If-Then Arguments, Conversion of Statements, and the First Figure of the Syllogism

Orientation #

He opens with “just review a little bit here” of the simple and compounded statements and the either-or and if-then arguments, and says of the three figures “you didn’t get that part” last time (prev. lecture). He says next they will “go through these three figures” and that the second figure will syllogize the universal negative but not the universal affirmative. The previous class treated the simple statement and the either-or and if-then compound statements; the next works through the second and third figures.

The class, in order #

1. Review: why either-or and if-then matter for reasoning (“looking before and after presupposes looking for distinctions”) #

Point: of compound statements, the either-or and the if-then are the ones most important for reasoning, because the first involves seeing a distinction and the second a before and after (“this is so, then that is so”).

• Conjunctive statements (“Berquist is a philosopher and a grandfather”) have little importance for reasoning.
• Thomas reasoning that the Father is not before the Son: an either-or statement (before in time or duration, in being, in cause and effect, in knowledge, in goodness), then eliminating all possibilities to conclude a negative.
• The geometer: this straight line is greater than, less than, or equal to that one; eliminate two, conclude the third — “kind of common sense.”
• Either-or reasoning summed up later: eliminate all possibilities for a negative conclusion, all but one for an affirmative.

2. The four cases of the if-then argument (“if A is so, then B is so”) #

Point: with letters standing for simple statements, the second premise may affirm or deny A or B; of the four combinations only two conclude necessarily by form.

1. A is so → B is so. Obvious: the first premise does not assert A or B, only that if the first, then the second; once you also admit A is in fact so, you must admit B.
2. A is not so → nothing about B. If A is not so, B may or may not be so.
3. B is not so → A is not so. Less obvious; he backs it with case 1: if A were so, B would have been so, which contradicts “B is not so.” This is a syllogism — he calls it the if-then syllogism and dislikes “hypothetical syllogism” because it suggests something hypothetical about it.
4. B is so → nothing about A. Many “will say at first” A is so, but not necessarily.

How he disproves cases 2 and 4: an if-then of its own — if a conclusion follows necessarily it follows always; so one example where the premises are true and B is so, and one where they are true and B is not so, shows nothing about B is always so, hence nothing necessary.

• Case 2: “If I am a dog, I am an animal; I am not a dog” — still an animal. “If I am a dog, I am a four-footed animal; I am not a dog” — not four-footed. So sometimes B is so, sometimes not.
• Case 4: “If I am a dog, I am an animal; I am an animal, therefore I am a dog” — no. “If I am a man, I am an animal; I am an animal” — happens to be so, but by the matter.
• Why the matter misleads: “if this number is two, it is half of four; it is half of four, therefore two” — justified only because two and half-of-four convert; the form is not necessary. Case 2 likewise fails when B is more universal (“if it is two, it is less than ten”).

Examples can disprove a form but never prove one. Examples show something is not always so; a million or an infinity of examples do not show it is always so. “A number is always odd: three, five, seven, nine…” — no proof. “Every man is white”: one black man disproves it; no number of white men proves it. To see case 1 is valid you must understand what “if A is so, then B is so” means, and that shows it universally.

• A student raises “if A is not so, then B is not so”; he sets it aside: “just really consider these four.”

3. The three figures of the simple syllogism (“this common term, because of the first figure”) #

Point: Aristotle uses “syllogism” for an argument from two simple statements, and the letters now stand for terms; there are exactly three arrangements of the term common to both premises.

• No common term (“every B is A, every X is Y”) gives no connection, affirmative or negative.
• With “every B is A, every C is B” it follows that every C is A; the shared term is the middle term — compared to the middleman in economics, who knows producer and customer who never meet.
• First figure: middle is subject in one premise, predicate in the other (“slant position”). Second figure: predicate in both (every A is B, every C is B). Third figure: subject in both (every B is A, every B is C).
• Why three and not four: the middle term is either between the major and minor, above them (as a predicate is said to be above), or below them; “crazy guys that have four figures” are “playing checkers.”
• Not by chance called first, second, third: the first figure yields universal affirmative and negative, particular affirmative and negative; the second yields universal negatives but no universal affirmatives; the third no universal conclusions — first “more powerful” than second, second than third. He says he is “looking before and after.”

4. The said-of-all and said-of-none (“all reasoning goes back to statements that are obvious”) #

Point: the syllogism is based on the dictum de omni and dictum de nullo (“said of all,” “said of none,” his short names), restated by him as if-then statements.

• Aristotle’s form: if A is said of all B, A is said of whatever B is said of (Categories: substance said of animal, animal of dog, so substance of dog). Said of none: if A is said of none of the Bs, A is said of none of what B is said of.
• His grammatical turning: “if every B is an A, then whatever is a B is also an A”; “if no B is an A, then whatever is a B is not an A.” Obvious if you understand “every” means no exception, “no” means none.
• Most statements are proven by others, but not forever; some must be immediate.
• In the first figure both are visible as it stands: every B is A, every C is B → every C is A; no B is A, every C is B → no C is A. With only “some C is B,” those some Cs must be (or not be) A.
• In the second and third figures the form is not one to which said-of-all/none applies as it stands, so you must ask whether the premises can be turned around: hence conversion. “Every dog is an animal” does not turn into “every animal is a dog” unless the matter is convertible.

5. Conversion of the four kinds of statement (“the universal negative converts simply”) #

Point: conversion is turning a statement around; what converts necessarily, and with what loss of power, decides what the later figures can conclude.

• Universal affirmative does not convert simply: one example suffices (every woman is a human being; every saint is a man; every dog is an animal). It converts partially, to “some A is B” — a student thinks this “true necessarily”; proof given after the next.
• Universal negative converts simply (“Aha! Got you now”): if “no A is B” were not necessary, then by the square of opposition some A is a B; name that A “x”; x is both an A and a B; so some B is an A — contradicting “no B is A.” So “no B is A” always gives “no A is B.”
• Universal affirmative converts partially, proved from that: if “some A is B” is denied, then “no A is B” could be true, which converts to “no B is A,” contradicting “every B is A.” It converts but “drops” to particular — “very important,” the reason the second figure can syllogize the universal negative but not the universal affirmative; converting the universal negative lands you “like magic” back in the first figure, but you “lose power” with the affirmative; “the battery runs down.”
• Particular affirmative converts and stays particular (a student’s “first impression, no”): if “some A is B” were false, “no A is B” would be true, converting to “no B is A,” which cannot stand with “some B is A.”
• Particular negative does not convert at all: “some animal is not a dog” does not give “some dog is not an animal.” “He’s a lost soul.”
• Ranking: universal negative “the winner of this prize” (converts, stays universal); universal affirmative converts but drops to particular; particular affirmative converts, stays particular; particular negative not at all. Universal negative beats universal affirmative, particular affirmative beats particular negative, “as far as conversion is concerned.”

6. Aside: the “inward parts of philosophy” (“what does the great Boethius mean by the inward”) #

Thomas commenting on Boethius (De Trinitate, he thinks): Boethius says he draws from the inward parts of philosophy; Thomas takes this as logic and wisdom. Mathematics has the imagination, natural philosophy the senses, but wisdom treats immaterial things and you must go into your mind, and logic does the same. So logic is like wisdom in universality and immateriality, while natural philosophy is like wisdom in knowing causes (wisdom knows first causes, natural philosophy goes further than second causes). “So I’m an inward philosopher.”

• Geometry aside (his students, one from TEC): Euclid has four things for circle and square — inscribe and circumscribe each in and around the other; he thinks an oblong can be inscribed in a circle and a circle circumscribed around it, but a circle cannot be circumscribed in an oblong, and a pentagon cannot be circumscribed in an oblong around a circle; “these guys” do not agree with him.

7. The sixteen cases and the four universal moods of the first figure (“four times four is sixteen”) #

Point: in the Prior Analytics each figure has sixteen cases, since each premise can be universal or particular, affirmative or negative; he takes the four with two universals first.

• Joke: four guys and four girls give sixteen possible marriages — “nowadays” guys marry guys, which “destroys my whole experience” and “shows the whole illogic.”
• The four universal moods: both universal affirmative (every B is A, every C is B); both universal negative (no B is A, no C is B); mixed with the negative as major (no B is A, every C is B); mixed with the affirmative as major (every B is A, no C is B). He miswrites one at first — “how can I be so stupid” — and corrects it. [In the transcript “no B is A, every C is B” is spoken twice before the correction.]
• The question asked of each: does anything follow necessarily with C as subject and A as predicate, or nothing?
• Every B is A, every C is B → every C is A (said of all). No B is A, every C is B → no C is A (said of none). Aristotle calls first-figure syllogisms perfect because said-of-all/none is seen as it stands; the second and third figures are imperfect, needing rearrangement.

8. Two negatives, then affirmative major with negative minor (“two negative parents; I can’t have any children”) #

Point: no B is A, no C is B, and every B is A, no C is B, yield nothing; shown by examples meeting two conditions.

• Method: find terms for A, B, C such that (1) both premises come out true, and (2) in one case every C is A, in another no C is A. If the universal affirmative is once true, both negatives are once false; if the universal negative is once true, both affirmatives are once false; so nothing is always true, so nothing necessary. His shortcut: keep A and B, vary only C.
• EE: A = animal, B = stone; C = dog (no dog is a stone, every dog is an animal — circled) and C = tree (no tree is a stone, no tree is an animal — underlined). Not a syllogism. Reason: you have “no B is A” but nothing is said to be a B, so the said-of-none has nowhere to apply; you need some affirmative. In any figure, two negatives give nothing.
• AE (he is “a little bit more hesitant”): said-of-all needs two affirmatives; said-of-none as it stands needs something said to be a C, but nothing is. A student converts “no C is B” to “no B is C” and derives “some A is not C”; he replies this has A as subject and C as predicate, “reading back in the form of the first figure” — “just playing checkers”; the question is C as subject, A as predicate. Examples: A = animal, B = dog; C = cat (no cat is a dog, every cat is an animal — circled), C = tree (no tree is a dog, no tree is an animal — underlined). Both premises true, so no defect in the matter; nothing follows necessarily — “it’s necessarily not necessary.”
• Closing aside: “Peter Piper picked a pickle pepper.”

His words #

• either-or argument / if-then argument: the compound-statement arguments that matter for reasoning.
• if-then syllogism: his name for the valid if-then argument; he rejects “hypothetical syllogism.”
• form vs. matter: what follows by the arrangement of letters vs. by the terms substituted (convertible matter can mislead).
• middle term: the term common to both premises; the middleman.
• first / second / third figure: middle as subject-and-predicate, predicate in both, subject in both.
• said of all / said of none: his short renderings of dictum de omni, dictum de nullo.
• immediate: a statement not proven by another.
• perfect (syllogism): first-figure, where said-of-all/none is seen as it stands; second and third “imperfect.”
• conversion: turning a statement around; “equivocal by reason” [ed.: unclear phrase in recording].
• converts simply / partially: keeps its universality vs. drops to particular.
• square of opposition: used in the conversion proofs.
• major / minor premise (and term): the first and second premise; middle “between, above, below” the major and minor terms.
• universal moods: the four cases with two universal premises.
• inward philosophy: Thomas’s reading of Boethius: logic and wisdom.

Texts #

Read in class

• Aristotle, Prior Analytics, on the figures and the four universal moods of the first figure (handout “The Form of the Simple Syllogism”, pp. 3–8, per verified sources).
• Handout “The First Definition of Reason”, p. 103 (per verified sources; not named in the transcript).

Mentioned

• Aristotle, Categories (substance said of animal, animal of dog).
• Thomas Aquinas, commentary on Boethius, De Trinitate (he says “I guess the De Trinitate or something like that”), on the “inward parts” of philosophy.
• Thomas on the Father not before the Son (locus not stated).
• Euclid, inscribing and circumscribing circle and square (locus not stated).

His questions #

• Of the four if-then combinations, in how many does something follow necessarily? Two (affirm A, deny B).
• If A then B, and A is not so — does it follow that B is not so? No; examples (dog/animal, dog/four-footed) show B sometimes so, sometimes not.
• Can you show a form is a syllogism by examples? No; examples disprove, only understanding the form proves.
• Any other possibility for the middle term besides between, above, below? No; hence exactly three figures.
• If every B is A, must every A be B? No; but some A is B, necessarily.
• If no B is A, must no A be B? Yes; proved by naming the supposed A-that-is-B “x.”
• If some B is not A, does some A is not B follow? No (animal/dog).
• No B is A, no C is B — does anything follow about C and A? Nothing; two negatives give nothing.
• Every B is A, no C is B — anything with C as subject, A as predicate? Nothing; cat/tree examples.