The Syllogism: Set of All, Set of None, and the Sixteen Moods of the First Figure

Orientation #

He opens by contrasting today’s “simple syllogism” with the either-or and if-then syllogisms already treated (prev. lecture), and says the second and third figures could not be done until conversion had been studied. He promises to put all the valid forms back on the board “when you get through all 16” and to show later that only one of the 48 forms concludes “every C is A”. The previous class covered disjunctive and conditional syllogisms; the next goes systematically through the second figure.

The class, in order #

1. The simple syllogism: the conclusion is “another statement” (“in the power of the premises”) #

Point: the definition of syllogism (speech in which, some statements laid down, another follows necessarily because of them) is more true of the simple syllogism than of the either-or or if-then.

• Either-or: the conclusion is “already there, but not yet asserted”; if-then: already there but “not yet said in fact to be so”; simple syllogism: “every animal is alive, every man is an animal” — “every man is alive” is “in no way really in the premises actually”, only in their power.
• Example of a fine either-or seen “this morning” in Thomas: the relation of God to creatures is a relation of reason, not real. If real, it is either an accident in God or his very substance; “the first book, the previous part” showed no accidents in God; if his substance, God would be towards the creature and depend on it, against his being necessary through himself. Both branches excluded, so it is not really in God.

2. Analysis, the three terms, the middle man (“we call this the syllogism … analysis”) #

Point: the two Analytics take the syllogism apart; the first checks whether the conclusion follows, the second whether the premises are true.

• “Prior” = before; “analytic” = take apart. Proportion: checking whether you added correctly, then checking whether you had the right numbers.
• Taking a syllogism apart: two premises; subject and predicate give not four terms but three, one found in both premises.
• Middle term vs. middle man: the shopkeeper who has contact with producer and consumer and “brings them together”; the matchmaker who threw a party at which everyone else was married so the two had to talk (they married and had children) — the affirmative statement. Someone stepping in to separate two fighting men — the negative statement (“This is not that”).
• Three relations of the middle term to the others = three figures: (1) in between, subject of one premise and predicate of the other; (2) predicate in both; (3) subject in both. Aristotle uses different Greek letters for each arrangement; we use A, B, C for all three, so “the order of the letters doesn’t necessarily correspond to the order of the terms” except in the first figure. The predicate is usually the more universal (“man is an animal” rather than “an animal is man”).

3. The question and the sixteen cases (“four times four, right?”) #

Point: in each figure we ask whether any statement with C as subject and A as predicate follows necessarily — universal or particular, affirmative or negative, the four of the square of opposition.

• Each premise has four possible forms; 4 × 4 = 16 combinations, “not four plus four”: four men and four women give sixteen possible marriages.
• Grouping: four cases with two universals, four with two particulars, eight mixed. The universal cases are the most important.
• In most cases nothing follows: “Bad for most part. The good is rarely.”

4. The two obvious principles: set of all, set of none (“the set of all and the set of none”) #

Point: reasoning must start from something obvious; two such principles underlie every genuine syllogism.

• Set of all [ed.: dictum de omni], in Aristotle’s grammatical form: if A is said of all B, A is said of whatever B is said of (animal said of all dogs → said of whatever dog is said of). Our form: if every B is A, whatever is a B is an A. Obvious? Yes — to deny it is to say something is a B and not an A, so not every B is A.
• Set of none [ed.: dictum de nullo]: if A is said of none of B, A is denied of whatever B is said of (cat said of no dog, dog said of all cocker spaniels → cat denied of all of them). Our form: if no B is A, whatever is a B is not an A.
• Both require (a) a universal statement and (b) an affirmative statement placing something under its subject. Hence two guesses before any proof: two particular premises give no syllogism (no universality); two negative premises give none (even set of none needs an affirmative). “But we won’t just guess that”: examples will show it.
• First figure: if valid, the principle is obvious “as it stands”; second and third figures: never obvious as they stand, statements must be turned around, so Aristotle calls them imperfect — the reason conversion had to be studied first.

5. How to disprove by examples, and why it suffices (“check it out”) #

Point: to show “nothing follows necessarily” you need examples satisfying two conditions.

• Names: predicate of the conclusion = major term (the more universal in the first figure), subject = minor term, the term in both premises but not the conclusion = middle term; major premise contains the major term, minor premise the minor.
• Worked case: every B is A, no C is B. Neither principle applies: no set of all (a negative is present), no set of none (the universal negative “no C is B” has nothing placed under C). So suspect no syllogism.
• Naïvely you would need four examples (one falsifying each possible conclusion); simplify: one example where “every C is A” is true falsifies both negatives, one where “no C is A” is true falsifies both affirmatives. So keep A and B, find two C’s.
• Condition 1: premises true on substitution. Condition 2: one C making “every C is A” true, one making “no C is A” true.
• Examples: animal / dog; C = cat (always animal, never dog) and stone (never dog, never animal). “Now check it out”: every dog is an animal, no cat is a dog, no stone is a dog — true; every cat is an animal; no stone is an animal.
• Why this suffices (the chain he wants students to give in “a separate question” on the exam): if syllogism, something is necessarily so; if necessarily so, always so; the examples show nothing is always so; therefore nothing necessarily so; therefore no syllogism.
• Aside: “Wisely and slow; they stumble to run fast” (Shakespeare) — pause and verify the two conditions.

6. The four universal cases of the first figure (“Does the set of all or the set of none apply”) #

Point: two of the four are syllogisms, two are not.

1. Every B is A, every C is B → every C is A, by the set of all (a universal affirmative with every C placed under the B’s).
2. No B is A, every C is B → no C is A, by the set of none.
3. Every B is A, no C is B — “a form that is apt to deceive people”. No principle applies (nothing under the C’s). Examples: plant / tree; bush (never tree, always plant), stone (never tree, never plant). Not a syllogism.
   • The deceptive matter: every mother is a woman, no man is a mother — students conclude “no man is a woman”, which is true but does not follow; the true matter deceives, whereas “every dog is an animal, no cat is a dog” would deceive nobody. “A crafty logician” uses this form with all-true statements to expose those who don’t really know logic.
4. No B is A, no C is B — two negatives, no principle possible. Examples: animal / plant; cat, stone. Not a syllogism.

• People call these “invalid syllogisms”; better “not a syllogism at all”.
• Very important for demonstration: case 1 is “the only one of all the 48” that concludes every C is A (more than one concludes no C is A), because you prove a property of a subject through the subject’s definition; “we tend to use this form as our style of syllogism” — “watch your habits”, he says, citing the Posterior Analytics.
• Aside (to the cat): “Hi kitty, you want some logic?”

7. The four cases with two particular premises (“if you’re lazy”) #

Point: no universal, so no principle; one set of examples can dispose of all four forms at once.

• Forms: some B is A / some C is B; some B is A / some C is not B; some B is not A / some C is B; some B is not A / some C is not B.
• From Porphyry’s Isagoge (genus, difference, species, property, accident): an accident can be present or absent, so it belongs to some and not to some. A = animal; C = dog (always animal) and stone (never animal); B = white: some white things are animals, some not (works for either major); some dogs are white, some not; some stones are white, some not (works for either minor).
• Check: condition 1 met for all four; condition 2: every dog is an animal (circled) knocks out the negatives, no stone is an animal (underlined) knocks out the affirmatives. “You ain’t got a syllogism.”

8. Mixed cases: universal major, particular minor (“Two negatives, yeah”) #

Point: of the four (every B is A / some C is B; every B is A / some C is not B; no B is A / some C is B; no B is A / some C is not B), the double negative is known at once to lack both principles.

• Reminder for building examples: “some C is not B” is true even when no C is B — “some” does not mean “some are and some are not”.
• Examples for no B is A, some C is not B: animal / plant; cat, stone: no plant is an animal, some cat is not a plant, some stone is not a plant — condition 1 satisfied. He does not spell out condition 2 for this case before the digression begins; the other three mixed forms of this group, and the second group of four, are not reached.

9. Digression: the conversion rules re-proved (“You yourself can do infinity”) #

Point: “a little induction” from the example turns into a full review of conversion.

• Universal negative: if true, its converse is true; if false, its converse is false (else, by conversion, the original would be true). This holds for an infinity of true and an infinity of false universal negatives — “Amazing, man.”
• Universal affirmative does not convert to universal: one example suffices — every dog is an animal / every animal a dog; every woman a human being / every human being a woman. Cases where both are true: every two is half of four and vice versa; a thing and its definition (“every square is a quadrilateral, and every quadrilateral is a square, right?” [ed.: as spoken; a few lines later he says not every quadrilateral is a square, giving it as the counter-case]). A definition should be convertible — Socrates turns definitions around: “dog = four-footed animal” — “but is every four-footed animal a dog?”; likewise a property in the strictest sense. But B may be less universal than A.
• Universal affirmative converts partially, by reasoning (you cannot survey all cases): if “some A is B” could be false when every B is A, then by the square of opposition “no A is B” would be true; by the converted universal negative, “no B is A” would be true; but every B is A — two universals can both be false, never both true. So it drops from universal to particular. This is part of why the second figure yields only negative conclusions: converting the universal negative keeps universality (set of none possible), converting the universal affirmative loses it (no set of all).
• Particular affirmative converts, by the same route (if “some A is B” could be false, no A is B, hence no B is A, impossible when some B is A).
• Student: doesn’t proving the universal negative’s conversion already use the particular-affirmative conversion? Answer: no, that was shown by the example — taking something that is both A and B and giving it a name; you could prove it this way, but Aristotle shows the universal negative first as the most important and derives the other two from it. Student: but implicitly, since “we had an X” and named it — “yeah, yeah.”
• Particular negative does not convert: some animal is not a dog / some dog is not an animal; some number is not odd / some odd number is not a number. One example is enough.
• The asymmetry: universal negative is most useful (keeps universality), particular negative useless, the two affirmatives in between (keep at least a particular).

10. Beginning the second figure (“your middle term is predicated in both cases”) #

Point: in the second and third figures you must “fiddle around” to reach the set of all or set of none by conversion; if you find examples satisfying the two conditions, you know conversion “never gets anywhere”.

• Four universal cases: every A is B / every C is B; every A is B / no C is B; no A is B / every C is B; no A is B / no C is B.
• Two negatives known at once to be no syllogism. Examples: animal / dog, stone; middle B = tree: no animal is a tree, no dog is a tree, no stone is a tree; every dog is an animal (knocks out the negatives), no stone is an animal (knocks out the affirmatives). Not a syllogism — “that’s in the very definition of syllogism.” The recording ends here.

His words #

• syllogism — speech in which, some statements laid down, another follows necessarily because of those laid down.
• in the power of the premises — how the conclusion of a simple syllogism is present, as against “already there” in either-or / if-then.
• analysis / Analytics — “take apart”; Prior Analytics takes arguments apart for whether the conclusion follows, Posterior for whether the premises are true.
• middle term / middle man — the term in both premises that puts the other two together or separates them.
• figure — arrangement of the middle term: in between (first), predicate of both (second), subject of both (third).
• moods / modes — the sixteen cases in each figure.
• set of all — if A is said of all B, A is said of whatever B is said of [ed.: dictum de omni].
• set of none — if A is said of none of B, A is denied of whatever B is said of [ed.: dictum de nullo].
• as it stands — the principle visible without conversion (first figure only).
• imperfect syllogism — Aristotle’s name for second- and third-figure syllogisms, “not clear” as they stand.
• major / minor / middle term; major / minor premise — predicate of the conclusion, subject of the conclusion, term in both premises; the premise containing each.
• the two conditions — (1) premises true on substitution; (2) one example where every C is A, one where no C is A.
• not a syllogism — his preferred phrase for what others call an “invalid syllogism”.
• converts partially — the universal affirmative drops to a particular affirmative on conversion.

Texts #

Read in class

• handout, “The Form of the Simple Syllogism”, pp. 3–7 (verified) — the forms worked on the board.
• handout, “Reasoning and the Four Kinds of Argument”, p. 6 (verified) — definition of syllogism and its comparison with either-or / if-then.

Mentioned

• Aristotle, Prior Analytics (definition, figures, the sixteen cases, conversion; “Aristotle shows the universal negative first”).
• Aristotle, Posterior Analytics (proving a property through the definition; “watch your habits”).
• Porphyry, Isagoge (genus, difference, species, property, accident; accident may be present or absent).
• Thomas Aquinas, “first book / previous part” of an unnamed work (no accidents in God; God necessary through himself; relation of God to creatures a relation of reason) — Summa Contra Gentiles (inferred).
• Shakespeare, “Wisely and slow; they stumble to run fast” (quoted).
• Socrates turning a proposed definition around (“four-footed animal”).

His questions #

• Is the “set of all” obvious, or does it need a proof? Obvious: to deny it is to say something is a B and not an A, i.e. not every B is A.
• How many possible combinations of two premises are there in a figure? Sixteen — four times four, like four men and four women.
• Can you think of something always an animal but never a dog? never a dog and never an animal? Cat (or man, horse); stone.
• Every mother is a woman, no man is a mother — does “no man is a woman” follow? No; the matter is true but the form is not a syllogism.
• Why do examples satisfying the two conditions suffice to show it is not a syllogism? A universal affirmative true once excludes the negatives, a universal negative true once excludes the affirmatives; nothing always so, so nothing necessarily so, so no syllogism.
• If “no B is A” is false, what about “no A is B”? False too — otherwise, by conversion, the original would be true.
• If every B is A, is it necessarily true that every A is B? that some A is B? Not universally (dog/animal); yes, some A is B, by the square of opposition and the universal-negative conversion.
• Can a particular negative be turned around? No: some animal is not a dog, but not some dog is not an animal.
• In the second figure, which universal case is known at once not to be a syllogism? The two negatives — not even the set of none is possible.