Demonstration versus Dialectic: Necessity, Per Se Properties, and Opinion in the Posterior Analytics and Topics

Orientation #

He refers back to “the logic of the second act” (contradictory statements) and to “the forty-eight cases and the three figures” of the Prior Analytics as work already done, and says of sophistical syllogisms “we’ll talk about those later on a little bit.” (prev. lecture: the class had been analysing syllogisms in Euclid and had begun the “matter” of syllogisms.) The neighbouring classes treat Euclid’s syllogisms and then the Topics and its four tools.

The class, in order #

1. Demonstration: from premises necessarily true and seen so (“apodeixis and demonstration come from”) #

Point: demonstration is a syllogism whose premises are necessarily true “and seen as necessarily true”; hence a very strong argument.

1. In any syllogism the conclusion follows necessarily from the premises.
2. In demonstration the premises are themselves necessarily true (and seen so).
3. Therefore the conclusion is necessarily true.

• Etymology: Greek apodeixis (hence “apodeictic”, “the horrible word”) and Latin demonstratio both mean “to show”; English “show” likewise implies making something very clear. Missouri’s “show me” and Cebes in the dialogue insisting on a necessary argument illustrate the demand for showing.
• In Euclid each demonstration begins with what he proposes to prove (English “proposition” is weaker than “propose”) and ends with an epilogue, Q.E.D., “what was to be demonstrated.” William F. Buckley used Q.E.D. ironically of opponents’ bad reasoning; in Euclid it is meant literally.
• Clearest examples are in geometry and the science of numbers. Prior Analytics: how to take an argument apart to see if the conclusion follows (the forty-eight cases, three figures). Posterior Analytics: how to examine the premises to see if they are necessarily true; its examples are mainly (not only) mathematical because the demonstration is clearest there.
• Aside: Einstein, in the autobiographical sketch in the Harper volumes Einstein: Scientist, Philosopher (a collection of essays with the subject’s reply), says that if Euclid did not arouse your youthful enthusiasm you were not born to be a scientist: the rigour of one thing shown from another.

2. Why Plato talks demonstration with a mathematician (“the dialogue is called the Theaetetus”) #

Point: Plato’s dialogues are named after the interlocutor because he is someone worth talking to about the topic; so the dialogue on episteme, the knowledge that is the effect of demonstration, is with Theaetetus, an historical mathematician who contributed to Euclid’s Elements.

• Examples of the pattern: Hippias, Protagoras, Gorgias (sophists teaching rhetoric) about rhetoric; Euthyphro, who charged his own father with impiety, about piety; Laches, an old general with experience of courage and its opposite, about courage.
• Demonstrations exist also in natural philosophy “in the beginning,” in logic itself, in wisdom; but the easier ones to get hold of are mathematical.
• Mathematics is named from learning (mathetes, the learner) because there the student most of all can learn rather than believe the teacher; Sacra Doctrina is named from the teacher (doctor), whose teacher is ultimately God, Christ, and belief is greatest there. In geometry belief is only an initial stage: once you see the demonstration you no longer believe Euclid, you know yourself, and you hold on even if the teacher “loses his mind” and starts saying strange things.

3. Dialectical syllogism and the definition of the probable (“syllogism from probable statements”) #

Point: the other main good syllogism is dialectical; just as much a syllogism, but its premises are not necessarily true, or not seen so, only probable.

• Its book, Topica/Topics, is a transliteration, not a translation (compare apologia, a legal defence in court, which English “apology” does not mean; Newman’s Apologia Pro Vita Sua defends the reasonableness of his becoming a Catholic). Greek topos = place; a place is where you look to find a dialectical argument.
• Aristotle’s definition of the probable: opinions of (1) all men, (2) most men, (3) all or (4) most men in a given art or science speaking of its matter, or (5) the most famous in that art or science. He divides these five into two: probability from the number (quantity) of those thinking so vs. from the quality of those thinking so.
• Examples of quality: all or most geometers on the square on the side opposite the right angle; Einstein (three Nobel-level papers in one year) saying scientific hypotheses are not reasoned out but freely imagined; Shakespeare on the purpose of playing being to hold the mirror up to nature; Mozart’s letter to his father on Osmin’s anger in The Abduction from the Seraglio: music must imitate anger but never displease the ear or cease to be music.
• Men most readily follow the experts when they do not contradict most men; but the famous can be ahead of, or opposed to, most men. Boundary case: most men think sense pleasure the greatest thing in life, most moral philosophers (Aristotle) do not; so both sides have some probability.

4. Reasoning to contradictory conclusions (“with fear of the opposite”) #

Point: because a probable statement’s contradictory need not be manifestly false, dialectic can reason from probable opinions to contradictory conclusions; demonstration never can.

1. Contradictories (same subject and predicate, one affirmative, one negative; from the logic of the second act) cannot both be true nor both false.
2. If one side is seen necessarily true, the other is seen necessarily false.
3. If one side is only probable, the other cannot be manifestly false (else the first would be more than probable); so there is probability on both sides, and even when we hold the more probable there is “some fear that the other side might have something in it.”

• Meno, third part: Meno wants to know whether virtue can be taught before knowing what virtue is; Socrates argues both sides. For: virtue directs us to the good; what directs to the good is knowledge; so virtue is knowledge, and can be taught. Against: given virtue’s importance for civic life there would be teachers, but Meno admits none except the sophists, whom he takes for quacks. Then Socrates asks whether one side is weaker: the major “what directs to the good is knowledge” is not necessary, since right opinion, even a correct hunch, also directs.
• Fork in the road to Boston or Providence: the man who merely thinks the right fork goes to Boston arrives as well as the man who knows. The great men of Athens perhaps directed the city by right opinion, not knowledge. MacArthur at Inchon had contingency plans and the Navy and chiefs of staff opposed him; he had at least the right opinion, whether he knew is unclear.
• In another dialogue (the Protagoras, he thinks) the weakness on the other side appears: “no teachers of virtue” is like “no teachers of Greek.” He can name who taught him French; his English was taught diffusely by mother, father, brothers, uncles, aunts; he hired a piano teacher for his children but nobody to teach them English, yet English is teachable.
• Thomas, in the proemium to Wisdom: opinion is held cum formidine, with fear of the opposite. MacArthur going over the whole plan again with a subordinate, then reading his Bible, shows that fear.

5. Sophistical and falsigraphical; demonstrator vs. dialectician (“gives you what a choice”) #

Point: at the beginning of the Topics Aristotle also names the sophistical syllogism (appears to be a good dialectical syllogism, is not) and, from geometry, the falsigraphical, “falsely drawn” (appears to be a demonstration, is not); the two good forms are demonstration and dialectic. Sophistical syllogisms deferred to later.

• Prior Analytics beginning: the demonstrator takes one half of the contradiction as necessarily true and discards the other as manifestly false; the dialectician in conversation gives you the choice, “do you think this is or is not so?”, because he does not see either side as necessary.

6. The necessary and the “through itself” or “as such” (“two is half of four”) #

Point: from the Posterior Analytics, the necessary is connected with the kath’ hauto, per se, “through itself” or “as such”; abstract in statement, clear in examples.

• Two is a number: necessary, and number pertains to what two is (definition). Two is half of four: necessary, not part of the definition but a property in the strict sense, belonging only to two, to every two, always; so also “as such,” in another way.
• Triangle is a rectilineal plane figure contained by three straight lines: genus and difference, anything in the definition belongs to it as such. Interior angles equal to two right angles: not in the definition but follows on its nature; the geometer shows it (draw through one vertex a parallel to the opposite side; alternate angles equal; the three make two right angles). Aristotle says both: if necessary then as such, and if as such then necessary.
• Importance for definition: the definition gives what belongs to a thing as such, and is needed to see what follows on its nature; if the nature is not understood clearly enough, you will not see what follows. Plato first brought out the importance of the kath’ hauto for episteme.

7. What cannot be demonstrated: green triangles, white snow (“all snow is white”) #

Point: what is neither in the definition nor follows on the nature can never be reasoned out, only sensed.

• A triangle is green: could be seen with the senses, never demonstrated.
• Snow is white: he holds it by induction (all the snow he has seen), not because he sees the nature of snow requires it (maybe someone else does); green snow on Mars would surprise him, not be impossible. A man in darkest Africa who had seen only black men would say “all men are black” as we say “all snow is white,” but black is not in the definition of man, animal with reason. So “all snow is white” is for him probable, not seen as necessary.
• Boundary case: someone claiming a triangle without two right angles is crazy or equivocating. The “triangle” from the North Pole to the equator and back has more than two right angles, but is not a plane figure: equivocation on “triangle,” the first sophistical way (to be treated later).

8. Does Aristotle argue in a circle? (“the real demonstration is from necessary”) #

Point: Aristotle reasons from necessary to per se and the reverse; Thomas asks whether that violates his own rule against circles and answers that the real demonstration runs from necessary to through-itself, while the reverse takes as a probable opinion (Plato’s, a common one) that necessary knowledge is about the as such.

• Student: is that because demonstration has to be certain? He: yes; and the stock reasons both ways show how closely the two are connected.
• His own parallel: the first road and the natural road in our knowledge, argued each from the other, because the nature is what is first in a thing (you must be what you are before being anything else); hence “first” stands in the definition of nature in Physics II.
• His term for episteme, “reasoned out knowledge,” helps: one can reason out the two right angles from what a triangle is, never that it is green.

9. Two senses of “as such”; risible and white (“watch out for black cats”) #

Point: the two most basic senses of as such are (A) a part of the definition belongs to the defined as such, and (B) a property belongs to its species as such; in A the predicate defines the subject, in B the property is defined by the species, “just the reverse.”

• Man, animal with reason, is risible (the medievals’ stock property): the incongruity that provokes laughter is seen by reason, the bodily movements of laughing are the animal side; so a connection is seen. White: no connection with animal (black cats) nor with reason; a matter of looking in the mirror.

10. Must a philosopher love reason? (“Must a philosopher love popcorn”) #

Point: yes to reason, no to popcorn; the necessity comes from an as such.

1. Philosopher by definition is a lover of wisdom.
2. Wisdom names the highest perfection of reason (as Homo sapiens, the “wise ape,” is named from excellence of reason).
3. One cannot love the good of a thing without loving the thing, nor love the thing without wanting its good: to love health is to love the body and the reverse; to want your good is to love you.
4. So the philosopher must love reason, and perhaps conversely whoever loves reason must love wisdom.

• Corollary he gives students: man is an animal with reason, so whoever does not love reason, or wisdom, does not love himself.
• Asides: Shakespeare, “love and reason keep little company nowadays,” refers to romantic love. The Naked Ape: man has less hair than apes, but that is not his scientific name.

11. Angels, understanding and reason; purgatory (“Why can’t we be born knowing everything”) #

Point: an angel need not love reason because he has something better, understanding; “understanding” names the act, the faculty, and is sometimes divided into understanding and reason.

• Man has the ability to understand only weakly, so it gets a new name, like the kitten, while the adult cat keeps the name because it is fully a cat. His son Marcus: “Why can’t we be born knowing everything we need to know?”; “You want to be an angel”; that is what an angel is.
• Shakespeare defines reason as the ability for large discourse, looking before and after: looking is trying to see, and man tries to understand more than he understands. Augustine: more things in the Bible he does not understand than does.
• If the angel loves the ability to understand he loves truth and wisdom; hence the fallen angels’ torment, naturally wanting to see God face to face and never doing so; the souls in purgatory are extremely anxious to see God and must first be purged. St John of the Cross asked a friend to pray that he have his purgatory on earth, “a very wise request.” Recording ends here.

His words #

• demonstration; syllogism from premises necessarily true “and seen” as necessarily true; from apodeixis, “to show”
• Q.E.D.; “what was to be demonstrated”
• Prior Analytics vs. Posterior Analytics; taking apart the argument vs. examining the premises
• episteme; “reasoned out knowledge” [ed.: science, scientific knowledge]; effect of demonstration
• mathetes / doctor; mathematics named from learner, Sacra Doctrina from teacher
• dialectical syllogism; syllogism from probable statements/opinions
• Topica, topos; place “where you look to find a dialectical argument”; transliteration vs. translation
• probable; opinions of all, most, all or most in an art, the most famous in it; by number (quantity) or quality
• contradictory statements; same subject and predicate, one affirmative one negative, one true one false
• cum formidine; with fear of the opposite (Thomas)
• sophistical syllogism; appears a good dialectical syllogism but is not
• falsigraphical; “the falsely drawn”; appears a demonstration but is not
• kath’ hauto, per se; through itself, as such
• property in the strict sense; belongs only to the species, to every one, always
• induction; “all snow is white” from the snow he has seen
• first road / natural road; his terms for the order of our knowledge
• risible; capable of laughter
• Homo sapiens; “the wise ape”
• understanding; act, faculty; divided into understanding and reason

Texts #

Read in class (quoted or commented):

• Aristotle, Topics, beginning: definition of the probable; sophistical and falsigraphical syllogisms
• Aristotle, Prior Analytics, beginning: demonstrator takes one side of the contradiction, dialectician offers a choice
• Aristotle, Posterior Analytics: the necessary and the kath’ hauto
• Plato, Meno, third part: arguments that virtue can and cannot be taught
• Plato, Protagoras: teachers of virtue like teachers of Greek (he says “I think”)
• Shakespeare, Hamlet (inferred): the purpose of playing, to hold the mirror up to nature; “love and reason keep little company”; reason as “large discourse, looking before and after”
• Mozart, letter to his father on Osmin’s aria (Abduction from the Seraglio)
• Thomas Aquinas, proemium on Wisdom (inferred: Metaphysics commentary): cum formidine
• Einstein, autobiographical sketch (Harper volumes): Euclid and youthful enthusiasm; hypotheses freely imagined

Mentioned:

• Euclid, Elements (proposal, epilogue, Q.E.D.; angle-sum of triangle)
• Plato, Theaetetus, Euthyphro, Laches, Hippias, Gorgias
• Aristotle, Physics II, definition of nature (“first”)
• Newman, Apologia Pro Vita Sua
• Augustine on the Bible; St John of the Cross; The Naked Ape

His questions #

• Is two through itself a number? Is two through being two half of four? Yes to both: number is in its definition; half of four is its property.
• Does it belong to triangle as such to be a plane figure, to be three-sided? Yes: genus and difference, anything in the definition belongs as such.
• Could you discover, by thinking what a triangle is, that a triangle is green? No; at best a matter of sensing, never of demonstrating.
• Does it belong to snow as such to be white? He does not see that it does; he holds “all snow is white” by induction, as probable.
• Is what directs us to the good necessarily knowledge? No; right opinion, even a correct hunch, also directs (fork in the road, MacArthur).
• Is Aristotle violating his own rule against circles by arguing both ways between necessary and per se? Thomas: no; the demonstration runs from necessary to per se, the reverse from a probable opinion.
• Must a philosopher love reason? Must a philosopher love popcorn? Yes; no. Lover of wisdom must love reason, wisdom being reason’s highest perfection.
• Must an angel love reason? No; he has something better, understanding, and naturally knows at once what he knows.