? Very strange conclusion.”

4. Third form: affirming the consequent (“how Homer taught the other poets”) #

The point: If A, then B; B is so; therefore A is so is the form “that deceives people the most” and is not a syllogism.

• Aristotle in the Poetics: this is how Homer taught the other poets to tell a good lie — take the truth of the expected consequence as a sign of the truth of what went before. Example: “if he’s losing the argument, he’d get angry; he got angry, therefore he’s losing” — but one can get angry for many reasons (the other denies the obvious, confuses the issue).
• Disproved again by examples. “If this number is two, it is half of four; it is half of four; therefore two” seems good — but only because of the matter: half of four is a property in the strict sense, convertible with two. From the form alone it does not follow.
• Typical case: B a genus, A a species. His cat Tabitha: “If Tabitha is a dog, Tabitha is an animal; Tabitha is an animal; therefore a dog” — no. With “cat” for A, the conclusion is true; with “dog”, false. So A might be so or not so with true premises; nothing always so, hence nothing necessarily so.

5. Fourth form: denying the consequent (“show what is not obvious through what is obvious”) #

The point: If A, then B; B is not so; therefore A is not so is a syllogism, shown through the first (obvious) form.

1. Either A is so or A is not so.
2. If A were so, then by the first form B would be so — but B is not so is given.
3. The three statements “if A then B”, “A is so”, “B is not so” are incompatible: one and three contradict two. Analogy from calculating: length 10, width 2, area 40 — “I don’t know which of those numbers is wrong, but they can’t all be right.”
4. Since “if A then B” and “B is not so” are given, “A is so” must be rejected; by the either-or, A is not so.

• “Often we reason this way”: from a statement known true and one known false, the unknown one must be false.
• Result: two forms of if-then speech are syllogisms and two are not; “you have to become very familiar with those four forms.”

6. Melissus and the need for logic (“Poor Melissus”) #

The point: the first book of “Natural Hearing” (the Physics) shows a philosopher using the invalid second form.

• Melissus: “If being had a beginning, then being has an end; but being does not have a beginning; therefore it has no end” — denying the antecedent. Aristotle says the matter is bad too; either defect suffices to overcome an argument, and Melissus has both.
• Logic is “the last part of philosophy to be discovered”, but the philosophers “begin to see the need for it just now.”

7. Reason looks before and after (“you look before… you look after”) #

The point: the two valid forms correspond to where reason looks relative to a statement B.

• To establish B, look before it for an A from which B follows; prove A, syllogize B (first form).
• To overthrow B, look after it for a C that follows from B; show C not so, syllogize B not so (fourth form).
• Two before-and-afters in syllogism: premises before conclusion (“pre-mise”), antecedent before consequent. Shakespeare’s phrase “looking before and after” fits; letters A, B, C mark where one looks.

8. The Meno: structure of the dialogue (“you can’t look for what you don’t know”) #

The point: Aristotle in the Metaphysics says Socrates tried to syllogize, the sign being that he tried to define, definition being the beginning of syllogism (especially the simple one); the Meno shows him syllogizing dialectically.

• Part 1 (examination): Meno asks “can virtue be taught?”; Socrates does not know, does not know what virtue is, has met no one who does. Meno gives examples — like answering “what is water?” with rain, lake, faucet. His definition does not separate the good man from the bad.
• Part 2: “put our two heads together”; Meno’s sophistical objection that you cannot look for what you don’t know; Socrates’ reply that learning is in a way recalling. Difficulty: the slave boy seems not to recall how to double the square but to come to know it for the first time through things he already knows — he begins mistaken.
• Part 3: Meno still wants to know whether virtue can be taught without knowing what it is. Brother Mark: “what do you do with a guy like that?” — people in academic life who want to know this without first knowing that. Student who wanted Pythagoras’ theorem (Euclid I.47) but “I don’t do 46.”
• Socrates proceeds dialectically, from probable opinions, even to contradictory conclusions; the demonstrator never does, since he knows what things are, but the dialectician is on the way, so the if-then forms suit his matter.

9. Socrates’ two arguments (“if virtue is knowledge, then it can be taught”) #

The point: for “can be taught” Socrates uses the first form; for “cannot” the fourth; he does not use the two non-syllogisms.

• For: If virtue is knowledge, then it can be taught (obvious on its surface: we teach what we know). Virtue is knowledge. Therefore it can be taught.
  • The second premise needs backing by a simple syllogism: what directs us to good things is knowledge (medical art → health, economics → wealth, military art → victory); virtue directs us to good things (vice to bad); therefore virtue is knowledge.
  • Terms: the main or chief syllogism is the one whose conclusion is the main conclusion; the backing one is the prosyllogism. Rule for examining reasoning: start from the main conclusion, then the statements next to it, and see which needs proof.
• Against: If virtue can be taught, then there are teachers of it. There are no teachers. Therefore it cannot be taught.
  • Why the if-then is probable (weaker than the first): the most obvious virtue is courage — virtus means manhood, Greek aretē similarly; a city without courageous citizens is taken, its men killed, women and children enslaved; temperance and justice likewise needed for men to live together. So if teachable, surely taught.
  • Why “no teachers”: two candidates — the sophists (unsavory to nobles like Meno and Anytus, who later brings the accusation against Socrates) and the great men of Athens, who were courageous and just yet had cowardly and thieving sons, hired horse-riding teachers but no teacher of justice — not because riding matters more. His own case: he can name his children’s piano teachers, hired no one for courage or temperance; “apparently there isn’t anybody.”
  • The other side, in the Protagoras: who taught you English? mother, father, brothers — diffused teaching; so “in a sense there are teachers.” Socrates in the Meno brings out only the one side.

10. Recap of the four forms, and scientific confirmation (“wishful thinking, but it’s not logical thinking”) #

The point: the two invalid forms recur in common life and in experimental science.

• Affirm antecedent → affirm consequent; deny consequent → deny antecedent. Not: deny antecedent (two/half of four seems to work only because convertible; “if I’m a dog, I’m an animal; I’m not a dog, therefore not an animal” fails), nor affirm consequent.
• Materially the two commonest deceptive cases: consequent more general than antecedent; consequent an effect with many possible causes. “If Berquist dropped dead last night, he’d be absent today; he’s absent, therefore he dropped dead” — the car could have broken down.
• Confirmation of a hypothesis has this form: “if my hypothesis is correct, such and such will happen; it happened” — never proves the hypothesis necessarily. Einstein: Newtonian physics predicted so much (unknown planets predicted from perturbations and found) that scientists thought it must be true; relativity predicted the same and more, so “it became crystal clear that we never know” — a theory is “a system of guesses”. Claude Bernard: doubt is intrinsic to experimental method; a hypothesis can always be tested again.
• More predictions → more probability, like induction (more frogs with three-chambered hearts), never necessity. More rigor in rejecting a hypothesis (predicted eclipse at ten; none; hypothesis false — the fourth form) than in confirming — “so it seems unfair.”
• Einstein: the hypothesis is freely imagined, like writing a novel; so it has no justification until tested, and even then is not sure.

11. Euclid I.6 uses all three syllogisms (“the part will be equal to the whole”) #

The point: a simple early theorem shows the three kinds working together.

• I.6 is the inverse [of I.5]: if angles B and C are equal, sides AB and AC are equal.
• Either-or: the sides are either equal or unequal.
• If-then: if unequal, one is longer (say AB); cut BD equal to AC (theorem 3); join DC; triangles DBC and ACB have an equal angle contained by equal sides (BC common, DB = AC), so by theorem 4 they are equal (a simple syllogism, like in I.1) — but that is the part equal to the whole, impossible. Therefore not unequal; therefore equal.
• Aristotle too sometimes uses all three; in geometry these three are what you will find.

12. The either-or syllogism and how he teaches the forms (“almost common sense”) #

The point: the disjunctive form is common sense but has two uses, and the if-then forms are where people go wrong.

• Two ways to reason from either-or: eliminate all but one member and conclude to the remaining one (you know the thing falls under the division but not which member; Euclid’s equal/unequal); or eliminate all members and deny the thing divided (you show it does not fall under the division). Thomas in the Summa Contra Gentiles: every name said univocally of two things is genus, difference, species, property or accident; all five eliminated; therefore no name is said univocally of God and creatures.
• Teaching method: four students at the board with the four forms; write the conclusion if something follows necessarily, “invalid” if not; someone always gets one wrong, sometimes all. “Your mind is in serious trouble” if it thinks something follows when it does not or vice versa — a sign you need logic. Always state it “If A is so, then B is so”; the mind may find A so, A not so, B so, B not so; ask what follows about the other.
• Order of teaching: either-or first (least important, common sense), then if-then (people get mixed up), then the simple syllogism, which needs 48 combinations because simple statements are affirmative/negative, universal/particular, “and other variations.”
• Names: Aristotle calls the simple one “the syllogism”; the if-then “we call syllogism from hypothesis”; used much in dialectic. He uses the Meno as an introduction to logic; the either-or “all the way through” Plato, Aristotle, Euclid.
• Closing recap: understanding which forms are not syllogisms explains why a scientific hypothesis “remains, as Einstein says, always a guess.” Student remark that the data suggests the hypothesis right away is left unanswered.
• Closing fragment [unclear in recording]: an anecdote about someone on a deathbed “having understood everything”, and one who wept and prayed over a passage in St Paul, then woke understanding it.

His words #

• syllogism, period — Aristotle’s name for the simple syllogism [ed.: categorical syllogism]
• if-then / hypothetical syllogism — “syllogism from hypothesis” [ed.: conditional syllogism]
• either-or syllogism — [ed.: disjunctive syllogism]
• false imagination — main cause of deception on the side of the knowing powers; makes the invalid forms look like the valid ones
• necessary → always — what is necessarily so is always so; one counterexample destroys necessity
• property in the strict sense — belongs only to the subject, to every one, and always (half of four to two); convertible
• Natural Hearing — [ed.: Physics]
• looking before and after — Shakespeare’s phrase for where reason looks: before to establish, after to overthrow
• antecedent / consequent — the if-part and then-part; affirming/denying them names the four forms
• main or chief syllogism — the one whose conclusion is the main conclusion
• prosyllogism — the backup syllogism proving a premise of the main one
• virtus / aretē — “manhood”; first sense close to courage
• system of guesses — Einstein’s phrase for scientific theory; hypothesis “freely imagined”
• 48 combinations — what the simple syllogism will require

Texts #

Read in class (worked through or quoted):

• Euclid, Elements I.1 and I.6 (with I.3, I.4 as steps)
• Plato, Meno (structure and the two arguments on whether virtue can be taught; Anytus)
• Handouts identified by verified sources: “Contradiction and the Growth of Reason” p. 22; “The First Definition of Reason” p. 103 (reading not audible in transcript)

Mentioned:

• Aristotle, Poetics (Homer taught poets the good lie)
• Aristotle, Physics I (Melissus)
• Aristotle, Metaphysics (Socrates tried to syllogize; definition)
• Plato, Protagoras (who taught you Greek)
• Thomas Aquinas, Summa Contra Gentiles (five predicables; no univocal name of God and creatures)
• Euclid I.46–47 (Pythagorean theorem anecdote)
• Shakespeare, “looking before and after” (Hamlet, inferred)
• Einstein; Claude Bernard (19th-century physiology)

His questions #

• If A is so then B is so; A is not so — does it follow necessarily that B is not so? No; examples (Socrates dog/animal, mother/woman) give B so and B not so, so nothing follows necessarily.
• Why do you need one example where B is so and one where B is not so? Because necessary means always; showing it is sometimes so and sometimes not destroys necessity.
• Can you show by examples that a form is a syllogism? No; examples cannot show even “always” (even numbers), let alone necessity — only that it is not.
• Does the truth of the prediction prove necessarily that the hypothesis is correct? No; it is the form of affirming the consequent, which is not a syllogism.
• Would you look before B or after B to prove that B is so? Before, for an A from which B follows; after, for a C, if you want to overthrow B.
• What sort of thing would virtue have to be in order to be taught? Knowledge, since we teach what we know.
• Is it because it’s more important to ride a horse (or play the piano) than to be just? No; so apparently there are no teachers of virtue.
• Is there more rigor in rejecting or confirming a hypothesis? In rejecting — denying the consequent is a syllogism; confirming is not.