Compound Statements and the Square of Opposition

Orientation #

He refers back to “as I was pointing out earlier today” (that “some B is A” does not deny “every B is A”) and to “the rule of two or three” and an argument of Thomas “I was giving an example there” (prev. lecture, not resolved by the context block). For next time: he will hand over a paper, “Reasoning and the four kinds of argument,” about seven or eight pages, to be reproduced for the class; “next Tuesday we’ll get done that paper.” Verified sources: the class works from the handout “The Art About Statement,” p. 6 (read from for about 10 minutes), and about 5 minutes matches Aquinas’s commentary on Posterior Analytics I, lectio 40 (probably the demonstration/dialectic contrast).

The class, in order #

1. The if-then and the either-or statement (“the simple statement in the if part is called the antecedent”) #

Point: two of the compound statements matter for reasoning, and both are built out of simple statements.

• If-then (“hypothetical conditional, they call it sometimes”): two simple statements joined by if and then; the if-part is the antecedent (ante, before), the then-part the consequent. Example: “if this number is two, then this number is even.”
• Either-or: at least two, possibly more, simple statements joined by either-or, trying to exhaust the possibilities: “a number is either odd or even”; “a triangle is either equilateral, or isosceles, or [scalene].” Usually we contract it, so the simple statements are not spelled out as they are in the if-then, but it helps to spell them out (“either a triangle is equilateral or a triangle is isosceles or…”) to see what you are actually doing.
• Euclid I.6 (if the angles are equal the sides are equal) rests on an either-or: either these sides are equal or they are unequal; he shows the unequal case ends with a part equal to the whole, so they must be equal.
• Other compounds: “John and Paul are students” is really two statements (conjunctive), but that kind is not important for reasoning. A student asks whether “man is the only animal that learns logic” is compound; he allows it seems compound: it implies man learns logic and no other animal does.
• Why these two are emphasised: we have reasoning in mind; they return in the syllogism.

2. True and false in the if-then statement (“Socrates is a mother”) #

Point: truth in an if-then cannot be judged from the truth of its simple parts; it means the consequent follows necessarily from the antecedent.

1. Mixing two sweet things you expect a sweet result; so one might expect two true simple statements to give a true if-then.
2. Counter-case: “Socrates is a mother” (false) and “Socrates is a woman” (false), yet “if Socrates is a mother, then Socrates is a woman” is true.
3. Counter-case the other way: “Mary is a woman” (true), “Mary is a mother” (true, Mary the Mother of God), yet “if Mary is a woman, then Mary is a mother” is false: not every woman is a mother.
4. Conclusion: truth and falsity here mean something else than in the simple statement; you must look for something else. Extreme cases were chosen to show this.

• Aristotle in the seventh book of Natural Hearing argues from an if-then whose simple statement is false; some criticise him, and Thomas says “so what,” that is not what truth means in an if-then. Euclid I.6 likewise: “if these sides are unequal, then…” is true though what follows is false; otherwise he could not use it further.
• The answer (a student gives it): the consequent follows necessarily from the antecedent; falsity: it does not follow necessarily. “If he is a mother” does not say he is one.
• Before and after matters (Shakespeare: reason is the ability to look before and after). Reversing “if Mary is a woman then a mother” gives a true statement; reversal is not always true. When it works both ways: “if this number is two, then it is half of four” and the converse, because half of four is a property of two in Porphyry’s strict sense: belongs only to two, to every two, and always. A looser property fails: “if two, then less than ten” is true, “if less than ten, then two” is false, since less than ten is not convertible. Definition and strict property convert (square vs. equilateral right-angled quadrilateral); something more universal does not (woman/mother; even/two).
• Shakespeare: “What is a man if his chief good and market of his time be but to sleep and feed? A beast, no more.” True as an if-then though both parts are false. His parallel: “what is a three if it be half of four?”: if three is half of four, then three is two, true though both parts false. Euclid does the same: if one side were longer you could cut off from it a part equal to the shorter, true, though in fact they are equal.

3. Why if-then truth is not the basic truth; the hypothetical syllogism (“If I won the lottery”) #

Point: the if-then alone leaves you dizzy (“if this, then that… have I won the lottery?”); it does not tell whether Socrates is in fact a mother. So reasoning must return to a simple statement.

• The hypothetical (if-then) syllogism: an if-then statement plus a simple statement, concluding to a simple statement. Shakespeare’s argument: if man’s chief good is to sleep and feed, then man is no more than a beast; but man is more than a beast; therefore his chief good is not to sleep and feed. Euclid: if unequal, then you could cut off and draw the line and get part equal to whole; but that is not so; therefore not unequal, so equal.
• Two if-thens together, concluding to an if-then, “doesn’t get us anywhere as far as knowing the way things really are.”
• Hence “statement is speech signifying the true or the false” is not the same definition for simple and if-then statements: not purely equivocal, but a word equivocal by reason; the connection is that “if Socrates is a mother, then a woman” is true because every mother is a woman, and the Mary case false because not every woman is a mother. Student exchange: is a true if-then knowing the way things are? He answers: not really saying things are or are not; better to say it is based on the way things are. That is the connection.

4. True and false in the either-or statement (“emptied the basket out”) #

Point: a third meaning of true: the members exhaust the possibilities; this depends on the logic of division.

• “A triangle is either equilateral or isosceles” is false, “a number is either odd or even” true. Some derive divide from “empty” (he does not vouch for the etymology): a division should empty out everything in the thing. Basket of triangles: pouring out the equilateral and isosceles leaves something in the basket.
• Two members may be enough or not; the rule of two or three usually holds but is not universally true.
• Thomas, following Porphyry: every name said univocally of many is either genus, species, difference, property or accident; he eliminates each and concludes no name is said univocally of God and creatures. The argument’s goodness depends on that five-fold division being exhaustive, harder to see than the triangle case.
• Even in mathematics you sometimes have to think: crisscrossing equilateral/isosceles/scalene with right/obtuse/acute does not give nine kinds (unlike good/bad crossed with male/female, four real things): there is no right-angled equilateral triangle, but you must reason from interior angles equal to two right angles and the equal angles of the equilateral (three right angles would result); an isosceles can be right-angled (half a square) or obtuse.
• Easy cases: every simple statement is affirmative or negative; two straight lines are equal or unequal (Euclid I.6 again).
• Thomas on how the persons of the Trinity are distinguished: distinction is formal or material; material rests on division of the continuous, nothing continuous in God, so formal; formal rests on opposition; four kinds of opposition from Aristotle’s Categories (contradictory, having and lack [privation], contraries, relatives); three eliminated, so relative distinction. “Are there only four kinds of opposition? That’s harder to see.” Likewise “God is a cause; only four kinds of cause: matter, form, mover, end.”
• Some either-ors are only probable: a man is white, black, brown, red or yellow; a green man on Mars would surprise him, but he does not see these are the only possibilities. Even a simple statement can be only probable.
• We are on page six of the handout. “The end of reason is to know the truth of a simple statement”: the courtroom oath wants the truth of the simple statement, guilty or not guilty, not the if-then (everyone knows if he murdered he should go to jail).

5. Contradictories with a universal subject; the square (“women are beautiful, Mr. Berquist”) #

Point: the bottom of page six extends contradiction to universal subjects, where the plain affirmative/negative pair is not enough.

• “Socrates is wise / is not wise”: no problem. “Man is wise / man is not wise”: does “man is wise” mean every man? does “not wise” mean no man or some men not? Aside: “women are beautiful”; “you mean every one, Mr. Berquist?”
• Square of opposition (“we allow squares in logic, but no circles”): every B is A (every woman is beautiful); no B is A; some B is A; some B is not A.
• Definition recalled: same subject and predicate, one affirmative, one negative, opposed so that both cannot be true and both cannot be false. Singular subject: “President Bush is sitting / is not sitting” is obvious.
• Universals can both be false: every woman beautiful / no woman beautiful; every man wise / no man wise; every man sitting / no man sitting (in this room); every man white / no man white; every human being male / no human being male. Sometimes one is true (every two is half of four; no two is half of four false), but the form does not guarantee it.
• Particulars can both be true: some man sitting / some man not sitting; wise; good; some human being male / not male.
• Argument for the diagonals, regardless of what B and A are: if every B is A, without exception, “some B is not A” must be false; if even one B is not A, “every B is A” cannot be true. “No B is A” denies A of all the Bs, so “some B is A” must be false, and if even one B is A, “no B is A” is false. “That’s kind of obvious stuff, but think about it.”
• Why the square matters: reasoning concerns universal subjects more than singulars.
• “Some B is A” being true does not deny “every B is A” nor assert “some B is not A”: alone at home sitting, “some man is sitting” is true, and for all he knows everyone else is sitting too.

6. Exercise: true, false or unknown from each corner (“one way of kind of exercising your mind”) #

Point: an exam exercise: take one corner as true (then as false) and say of the other three whether true, false or unknown.

• Every B is A true: no B is A false, some B is not A false, some B is A true (every student sitting, so some student sitting).
• No B is A true: some B is A false, every B is A false, some B is not A true.
• Some B is A true: no B is A false; every B is A unknown; some B is not A unknown (“if one is unknown, the other is unknown”).
• Some B is not A true: every B is A false; the other two unknown.
• Every B is A false: some B is not A true; others unknown (false because some are and some are not, or because none are). No B is A false: diagonal true, others unknown. Some B is A false: no B is A true, some B is not A true, every B is A false.
• Summary rule: universal true → particular true, not the reverse; particular false → universal false, not the reverse. “The most important thing to see is that the diagonals are opposed as contradictories.”

7. Demonstration, dialectic, rhetoric (“you argue to only one half of a contradiction”) #

Point: Aristotle’s contrast of demonstration and dialectic uses the opposition of contradictories.

• Demonstration: you know which contradictory is true and must be so, hence the other must be false. Dialectic: only probability, inclined to one side with some fear the opposite may be true; one sometimes reasons to both affirmative and negative, and one side may outweigh the other. Rhetoric even more so.
• Example: the Inchon landing; the chief of naval operations and the chief of staff opposed it, MacArthur for it, dangerous, with a backup plan (“nothing will be lost except my reputation”): the nature of rhetoric, arguing on both sides.
• If one half of a contradiction is seen only as probable, the other cannot be seen as necessarily false; if it were, the first would be necessarily true, not just probable.
• In the third book of Wisdom Aristotle reasons for and against before determining; in geometry, where demonstrations are clear, only one side.

8. Contraries; against circles; universal vs. class (“the trapezium of opposition”) #

Point: the two universals are sometimes called contraries, and diagrams of universals mislead.

• The man who thinks everyone beautiful is contrary in thinking to the man who thinks no one is; further apart, yet not contradictories. Student joke: should be a trapezium of opposition, top longer than bottom; he: the square has been around a long time.
• Circles / Venn diagrams are “very bad”: colouring the part that is makes “some” mean some are and some are not, yet “every man is an animal” makes “some men are animals” still true; you “dissolve into the imagination,” as young Socrates in the Parmenides, imagining the universal spread over everyone like a sail, each having only part of man, when the whole of what man is is found in you.
• Moderns speak of class (a collection) rather than universal (one said of many, not a collection): man vs. mankind; the biologists’ “animal kingdom” as a collection.
• Porphyry’s two pre-logical meanings of genus (Aristotle gives them in the fifth book of Wisdom): the one man from whom a multitude descend (Adam), and the multitude descended (the Adamites). The logical genus is like the first in being one, but not individual; like the second in reference to many, but not a multitude: one said of many. His grandchildren: he is an individual, not said of the multitude descended from him. One must transcend the imagination, which wants a collection.

9. Closing and the “some man / some men” question (“some man is wise, rather than some men”) #

• The logic of the second act is easier than the first, much easier than the third. Next: a handout on reasoning and the four kinds of argument (syllogism, induction, enthymeme, example), ordered, imitating Monsignor Dionne, from those closest to the senses up to the more profound; the syllogism is the main thing studied, but the others matter too.
• Student: why “some man is wise” not “some men are wise”? Because with the plural both can be false: every man is God false, some men are God false, but some man is God true; every man is my father false, some man is my father true; every man here is the abbot false, some men here are the abbot false (only one), some man here is the abbot true; every bishop is the pope false, some bishop (the bishop of Rome) is the pope. If “every B is A” is false, at least one B is not A; it need not be three or four. He admits he formerly said “some men” himself; that gets a mistake.
• Aside: a student recalls yesterday’s argument that “when you define something, you exhaust it”; he mentions the pope’s remarks on the Transfiguration.

His words #

• hypothetical conditional / if-then statement; antecedent (ante, before) and consequent.
• either-or statement; usually contracted; to be true it must exhaust the possibilities.
• conjunctive statement (“John and Paul are students”); not important for reasoning.
• true in the if-then: the consequent follows necessarily from the antecedent; false: it does not follow necessarily.
• Natural Hearing [ed.: Physics].
• property in the strict sense (Porphyry): only, every, always; convertible, as is a definition.
• equivocal by reason [ed.: analogous]: “true” of simple, if-then and either-or statements, not purely equivocal.
• hypothetical syllogism / if-then syllogism: if-then plus simple premise, simple conclusion.
• divide / empty: a division should empty out the thing.
• rule of two or three: usually enough members, not universally.
• formal vs. material distinction; four kinds of opposition: contradictory, having and lack, contraries, relatives.
• square of opposition; universal affirmative/negative, particular affirmative/negative; diagonals; contraries (the two universals).
• Wisdom [ed.: Metaphysics].
• demonstration vs. dialectic vs. rhetoric: necessity vs. probability with fear of the opposite.
• class (a collection) vs. universal (one said of many).
• genus, two pre-logical meanings: the one ancestor; the multitude descended.

Texts #

Read in class

• handout “The Art About Statement,” p. 6 (true and false in either-or; bottom of p. 6 to end: contradictories with universal subject) (verified source)
• Aquinas, Commentary on Posterior Analytics I, lectio 40 (verified source, probably; the demonstration/dialectic contrast)
• Euclid, Elements I, theorem 6
• Shakespeare, “What is a man if his chief good and market of his time be but to sleep and feed? A beast, no more” (Hamlet, inferred)

Mentioned

• Aristotle, Natural Hearing (Physics), book 7, with Thomas’s reply to critics
• Thomas, argument that no name is said univocally of God and creatures (following Porphyry); Thomas on the distinction of the persons of the Trinity (loci not stated)
• Aristotle, Categories (four kinds of opposition); Wisdom (Metaphysics) book 3 (reasoning for and against) and book 5 (meanings of genus)
• Plato, Parmenides
• Porphyry (properties; meanings of genus) (Isagoge, inferred)
• Shakespeare, “reason… look before and after” (Hamlet, inferred)

His questions #

• Is “man is the only animal that learns logic” a compound or a simple statement? Seems compound: man learns logic and no other animal does.
• Is “if Mary is a woman, then Mary is a mother” true? False, though both parts are true; two false parts can make a true if-then (Socrates a mother, a woman).
• What does truth mean in the if-then statement? The consequent follows necessarily from the antecedent.
• “If this number is half of four, then this number is two”: also true? Yes, because half of four is a property of two in the strict sense; “less than ten” would not convert.
• Is “a triangle is either equilateral or isosceles” true, and “a number is either odd or even”? The first false, the second true: truth is exhausting the possibilities.
• Can you crisscross the two divisions of triangles and get nine kinds? No; there is no right-angled equilateral triangle, shown by reasoning from the angle sum.
• What is the contradictory of “every B is A”: “no B is A” or “some B is not A”? The diagonal, “some B is not A”; the two universals can both be false, the two particulars both true.
• If “some B is A” is true, what of “every B is A”? Unknown; and if one diagonal-mate is unknown the other is unknown.
• Why say “some man is wise” rather than “some men are wise”? With “some men” both could be false (every man here is the abbot; some men here are the abbot); “some man” and “every man” cannot both be false.