Analyzing Arguments in Euclid and Aquinas: Main Syllogisms, Continuous Syllogisms, and the Matter of Syllogism

Orientation #

He refers to “the first page” (all first-figure syllogisms, incl. Euclid I.1 and the Thomas example) and to “the last time I was here”, when he mentioned the Summa Contra Gentiles argument; the eight forms recur (prev. lecture). Next time: “a little bit about the matter of syllogism”, the difference between dialectic, demonstration and sophistical argument, plus an exercise on equivocal names. The previous class worked the eight syllogistic forms through Euclid I.1 and Aquinas; the next treats demonstration vs. dialectic and the notion of per se.

The class, in order #

1. Euclid I.6: the main syllogism is either-or (“uses all three kinds of syllogisms”) #

Point: proposition six is perhaps the first in Euclid that uses all three kinds of syllogism: the “syllogism period” (what some call the simple categorical syllogism), the if-then, and the either-or.

• To prove: in triangle ABC, if angles ABC and ACB are equal, sides AB and AC are equal.
• “Main syllogism” = the one whose conclusion is the main conclusion, not the most interesting one. Here it is either-or: two straight lines are either equal or unequal, no other possibility; exclude unequal, conclude equal. He does not prove equality “directly” but shows the other possibility impossible.
• He notes this form is so obvious it is often not spelled out: “it sounds almost childish, but in a sense that’s what he’s arguing.”

2. Eliminating “unequal”: if-then and the simple syllogism (“the lesser can’t be equal to the greater”) #

Point: the alternative is removed by an if-then argument whose consequent is shown by a categorical syllogism resting on Prop. 4 and an axiom.

1. If AB, AC are unequal, one is longer; say AB. Then (by an earlier theorem, cutting off from a greater line a line equal to a given line) cut BD on AB equal to AC, and join DC.
2. Triangles DBC and ACB: angle DBC = angle ACB (given), sides DB, BC = sides AC, CB (DB constructed equal to AC; BC common). So they have “equal angles contain[ed by] equal sides.”
3. Simple syllogism: triangles having equal angles contained by equal sides are equal (Prop. 4, the same one used in Prop. 1); DBC and ACB are such triangles (from the constructions); therefore they are equal.
4. But DBC is clearly a part of ACB, hence lesser; the less cannot equal the greater. Absurd; so the lines cannot be unequal.

• Student: is the middle term Proposition Four? He: Prop. 4 is the premise; “triangles having equal angles contained by equal sides” is the middle, and the constructions show these triangles fulfil the two conditions.
• He stresses that Euclid resolves “all the way back to the axioms” — the whole is greater than the part — “the statements known to themselves by all men.”
• Aside: the triangles are “turned around”, but one can turn them and see they are the same.

3. Euclid, prop. 29: prime numbers (“prime to any number which it does not measure”) #

Point: a second elementary theorem (“not particularly interesting or profound”) again uses all three kinds, and here the categorical syllogism is second figure, unlike the first page where all were first-figure affirmative.

• Definitions he gives: a prime number is measured only by one and by no other number (2, 3, 5, 7, 11; 4, 6, 8, 9, 10, 12 are composite). Two numbers are prime to one another when no number measures both; 3 and 9 are not, since 3 measures both — “you got to understand that way of speaking.”
• Theorem: let A be prime and not measure B; then A and B are prime to one another.
• Main syllogism (student first guessed if-then; he: either-or): A, B are either prime to one another or not (“prime or composite to one another”); they cannot be not-prime; therefore prime.
• Elimination, by if-then: if not prime to one another, some number C measures both.
• Second-figure syllogism: C measures B; A does not measure B; therefore C is not the same as A — “affirming and denying the same thing” (measures B) of C and of A.
• Then: C measures A, which is prime; but no prime number is measured by any number except perhaps itself (3 measures 3), and that possibility is excluded since C is not A. Impossible; so no number measures both; so the other alternative holds.

4. Thomas: relation of God to creatures (“not really in God”) #

Point: the same three kinds appear in theology.

• SCG argument: if the relation were really in God, it would be either God’s substance or an accident of God. Accident: eliminated by a previous chapter (no accidents in God). Substance: then God would be his very substance towards creatures and so depend on creatures; but God in no way depends on creatures. So either-or, if-then and categorical all present.
• Aside: such analyses “drive students crazy… that’s laudable too,” but the real reason is to show that even elementary theorems may need all three kinds.

5. Continuous syllogisms and continuous definitions (“the end of one is the beginning of the next”) #

Point: when the conclusion of one syllogism is a premise of another, he calls them continuous syllogisms.

• Likeness to the mathematician’s continuous proportion: 4:6 and 6:9 is continuous, the end of one being the beginning of the next; 2:3 and 4:6 is not. Metaphorical likeness: the end of America is the beginning of Canada or Mexico.
• Continuous definitions: one definition inside another, as square is defined by quadrilateral, which has its own definition.
• Recap of the examples: Euclid I.1, two syllogisms whose conclusions were needed for the minor of a third; Thomas, a syllogism with one backing the major and one the minor; the two Euclid theorems today, an either-or backed in a premise by an if-then, backed in a premise by a simple syllogism. “Those eight forms are going to be used again and again, sometimes just one, sometimes two or three.”

6. Meno and the use of the forms (“know a few basic things and to use them”) #

Point: the Meno’s first argument shows a simple syllogism backing the second premise of an if-then.

• If virtue is knowledge, it can be taught (obvious); virtue is knowledge; therefore it can be taught. “Virtue is knowledge” shown by: virtue directs us to the good; what directs us to the good is knowledge; therefore virtue is knowledge.
• Euclid Book I: one continuous syllogism after another, A proving B, B proving C, a long chain at the end.
• He stresses Aristotle’s counsel in the logic: better to know a few basic things and use them often and well, since they serve so many things; one need not always stop and analyze, but suddenly “Oh, I see.”
• The forms run “from Euclid, the first science, to theology, the queen of the sciences.”

7. Induction and the return to the senses (“the mind wants to return always to the senses”) #

Point: besides syllogism, induction is used, and in two senses.

• In the Phaedo (inferred; ASR “theater”) Socrates induces the change between contraries, then syllogizes to the immortality of the soul (first argument).
• First meaning of induction: from many singulars to one universal. Second sense: from many less universal things to one more universal — a “kind of philosophical example”, from one particular kind to another kind rather than singer to singer.
• As Dionne [ed.: name uncertain in recording] often pointed out, Aristotle gives the syllogism and then a sign, a weaker argument: to take us back to the senses, “the alpha and the omega,” the beginning of our knowledge.

8. The matter of syllogism: necessary, probable, false (“the rigor of the syllogism doesn’t mean”) #

Point: form’s rigour does not make the conclusion true; premises may be necessarily true (even seen as such), only probable, or simply false.

• Comparison to calculating: correctly adding does not give the right number if the numbers added were wrong — which is why he defines calculating as “coming to know or guess a number from other numbers.”
• Example of valid form with false premises: every man is a stone, every stone is an animal, therefore every man is an animal — “not a good argument because the premises are clearly false.”
• Aristotle takes Melissus apart in Physics I: the conclusion does not follow and the premises are neither true nor probable; Aristotle is “very kind” — Melissus did not have logic.
• Meno: Socrates reasons that virtue can be taught and that it cannot; contradictory conclusions cannot come from necessary reasoning, but dialectic reasons from probable opinions to contradictory conclusions. Demonstration, from necessary statements, “always reason[s] to one half of a contradiction.”

9. Next time: equivocals, fallacies, three distinctions (“the three that are most common”) #

Point: assignment and preview.

• Exercise: is the first name said equivocally or univocally of the second? In the Isagoge’s five predicables a name is said of many with the same meaning; sometimes with different meanings.
• Fallacies: thirteen, all eventually to be learned; he does the first in language and the first two outside language. Most common: equivocation (mixing senses of a word), the accidental (accidental vs. as such), and simply vs. in some respect. These correspond to three distinctions (“not really divisions”) met “again and again” in philosophy; not understanding them makes philosophy hard.
• Aside: his Love and Friendship course; he refused a demand to “put sex in there”; scheduled at 8:30 a.m. to cut numbers; students expecting fluff found it “all real loud” [unclear in recording].

His words #

• “syllogism period” / simple syllogism: Aristotle’s name for what others call the categorical syllogism.
• if-then syllogism [ed.: hypothetical]; either-or syllogism [ed.: disjunctive]: eliminate all but one possibility when one must be the case.
• main (chief) syllogism: the one whose conclusion is the main conclusion.
• prime number: measured only by one and no other number.
• prime to one another: no number measures both.
• axioms: statements known to themselves by all men.
• continuous syllogisms: conclusion of one is a premise of the next (by likeness to continuous proportion); continuous definitions likewise.
• induction: from many singulars to one universal; second sense, from many less universal to one more universal.
• calculating: coming to know or guess a number from other numbers.
• matter of syllogism: the premises as necessary, probable or false.
• dialectic vs. demonstration: probable opinions allowing contradictory conclusions vs. necessary truths yielding one half of a contradiction.

Texts #

Read in class

• Euclid, Elements I.6 (with I.4 and the axiom “the whole is greater than the part”)
• Euclid, Elements, “number twenty-nine”: any prime number is prime to any number it does not measure (Book VII, inferred) Mentioned
• Euclid, Elements I.1; the earlier theorem cutting off a line equal to a given line
• Thomas Aquinas, Summa Contra Gentiles, relation of God to creatures not really in God
• Plato, Meno, virtue teachable / not teachable
• Plato, Phaedo (inferred), argument from contraries to immortality
• Aristotle, Physics I, against Melissus; Aristotle’s logic on knowing few things and using them
• Porphyry, Isagoge, five predicables

His questions #

• What kind of syllogism is the main syllogism in I.6? Either-or: the lines are equal or unequal; unequal is excluded.
• How does he eliminate the possibility that they are unequal? By an if-then argument: if unequal, the lesser triangle equals the greater, which is impossible.
• Where is the simple syllogism? Triangles with equal angles contained by equal sides are equal (Prop. 4); DBC and ACB are such; so equal.
• Is the middle term Proposition Four? Prop. 4 is the premise; the middle is “having equal angles contained by equal sides.”
• Are three and nine prime to one another? No, in Euclid’s way of speaking: three measures both.
• What kind of syllogism, and what figure, is “C measures B, A does not measure B, so C is not A”? A syllogism in the second figure, affirming and denying “measures B.”
• How do you know A and B can’t be composite to one another? Some C would measure both, C is not A, so C measures the prime A — impossible.
• Why does Aristotle add a sign after the syllogism? To take us back to the senses, the beginning of our knowledge.