Second Figure Syllogisms, the Power of the Figures, and Why Universal Negatives Convert

Orientation #

He refers back to “the previous argument” in the Phaedo and to “the four forms that we showed were syllogisms” (prev. lecture); looking ahead he says “we have to study the topic of conversion” and “we’ll see” that the universal negative converts. Verified sources: the class works from the handout “The Form of the Simple Syllogism”, pp. 4–8, and the Phaedo paper, p. 11. The previous class set up the three figures and the dictum de omni / de nullo; the next continues with conversion and the second figure.

The class, in order #

1. Second figure, two universal premises (“by conversion, I can get the set of none”) #

Point: in the second figure (middle term predicate in both) the set of all/none never applies “as it stands”; you must convert.

• No A is B, every C is B. Nothing comes under A (subject of a negative), so no set of none as it stands. But the universal negative turns around: no B is A; keep every C is B; “lo and behold” you are back in the first figure, and no C is A. One conversion.
• Every A is B, every C is B. Would need to convert the universal affirmative to “every B is A”, “but we saw you can’t necessarily do that. The syllogism requires necessity.” Suspicion: not a syllogism; confirm by examples. B = animal; A = dog; C = spaniel and cat. Condition 1: premises true (every dog is an animal; every spaniel / every cat is an animal). Condition 2: one case where every C is A (every spaniel is a dog — circled — kills both negatives) and one where no C is A (no cat is a dog — underlined — kills both affirmatives). Nothing always so, so nothing necessarily so.
• No C is B, every A is B. “More involved”: convert to no B is C, bring every A is B under it → no A is C; but we want C as subject, so convert again → no C is A. Two conversions.
• Conversion “is a way of making clear what is not clear”.

2. How common these negative syllogisms are (“said of one and denied of the other”) #

Point: whenever something is said universally of one term and denied of another, you syllogize this way.

• Example: what changes is composed; God is not composed; therefore God is unchanging.
• Phaedo, soul not the harmony of the body: the soul resists the body (the man fasting), the harmony of the body does not resist the body; there is a harmony of the soul, no harmony of the harmony (as he puts it: “there’s a harmony of the soul, but there’s no harmony of the body”); the soul existed before the body (from the previous argument), the harmony does not. Each: therefore the soul is not the harmony of the body.
• So there are two ways to conclude no C is A in the second figure (said of A, denied of C; or vice versa), one way in the first figure — three ways in all to the universal negative, only one to the universal affirmative. “Those four are the most important.” He adds the two rules not yet proven but suspected: two particulars, no syllogism; a particular and a universal cannot give a universal conclusion.

3. Second figure, two particular premises (“conversion never gives you a syllogism”) #

Point: the four particular pairs (some/some, some/some not, some not/some, some not/some not) are all non-syllogisms.

• Reason: the particular negative does not convert at all, the particular affirmative only to itself; so conversion cannot bring you to the first figure.
• One set of examples “to eliminate all of them”: A = animal, C = dog and stone, B = white — a predicable accident, “present or absent” (his teacher Porphyry, “maybe he’s not my dear friend because he’s not a Christian”). Some animals / dogs / stones are white and some are not, so all four particular premise-pairs are true, while every dog is an animal and no stone is an animal.

4. Second figure, major universal, minor particular (“by magic, you’re back in the first figure”) #

Point: two of the four are syllogisms, but one of them cannot be shown by conversion.

• No A is B, some C is not B. Two negatives — not a syllogism. Examples: animal / tree / dog and stone (no animal is a tree; some dog, some stone is not a tree — “that doesn’t mean some are and some are not, just some”); every dog is an animal, no stone is an animal.
• No A is B, some C is B. Convert the major to no B is A; some C comes under B; Ferio; some C is not A. Again a negative conclusion: “all we’ve had so far in the second figure is negative conclusions.”
• Every A is B, some C is B. Converting the affirmative gives only a particular, so not a syllogism; examples (student: dog for A, animal for B; spaniel and cat for C). Both conditions checked as before.
• Every A is B, some C is not B. Neither set of all (a negative present) nor set of none (no universal negative, particular negative cannot convert) — “according to the rules I’ve given you up to this point” you would guess not a syllogism, “but do you think you could find examples? No.” You can see that some C is not A “and it won’t go away.” Proof: take the contradictory of the conclusion, every C is A; join it to every A is B; by the first figure every C is B, which contradicts some C is not B. The three statements are incompatible, so with the two premises laid down you must reject every C is A and accept its contradictory (contradictories: both cannot be true, both cannot be false). “That’s kind of the exception to the rules I gave.” Student: does this hold of all valid syllogisms? — Yes, “same exact idea”; this one is just harder. He compares the if-then case: “if A is so, B is so; B is not so, therefore A is not so” is shown the same way, since A-is-so with the if-then would include B-is-so.
• Aside: students who “disprove” a real syllogism never satisfy the conditions (“they would have made history”); always go back and check your examples slowly — Einstein’s miscalculations, Heisenberg calculating excitedly with “stupid mistakes”, then slowly, “on the way to the Nobel Prize”.

5. First figure, major universal, minor particular (“all you can conclude necessarily is some C is A”) #

Point: (the transcript resumes on the first figure) two syllogisms, two not.

• Restates why the two conditions suffice, “in the form of a syllogism”: if necessarily so, then always so; nothing always so; so nothing necessarily so; and if a syllogism, something necessarily so.
• Every B is A, some C is B: set of all applies (universal affirmative with something under its subject), but only some C comes under B → some C is A. No B is A, some C is B: set of none → some C is not A.
• Every B is A, some C is not B: nothing is said to be a B → not a syllogism. Examples: dog/animal; cat (every cat is an animal, no cat a dog — “we get every C is an animal”), stone (not a dog, not an animal). (The fourth, two negatives, is dismissed with the tally: “two cases that are syllogisms and two that are not”.)

6. First figure, major particular, minor universal (“the particular is on top as the major premise”) #

Point: none of the four is a syllogism.

• Some B is not A, no C is B: two negatives. Easier to take a case where no B is A: no stone is an animal, so some stone is not an animal. C: cat/dog (never a stone, always an animal), plant (never a stone, never an animal). Aside on “some”: “some students have failed” does not mean some passed; if he has corrected only some exams and they passed, all he can say for sure is “some have passed”. Later: “every student passed, but some students did not pass — that’s not good thinking.”
• Some B is A, every C is B: a universal premise, but nothing comes under its subject. Examples: some animal is a dog; spaniel (every spaniel an animal and a dog), cat (an animal, not a dog).
• Some B is A, no C is B: universal negative present, but nothing is said to be a C. Examples: some dog is an animal; cat (no cat a dog, every cat an animal), stone (neither).
• Why the two conditions suffice (asked separately of students): a universal affirmative true once shows no negative is always true; a universal negative true once shows no affirmative is always true; “go back to the square of opposition”; two examples “kill two birds with one stone” instead of one for each of four conclusions.
• Student: keep the same order but “some A is C, no C is B” — set of none? You could convert no C is B, but then you’d need to convert a particular negative for a conclusion with C as subject — no conclusion.
• Rule summary: first figure — set of all/none applies as it stands or it is not a syllogism and examples can always be found. Second and third figures — never as it stands; sometimes by conversion; if not, suspect invalid and find examples, “one case is a little tricky”, shown roundabout.
• Aside: his standing money challenge — he would wave bills, put “a lien on my salary” for anyone finding examples against the four valid forms, “telegram me from South Africa 40 years from now”; “a crude age: they don’t believe in truth, but they believe in money.”
• Some B is not A, every C is B: no set of all (negative present), no universal negative. Examples: first try plant/animal with a chlorophyll micro-organism as C; then, doubling A instead of C: B = white, C = snow, A = animate and inanimate (some white things are not animate, some not inanimate; all snow is white; snow-animate a universal negative). Then the usual: some animal is not a dog; every spaniel, every cat is an animal; circle “every cat is a dog” [ASR; likely “every spaniel is a dog”], underline no cat is a dog.

7. Tally and the four forms of the first figure (“out of the sixteen cases, then, we have four”) #

Point: 4 syllogisms, 12 not; two among the universals, none among the particulars, two among the mixed with universal major, none with particular major.

• The four: every B is A / every C is B → every C is A; no B is A / every C is B → no C is A; every B is A / some C is B → some C is A; no B is A / some C is B → some C is not A. All four possible conclusions with C subject and A predicate occur.
• Falling off of power: second figure only negative conclusions (no C is A, some C is not A); third figure only particular; a universal conclusion is more powerful, so first > second > third — “one reason why it calls it the third figure”; the first and second figures’ universal cases are what is used “over and over” in geometry, natural philosophy, theology.
• Induction on the premises: in the first figure the major premise (containing the conclusion’s predicate) is always universal, the minor always affirmative; 2 × 2 = 4, so no other of the twelve shares both conditions.
• Mnemonic from high school: in the second figure the middle term is in the second spot (predicate in both); the third is the reverse (subject in both).

8. Why conversion of the universal negative must be proved (“how can Aristotle show for an infinity”) #

Point: second-figure syllogisms depend on “no A is B” necessarily converting, and there are infinitely many true statements of that form, so examples cannot show it.

• Universal affirmative does not convert with necessity: one example suffices — every dog is an animal but not every animal a dog; every odd number a number but not the reverse.
• Aristotle’s proof, using the square of opposition (the logic of the second act presupposes that of the third): suppose no B is A is true but no A is B could be false; then some A is B could be true; let it be true and call the A that is a B X; X is both an A and a B; so there is a B, namely X, that is an A — impossible when no B is A. So you are forced to grant that if no B is A then no A is B. “This is something a monkey can’t do.”

His words #

• set of all / set of none — a universal affirmative/negative with something coming under its subject [ed.: dictum de omni / de nullo]
• as it stands — the set of all/none applying without conversion
• conversion — turning subject and predicate around; “a way of making clear what is not clear”; imperfect second/third-figure syllogisms made clear by returning to the first figure
• the two conditions — (1) premises true with the examples; (2) one example where every C is A, one where no C is A
• circle / underline — his marks for the universal affirmative and universal negative examples
• predicable accident — what may be present or absent (Porphyry)
• Ferio — the first-figure form with universal negative major and particular affirmative minor
• contradictory — opposed so that both cannot be true nor both false
• square of opposition; logic of the second act / third act — statement vs. reasoning
• mixed forms — one universal, one particular premise; major premise = the one containing the predicate of the conclusion
• falling off of power — universal conclusions more powerful than particular
• X — Aristotle’s name for “the A that is a B” [ed.: ecthesis]

Texts #

Read in class: handout “The Form of the Simple Syllogism”, pp. 4–8 (verified); Phaedo paper, p. 11 (verified); Aristotle’s proof of the conversion of the universal negative (Prior Analytics, inferred). Mentioned: Plato, Phaedo (soul not the harmony of the body); Porphyry on the predicable accident; God not composed, therefore unchanging (no locus given).

His questions #

• Which one are you sure right away is not going to be a syllogism? The double negative (no A is B, some C is not B).
• Have I satisfied the two conditions? Premises true with the examples; one case every C is A, one case no C is A.
• Why do those two conditions suffice to show it is not a syllogism? A universal affirmative true once excludes every negative as always so, a universal negative excludes every affirmative; nothing always so, so nothing necessarily so.
• Can you find examples for every A is B, some C is not B? No — it follows necessarily that some C is not A, shown through the contradictory.
• Must that (proof by the contradictory) hold of all the other valid syllogisms? Yes, same idea; this case is only harder.
• What can you say about the major and minor premise in a first-figure syllogism? Major always universal, minor always affirmative.
• If no A is B, is it necessarily true that no B is A? Yes — proved by supposing some A is B, naming it X, and deriving that some B is A.