If-Then and Either-Or Syllogisms; the Three Figures; Prior vs. Posterior Analytics

Orientation #

He opens by recalling “we spoke of the if-then syllogism” (prev. lecture) and continues from there. The class works from the handout “The Art About Statement”, p. 5 (verified source). At the close he says they will go through at least the four universal cases of each figure and has just set out the four of the first figure. The neighbouring classes treat the division of logic and compound statements before, and the dictum de omni / de nullo applied to the figures after.

The class, in order #

1. The if-then syllogism and what “true” means in an if-then statement (“if I am a giraffe, then I have a long neck”) #

Point: the if-then statement matters most for reasoning, and its truth is not the truth of its parts.

• The if-then syllogism: one statement of the form “if A is so, then B is so” (A, B standing for simple statements), plus the finding that A is in fact so; then you must say B is so. He calls this “formal logic” because you do not look at the matter. Example: if the students in this room are four, then they are an even number (a student objects “depends how many are in the room”; he replies he is not asserting the antecedent, only the connection).
• Can a true if-then be made of two false simple statements? Yes: “if I am a giraffe, then I have a long neck” (true, though he is no giraffe and has no long neck); “if I am a mother, then I am a woman” (true, both parts false of him). So truth means something different in the if-then statement: not “this is that” but “if this is that, then that will be that”.
• Why not stay with if-thens (“if A, B; if B, C; so if A, C — who cares?”)? Because you want to get back to the truth of a simple statement. Hence the if-then syllogism has one if-then premise, one simple premise, and a simple conclusion — “that’s amazing”.

2. Denying the antecedent: likeness is the cause of error (“likeness is the cause of error”) #

Point: “if A then B; A is not so; therefore B is not so” looks right by proportion but does not follow.

• Why people fall for it: it looks proportional, like the valid case. Illustrations of likeness causing error: Shakespeare’s Comedy of Errors (two sets of twins), twin girls switching dates, twin boys in the parish his wife could not tell apart, his own congratulating the wrong sister after a play.
• Counter-examples (a more universal B following from many particular As): if I am a dog / cat / horse, I am an animal; “I am not a dog” leaves “I am an animal” possibly true. B may be not so, but it may be so.
• Boundary case: with convertible matter it works — “if the number is two, it is half of four; not two, so not half of four” — but that is known from the matter, not the form.
• Rule: you can prove a form is not a syllogism by examples, but cannot prove it is one by examples — you would need all possible examples, “too many”.

3. Why one example disproves and no number proves (“if something is necessarily so, then it is always so”) #

Point: the disproof rests on an if-then: if necessarily so, then always so.

1. If man is necessarily white, he is always white; if a book necessarily has a blue cover, it always has one.
2. One black man, one red or green cover, shows “not always”, hence “not necessarily”.
3. But how many white men prove man is always white? Endless. “I maintain all numbers are odd”: one even number disproves; 1, 3, 5, 7, 9… never prove. A student says it “seems unfair”; he agrees it shows “the weakness of induction”.

• Applied: in the first arrangement (A so, so B so) the necessity is seen, not shown by examples — you have admitted the connection and admitted A. In denying the antecedent you are not told A is so; nothing follows.
• Asides: Shelley on poetry — reason tells differences, poetry sees likenesses (he doubts that makes poetry higher; “differences divide”); scientists say most discoveries come from seeing proportion, but likeness is “a slippery thing”; rhetoric’s word is “likelihood”.

4. Affirming the consequent: many causes of one effect (“if Perkis dropped dead last night”) #

Point: “if A then B; B is so; therefore A is so” does not follow, though science seems to reason so.

• Louis de Broglie’s wave mechanics, mocked in France (“La Comédie Française”), backed by Einstein, confirmed by experiment at Bell Telephone laboratories and again in Europe. “If my hypothesis is so, there will be an eclipse at such a time; there is; have I proven my hypothesis?”
• “If Perkis dropped dead last night, he’ll be absent today; he is absent; therefore he dropped dead” — wishful thinking, not logical thinking: his car broke down, he is sick, he went to Florida. His own missed Monday class at the monastery: dead battery, garage door left open, Triple A.
• Reason: many possible causes of the same effect; the more universal follows from the less universal, and given the more universal you cannot see which less universal holds. “If I am a woman, I am a human being; I am a human being; therefore a woman” — no. “If odd, then a number; you are a number…” — no. “This is no good, the form.”
• Aside: logic is divided into formal (A, B, C — “drives my wife crazy”) and material; Aristotle’s Prior Analytics is the formal, Posterior Analytics (Book B) the material.

5. Denying the consequent, and the four arrangements (“in these four arrangements of the if-then”) #

Point: “B is not so, therefore A is not so” does follow, though less obviously.

• Shown through the obvious case: if A were so, B would be so; but B is not so; so A cannot be so; therefore A is not so.
• Summary of the four: A so → B so (must); A not so → nothing; B so → nothing; B not so → A not so (must).
• He uses the disproof form for the ordinary syllogism: if it follows necessarily, it is always so; my examples show it is not always so; therefore not necessary, therefore not a syllogism.

6. What “syllogism” means: statements laid down (“some statements laid down, another follows”) #

Point: syllogism is speech in which, some statements being laid down, another follows necessarily because of them.

• Students guess the word’s meaning; he settles on “argument”, or “speech”. He likes the Latin but gives the English “laid down”: what is laid down is at rest, firm, as “I’ve laid down the law” and expect you to follow it. If I think of one premise, forget it, then think of the other, I cannot syllogize; both must be laid down together in the mind, then “this follows”.
• Aside: his father’s whiskey in the cellar, the children pretending with root beer, the dropped glass.

7. One if-then plus one simple; the either-or syllogism (“eliminate all but one of the alternatives”) #

Point: the compound syllogism ends in a simple statement because the primary meaning of truth is in the simple statement.

• Two if-thens can be chained (if A, B; if B, C; A; so C) but that is really two arguments. Truth in the simple statement: saying what is is and what is not is not; falsity the reverse. So you want to get back to it.
• Either-or syllogism, “not as deceptive”: arguing affirmatively you eliminate all alternatives but one. This straight line is greater than, less than, or equal to that one; not greater, not less; therefore equal. Thomas: if the Father is before the Son, he is before in time or duration, in being, in knowledge, or in excellence/goodness; he is not before in time, not in being, not in knowledge (knowledge of relatives, opposites), not better; therefore not before. “Common sense, but sometimes downright hard.”

8. Aristotle’s syllogism of simple statements: three figures (“Monsignor Dionne used to call that playing checkers”) #

Point: the syllogism from two simple statements has three figures, and only three.

• Aristotle calls this simply “syllogism”; the other kind is ex hypothesi. The two statements must share a term; B, the middle term, is truly in the middle only in the first figure. The predicate can be above both subjects, below them, or in the middle — three, and not by custom.
• Objection “what about C–B?” (a fourth figure): “playing checkers”, rearranging pieces; it is really the first figure with the terms swapped, set up for the said-of-all. “Forget about the fourth figure” — those who hold it do not know “two is the first number” [ed.: sense unclear in recording].
• Each premise universal or particular, affirmative or negative: four possibilities each, sixteen per figure, forty-eight in all (“like the number of states we used to have”).

9. Why form before matter: Prior before Posterior Analytics (“from confused knowledge to distinct knowledge”) #

Point: the Prior (“before”) Analytics treats form first because form is more universal and man goes from confused to distinct.

• Beginning of the books of “natural hearing” [ed.: Physics]: consider things in general before the particular, since the more universal is more known to us and more confused.
• Parallel: name said of many things → equivocal (one name, many meanings) → equivocal by chance (no connection) vs. equivocal by reason (an order among meanings; “order” starts Aristotle’s definition of reason).
• The same premises may turn out necessary (demonstration, science) or only probable (dialectic); the form is the same, so more general, so taken first.
• Aside: checking the checkbook against the bank — first the addition, then the numbers; his mother’s separate account; brother Mark on radios and cars that were “supposed to work”.

10. The four universal cases; perfect and imperfect syllogisms (“the set of all and the set of none”) #

Point: begin with the four all-universal cases of each figure.

• Preview: first figure yields a universal affirmative and a universal negative conclusion; second, universal negatives only, no universal affirmative (“a falling off”); third, only particular conclusions.
• In the first figure the said-of-all and said-of-none [ed.: dictum de omni / de nullo; ASR “set of all”] apply as they stand (he means to state them as if-thens); in the others you must convert first, so Aristotle calls those imperfect syllogisms, the first-figure ones perfect — “doesn’t need any help”.
• The four universal cases of the first figure: every B is A, every C is B; no B is A, no C is B; no B is A, every C is B; every B is A, no C is B. “Four, not three”: two variables (universal/particular, affirmative/negative) crisscrossed give four.

His words #

• if-then syllogism — one if-then statement plus one simple statement, simple conclusion [ed.: hypothetical / conditional syllogism]
• formal logic — reasoning without looking at the matter (his A, B, C)
• likeness is the cause of error — his account of why denying the antecedent seems valid
• laid down — his English for the premises of a syllogism: at rest, firm in the mind [ed.: posita]
• either-or syllogism — argue affirmatively by eliminating alternatives [ed.: disjunctive syllogism]
• ex hypothesi — Aristotle’s name for the non-simple syllogism
• middle term — B, “really in the middle” only in the first figure
• playing checkers — Monsignor Dionne’s name for the supposed fourth figure
• before / after analytics — his rendering of Prior / Posterior Analytics
• natural hearing — [ed.: Aristotle's Physics]
• equivocal by chance / equivocal by reason — meanings unconnected vs. ordered
• said of all / said of none (ASR “set of all”) — [ed.: dictum de omni / de nullo]
• perfect / imperfect syllogism — needs no conversion vs. must convert

Texts #

Read in class

• Handout, “The Art About Statement”, p. 5 (verified source; about 8 minutes)

Mentioned

• Aristotle, Prior Analytics (formal) and Posterior Analytics, “Book B” (material)
• Aristotle, Physics, beginning (“natural hearing”)
• Shakespeare, The Comedy of Errors
• Shelley, “a thing on poetry” (A Defence of Poetry, inferred)
• Thomas, on the Father not being “before” the Son (no locus given)

His questions #

• Does an if-then statement, to be true, have to have its parts be true? No; “if I am a giraffe, then I have a long neck” is true with both parts false.
• If A is not so, must B be not so? No; “if not a dog, not an animal” fails — B may still be so.
• Why can’t you prove a form is a syllogism by examples? You would need all possible examples, too many; one counter-example disproves.
• How many white men do you need to prove man is always white? Endless numbers — the weakness of induction.
• If B is so, does it follow by the form that A is so? No; many causes of one effect (“wishful thinking”).
• If B is not so, does something follow about A? Yes, A is not so — shown by the obvious first case.
• What does syllogism mean? Speech in which some statements being laid down, another follows necessarily.
• How do you argue from an either-or statement? Eliminate all but one alternative.
• Into how many figures does Aristotle divide the syllogism, and is there a fourth? Three; the “fourth” is the first with terms swapped, “playing checkers”.
• Why does the Prior Analytics come before the Posterior? Form is more universal, and we go from confused to distinct.