Form and Matter of the Syllogism: Either-Or and If-Then Syllogisms, Right Opinion, and Proportion

Orientation #

He picks up “laying down the law” from last time as the image for premises laid down firmly and ordered to a conclusion (prev. lecture). He says that this morning they did the first chapter of the Natural Hearing (the confused known before the distinct) and that next time they go to the next group of readings, “the form of the simple syllogism”, where instead of four cases there are 48. Verified sources indicate handout pages used: Aquinas’s commentary on Physics I, lectio 1; “Note for Book One of Natural Hearing, Reading I” p. 2; “Note on End or Purpose of Man” p. 4; “The Skopos or Aim of the Categories” p. 30. The previous class distinguished the three kinds of syllogism and the valid and invalid conditional forms with Euclid and the Meno; the next takes up the three figures of the categorical syllogism.

The class, in order #

1. Syllogism, premises, form and matter (“form and matter of the syllogism”) #

Point: the syllogism is the measure of arguments, and it can be studied formally, apart from what its terms stand for.

• Syllogism: speech in which a statement follows necessarily because of those laid down. “Laid down” implies firmness (like laying down the law) and looking forward to a conclusion; the Greek word for premise is more explicit than the English, having the idea of stretching forward.
• Albert the Great: the syllogism is the main subject of logic, since the first act is ordered to the second, the second to the third, and the syllogism is the chief tool of the third, the only argument where the conclusion follows necessarily once the premises are laid down.
• Formal logic vs. material logic: whether the conclusion follows vs. whether the premises are necessarily true, only probable, or false. He notes students often do fine “until we started using these letters”.
• Form already visible in the second act: Every B is A / Some B is not A, and No B is A / Some B is A, are contradictions whatever B and A stand for (one true, one false). Every B is A / No B is A cannot both be true but can both be false (every man is sitting / no man is sitting) or one true one false (every man is an animal / no man is an animal; no man is a stone / every man is a stone). The particulars cannot both be false. Reason for abstracting the form: the same form is found in an infinity of statements.

2. The either-or syllogism (“eliminate all but what? One”) #

Point: the disjunctive syllogism has two shapes, distinguished by whether the conclusion is affirmative or negative.

• Three syllogisms: the simple (what the text calls categorical; Aristotle just calls it the syllogism), the hypothetical (if-then), the disjunctive (either-or).
• Affirmative: X is either A or B; X is not B; therefore X is A. With three members (A or B or C) you must eliminate two, so an either-or syllogism may have more than two premises, unlike the categorical (always two) and the if-then (two).
  • Euclid I.6: the two sides under equal base angles are either equal or unequal; unequal is eliminated by a separate if-then argument (cut off from the longer a part equal to the lesser, join, two triangles with equal angle contained by equal sides are equal, so the part equals the whole, lesser to greater); therefore equal.
• Negative: Y is either A or B; X is not A; X is not B; therefore X is not Y (if X were Y it would have to be A or B).
  • Differences he stresses: here you eliminate all, not all but one; the subject of the conclusion is not the subject of the either-or statement but a new term, while the subject of the either-or becomes the predicate of the conclusion; in the affirmative the predicate is the remaining alternative.
  • Example from an objection in the Summa: every virtue is either moral or intellectual (Aristotle’s division); faith, hope and charity are neither; therefore not virtues. Thomas’s answer: that division is of the natural virtues acquired by natural powers, whereas these are built on grace. The form is still valid.
  • Summa contra Gentiles, no name said univocally of God and creatures: a five-part disjunction (genus, difference, species, property, accident), the complete division of names said univocally of many from Porphyry’s Isagoge; separate arguments eliminate each; therefore none.
• Everyday and detective cases: eliminating days for a meeting until Wednesday remains; Sherlock Holmes, when all possibilities but one are eliminated the remaining one “no matter how strange” must be it; three murder suspects, two with alibis.
• The problem is not so much in the form but in having a complete division (material logic): if someone outside the three did the murder, “you can go to prison, or worse”. Any two lines equal or unequal, every triangle equilateral/isosceles/scalene are easy; the fivefold division of univocal names is not, without the Isagoge. Thomas showing the Father is in no way before the Son eliminates every sense of “before” (basically those in the Categories); hard to be sure every sense is eliminated. Athanasian Creed: no before and after in the Trinity. Every number odd or even: same form, infinite uses.
• Student: could “Y is not X” distort the logic? — deferred to conversion, needed for the second and third figures; Aristotle will show with letters that whenever No B is A is true, No A is B is true, for an infinity of statements, and that this fails for Every B is A.
• Student: could you conclude negatively from X is either A or B, X is A, therefore X is not B? — Yes, “same thing, no difference”; the main point is that you eliminate all but one and conclude affirmatively to the remaining one, knowing from the first statement that X is one of them.

3. The if-then syllogism: four forms, two valid (“only two of these forms are syllogism”) #

Point: people are far more easily deceived here; of four forms of if-then speech only two are syllogisms.

• Letters now stand for simple statements: If A is so, then B is so. The if-part is the antecedent (ante = before, anteroom), the then-part the consequent.
• In both compound syllogisms only the first premise is compound; the other premises and the conclusion are simple, because truth about the way things are is in the simple statement. Courtroom: “if John murdered so-and-so, John should be punished” — but did he, and should he?
• Four possibilities: look at the antecedent and find A is so or A is not so; look at the consequent and find B is so or B is not so. Question in each: does anything follow necessarily?
• Classroom practice: four students at the board writing “follows necessarily” or “invalid”; he can hardly ever get four right, and some still get it wrong on the final. You are in deep trouble either way: thinking something true that isn’t, or missing what you could know. On the Soul II: the early Greeks explained how men and animals know but not how they are deceived, so their account was incomplete; men seem more deceived than knowing.
• One form is really obvious: A is so, therefore B is so. [Recording jumps here; a passage is missing.]
• Scientific hypothesis: if H is so then P is so; P is so; H does not follow — not a syllogism, “some weakness there”. Theories are found more interesting when they predict unknown things: Dirac’s mathematics yielding a negative electron, later the positron found. Sociology’s predictions may be very vague.
• Highest genera (used earlier in the course): if every genus had a genus above it, there would be no most universal names; but there are (being, something — could there be something that isn’t something?); therefore not every genus has a genus above it. Denying the consequent; also stateable as either-or (either every genus is a species or some genus is not). Also: if every genus had a genus before it, every definition would have a definition before it and we’d know nothing by definition.

4. Reasoning before the art: Socrates and Melissus (“men were syllogizing before Aristotle”) #

Point: men reasoned in these forms before Aristotle reflected on them; the dialogues show the valid forms used and part of the truth seen at a time.

• Socrates says “if these things are so, isn’t this necessarily so?” — anticipating the definition of syllogism; he uses the two valid if-then forms, whether or not explicitly aware.
• Meno: virtue can be taught / cannot be taught can’t both be true; mind can’t rest in contradiction, so one argument has a defect. In the Protagoras he questions the premise “there are no teachers of virtue”: parents give lessons — a child made to return candy taken from the counter (justice); “you’ve had enough for today” (temperance). The dialogues observe that we see part of the truth before the whole.
• Socrates also weakens “what directs us to the good is knowledge”: right opinion also directs. Fork in the road to Boston or Providence: the one who thinks correctly which road arrives as well as the one who knows. Lottery hunch wins as much as knowledge would. MacArthur at Inchon: Washington opposed, sent the chief of staff and an admiral, denied responsibility on the day; he reviewed the plan by phone then read his Bible; did he know or have the right hunch? Either way he led to the good.
• Socrates formalises: the great men of Athens led Athens to the good either by knowledge or by right opinion; if they knew they could teach, they couldn’t teach, therefore they didn’t know (denying the consequent); therefore by right opinion. Either-or backed by if-then; plus the categorical “what directs us to the good is knowledge; virtue directs to the good; therefore virtue is knowledge”. All three kinds in one passage.
• Melissus: if being had a beginning it has an end; being has no beginning; therefore no end — arguing from a form that is not a syllogism, as Aristotle points out in the first book of the Natural Hearing. (The transcript’s next phrase “arguing from the two that are… syllogisms” is garbled; his point is that Melissus’s form is invalid.)

5. Proving the forms: obvious to not-obvious, then the invalid ones (“if it isn’t always so, it isn’t necessarily so”) #

Point: the not-obvious valid form is shown through the obvious one, and then itself used to show the other two invalid.

1. From the obvious (A so, so B so) to the not-obvious (B not so, so A not so): either A is so or not; if A were so, then by the first form B would be so; “if A then B”, “A is so”, “B is not so” are incompatible; therefore A is not so.
2. Invalid forms: (a) A not so — dog/animal: “if I am a dog I am an animal; I am not a dog; therefore not an animal” fails, since I may be a man, another animal; (b) B so — “I am an animal” may hold because I am a man, not a dog. Why examples suffice: if it were necessarily so it would always be so; it is not always so; therefore not necessarily so — denying the consequent.

• Continuous syllogisms (domino effect): if A then B, if B then C, if C then D; reach something not so and everything before turns over; or “look before” until you reach something you know is so (A) and syllogize down to D. “Looking before and after”, Shakespeare. Euclid I.6 has several such steps. Often the intermediate “but B is so” is left unspoken; it is the same syllogism repeated, the conclusion of one a premise of the next.
• Student: Aristotle doesn’t do this explicitly? — He talks about it but there is no book on the hypothetical syllogism; later treatments complicate it beyond need by multiplying cases according to affirmative and negative statements; four possibilities, two valid, is enough.

6. Reasoning from proportion: likeness, whole and part, purpose of man (“what is to man as seeing is to the eye”) #

Point: a proportion has four terms and so cannot go into a categorical syllogism but can into an if-then.

• Fourth tool of dialectic, likeness: Aristotle says the mind is exercised more in seeing likeness between distant things and likeness of ratios.
• Natural Hearing I ch. 1 (this morning): the confused more known than the distinct — salad dressing, symphony, painting, parts not distinguished at first; comparison whole:part :: general:particular. “Particular” comes from “part”; Greek katholou from kath’ holou, “according to the whole”; the general is sometimes called the universal whole. So: if the whole is known before the parts, the general is known before the particular; and the whole is known first; therefore the general is.
• Categorical syllogism has three terms (middle term used twice: good/knowledge/virtue); a proportion has four, so put it as if-then. Mathematicians: if x:y as a:b, then if x = y, a = b; if x > y, a > b.
• Shakespeare, “What is a man if his chief good and market of his time be but to sleep and feed? A beast, no more”: proportion chief good of man : man :: chief good of beast : beast, alternated as in math; if the chief good of man is no more than the beast’s, man is no more than a beast; man is more (deny it and we’d cage you); therefore his chief good is more than sleep and feed — denying the consequent. Could also run it the obvious way (man is more, therefore his good is more); Shakespeare picks the not-obvious form and understands these things naturally.
• Purpose of man: what is to man as seeing is to the eye? — the act of reason; if seeing is the purpose of the eye, the act of reason is the purpose of man; and seeing is; therefore. Refined: seeing well (we wear glasses), so the act of reason done well.
• Three-term arguments can be recast as if-then (if three is odd, it is not even), but four-term ones cannot go into the categorical form.
• Closing: next time the form of the simple syllogism, 48 cases; he saves “the worst to last”. Aside: Brother Richard taught him logic in high school; he got college credit by exam; “do it in my sleep”.

His words #

• laid down: premises at rest firmly in the mind and ordered forward to a conclusion; Greek word for premise has the sense of stretching forward [ed.: protasis]
• formal logic / material logic: whether the conclusion follows vs. whether the premises are necessary, probable or false
• simple syllogism: what the text calls categorical; Aristotle calls it just the syllogism
• either-or syllogism [ed.: disjunctive], if-then syllogism [ed.: hypothetical]
• antecedent / consequent: the if-statement (ante = before) and the then-statement
• right opinion: true opinion, not knowledge, that directs to the good as well as knowledge does
• continuous syllogisms: the conclusion of one is a premise of the next; “domino effect”
• Natural Hearing [ed.: Physics]
• katholou: Greek for general, from kath’ holou, “according to the whole”; particular from “part”
• universal whole: the general as a whole with respect to its particulars
• middle term: the term used twice in a three-term syllogism
• tool of likeness: fourth tool of dialectic, likeness of things far apart and of ratios

Texts #

Read in class

• Euclid, Elements I, prop. 6 (either-or plus if-then structure retraced)
• Aristotle, Physics I ch. 1 with Aquinas, lectio 1 (handout; whole/part, general/particular)
• Shakespeare, “What is a man if his chief good and market of his time be but to sleep and feed” (Hamlet, inferred)
• Handout “Note on End or Purpose of Man” p. 4 (seeing : eye :: act of reason : man)

Mentioned

• Thomas Aquinas, Summa theologiae, objection in the article on faith, hope and charity as virtues (locus not stated)
• Thomas Aquinas, Summa contra Gentiles, no name said univocally of God and creatures (locus not stated)
• Porphyry, Isagoge (five names said univocally)
• Thomas Aquinas on the Father not before the Son; Athanasian Creed
• Aristotle, On the Soul II, early Greeks on knowing vs. being deceived
• Plato, Meno; Plato, Protagoras
• Aristotle, Physics I, on Melissus
• Aristotle, fourth tool of dialectic (Topics, inferred)
• Handout “The Skopos or Aim of the Categories” p. 30 (verified source; senses of “before”)
• Sherlock Holmes; Dirac and Heisenberg; MacArthur at Inchon

His questions #

• For an affirmative conclusion from an either-or, how many possibilities do you eliminate? All but one; for a negative, all.
• If two lines are unequal, what follows in Euclid I.6? You cut off from the longer a part equal to the lesser, get equal triangles, and the lesser equals the greater — impossible; so the lines are equal.
• Is the difficulty with the either-or argument in the form? No, it is having a complete division — material logic.
• Which of the four if-then forms is really obvious? If A then B, A is so, therefore B is so.
• Why are examples enough to show a form invalid? If necessarily so it would always be so; the examples show it isn’t always so.
• Did MacArthur know Inchon would succeed, or think correctly that it would? Left open; either way he led to the good, as Socrates’s great men of Athens by right opinion.
• Why can’t you put a proportion into a regular syllogism? It has four terms; the regular syllogism has three.
• What is to man as seeing is to the eye? The act of reason; and as seeing well is to the eye, the act of reason done well.