Conversion and the Second Figure Mixed Cases, Then Nature, Faith and Reason

Orientation #

He is continuing the second figure: the universal cases and the mixed cases with the particular premise on the bottom are already done; what remains is “one more set of four” with the particular premise on top. He announces the third figure “next time”, where only particular conclusions will be found, and that the group will not meet next week (community meetings) but two weeks from now. The previous class tested the sixteen second-figure combinations and compared the figures’ strength.

The class, in order #

1. Conversion of the universal negative holds for an infinity of cases (“no matter what you take for b and a”) #

Point: the conversion of “no B is A” to “no A is B” is shown necessarily, not by inspecting cases.

1. There is an infinity of true universal negatives (no three is a four, no three is a five…), so you cannot go through them one by one.
2. Aristotle’s reasoning shows that if you deny the converse, something impossible follows: if “some A is B” were true, take that A, call it X; X is both A and B; so some B (X) is A, which is the strict opposite of “no B is A”.
3. Conclusion: whatever B and A are, if the universal negative is true, so is its converse. “That’s kind of marvelous. You stop and think about that.” Examples: no dog is a cat / no cat is a dog; no man is a woman / no woman is a man — true in each case, but the cases never prove it always so. He notes this is why the second figure yields negative conclusions: the universal negative can be converted while staying universal.

2. Examples refute; they never establish (“one example is enough to show”) #

Point: examples can show something is not a syllogism, never that it is.

• Necessity: “man is necessarily white” — a million white men prove nothing; one black or yellow man destroys the claim.
• “Every number is odd”: 3, 5, 7, 9, 1 — an infinity of examples, still not shown always so; one even number and “poof!”
• Applied here: he cannot show the conversion by examples, but having shown it the way he did, no counter-example can exist. Aside: he would leave “my wad of dollars” and insurance policies on the table against anyone finding one.
• To see validity you must go back to something universally obvious: “the set of all and the set of none”, or the square of opposition.

3. The last four second-figure cases: particular major, universal minor (“the particular one is on top”) #

Point: none of the four is a syllogism; each is knocked out by examples satisfying two conditions (premises true; one set of terms with “every C is A”, another with “no C is A”).

• Some A is not B; no C is B. Two negatives, so no syllogism is expected. Terms: animal (A), dog (B); C = cat (every cat is an animal) and stone (no stone is an animal). Every cat an animal being true once knocks out the negatives; no stone an animal knocks out the affirmatives. “Everything’s knocked out.”
• Some A is B; every C is B. No negative, so no “set of none”; conversion of the universal affirmative is not necessary, so no “set of all”. Terms: dog (A), animal (B); C = cocker spaniel (every cocker spaniel is a dog) and cat (no cat is a dog). “If something were necessarily so, it would have to be always so”; nothing is.
• Some A is not B; every C is B. One negative excludes the set of all; no universal negative excludes the set of none. Finding examples is hard: student tries dog / white / animal, but “not every dog is white”, so condition one fails; tree / animal / dog fails on finding a C that is always an animal and never a tree. Working set: habit (A), good (B); C = virtue (every virtue is a habit) and a good act (every good act is good, no good act is a habit; beauty or God also offered). “At first you don’t succeed, try and try again… after a while it becomes kind of automatic.”
• Some A is B; no C is B. He converts “no C is B” to “no B is C” and gets “some A is not C”, but that has A as subject; the particular negative cannot be converted, so no conclusion with C subject and A predicate — unlike the two-universal case, which was converted twice. Terms: habit (A), virtue (B); C = vice (every vice is a habit, no vice is a virtue) and stone (no stone is a virtue, no stone is a habit). Set of all knocks out the negatives, set of none the affirmatives. Method flag: “the pause that helps you to avoid mistakes” — always stop and check that the premises are true with the chosen examples before checking the conclusions.

4. The four valid forms of the second figure (“let’s put down the four”) #

Point: recap of the second figure’s valid cases and how each is reduced.

1. No A is B, every C is B: convert the major, syllogize in the first figure to no C is A.
2. Every A is B, no C is B: convert “no C is B” to “no B is C”, syllogize “no A is C”, convert to no C is A.
3. No A is B, some C is B: convert to no B is A, “by magic you’re back in the first figure”; some C is not A.
4. Every A is B, some C is not B — the difficult one. Common sense: if every A is B without exception and some C is not B, that C cannot be A. Rigorous proof: deny that “some C is not A” is necessary; then by the square of opposition “every C is A” is possible; add it to “every A is B” and every C is B follows, which contradicts the premise; so “every C is A” is incompatible with the two premises, must be false, and since contradictories cannot both be false, “some C is not A” must be true. Count: none valid among the particular cases, two among the mixed, two among the universals — four in all.

5. Comparing the figures (“all conclusions are negative”) #

• Second figure: all conclusions negative; two ways to a universal negative, two to a particular negative. “Less powerful” because power consists in getting a universal conclusion, and here only a universal negative.
• First and second together: one way to a universal affirmative (first figure only), three ways to a universal negative, one to a particular affirmative, three to a particular negative. “The most important cases are the three ways of syllogizing a universal negative, and the way of syllogizing a universal affirmative.”
• Reasoning backwards: a universal negative conclusion means one of these forms; a universal affirmative means only the first form.
• Third figure (next time): only particular conclusions; so “easier to syllogize the negative than the affirmative, and the particular than the universal”.
• Premises in the second figure: the major is always universal; the minor can be any of the four.
• Asides: women students say they understood everything until A, B, C; he never bothered with the Latin mnemonic words for A, E, I, O and thinks it out at the board; the rules “major universal, minor affirmative” (first figure) are induced after going through the sixteen, not used to decide; some books talk of “distribution” of the predicate, but that is not the same as universality.

6. Singular terms and the second figure in theology (“God is not composed”) #

Point: with a singular subject a negative statement converts like a universal negative, so theology, which mostly denies things of God, uses the second figure constantly.

• Socrates is not a woman / no woman is Socrates; Perchus is not a dog / no dog is Perchus; Socrates is not a carpenter, not a major-league ballplayer.
• Natural philosophy (first book of “natural hearing”) shows what changes is composed; theology adds God is not composed (simple); therefore God does not change. This is the second valid form (every A is B, no C is B) with a singular in place of the universal term; “not too pedantic”, but one must see it is valid.
• A student recalls Father McGovern making them extract the syllogism when doing St. Thomas.

7. Socrates against soul as harmony (“said of one and denied of the other”) #

Point: the Phaedo argument is second-figure reasoning — find something affirmed of one and denied of the other, and you conclude one is not the other. Three things: the soul resists the body (fasting, fighting passions), the harmony of the body does not; the soul exists before the body (argued earlier), the harmony does not (Simmias admits); the soul has a harmony (philosophy aims at it), but there is no harmony of a harmony. It makes no difference which you affirm and which you deny. Practical summary: two negatives, no syllogism; two particulars, no syllogism; mixed forms give only particulars; the four universal forms (two first-figure, two second-figure) “you use again and again”; the third figure not much.

8. Student difficulty: trust wholly in God, distrust wholly yourself (“it seems to go against the hope”) #

Student: if God wants all saved and some are not, the failure is from themselves; but to distrust oneself completely is to trust one would reject grace, and there is no hope — either presumption or despair. He: “you got to be careful the way you state that.” He answers indirectly through the following discussion of nature and grace, and the student notes at the end that this was the difficulty he was meant to be answering.

9. As You Like It: the natural bond of sisters (“natural bond of sisters”) #

Point: nature is the measure of the good and bad in the play.

• The phrase implies a natural bond of brothers too. Rosalind and Celia are cousins yet close as sisters; the two pairs of actual brothers (the Duke and the usurping younger brother; Oliver neglecting Orlando’s education) are in conflict with the natural bond — the cousins’ closeness shows up the brothers’ unnaturalness the more, since the bond of cousins is by nature weaker.
• The forest of Arden as nature: the exiled Duke on their life “exempt from public haunt” finding tongues in trees, books in brooks, sermons in stones. Both pairs reconcile there; weddings (natural too); the usurper meets a monk and repents.

10. Why brothers among the first apostles (“why did he choose brothers”) #

Peter and Andrew, James and John: at minimum it shows harmony between the natural bond and the supernatural bond; perhaps more, that the natural bond disposes for the supernatural. Thérèse of Lisieux’s sisters all in the convent, yet she suppressed the urge to go speak to a sister. Moses, Aaron and Miriam. He does not think the supernatural bond of brothers is necessarily stronger than among non-brothers; Peter is associated with James and John, not Andrew.

11. Belief before knowledge: Thomas on faith in a teacher (“you don’t come to know very much unless you really believe”) #

Point: Thomas argues that requiring faith for the vision is in harmony with man’s nature, since a learner naturally believes his teacher.

• Example: Metaphysics V on “beginning” — first sense the start of a thing (of a table), second the most convenient place to begin (leaving the table from where you sit; halfway down the road to Boston). Aristotle’s “as in knowing we begin with what is more known to us” is a comparison, not the beginning of our knowledge, which comes only at the fifth sense — Thomas is clear. John Gallup misread it; Monsignor Dionne corrected him from Thomas; Cajetan’s mistake was commenting directly on Aristotle; the Greek commentators seem to make Gallup’s mistake. Read Aristotle, then Thomas on Aristotle, then think.
• He got further because he believed men like Dionne and Thomas; he read Shakespeare carefully because he believed he was a poet.

12. Nature disposes for grace: sacraments, fatherhood, reason, virtues, Psalm 99 (“belief seeking understanding”) #

• Sacraments: God uses the sensible to lead to the spiritual, as philosophy leads from motion to the unmoved mover; “brother”, “sister” in a monastery come from nature.
• Our Father: a bad human father makes it hard to see God as father; a good one (Teresa’s father) disposes for meditating “how sweet it is to call him father”.
• Fides et Ratio: distrust of reason impedes faith; Paul’s obsequium rationabile; Augustine’s letter — we are capable of faith because we have reason.
• 1 Corinthians (Thomas: mainly about sacraments — baptism, marriage, Eucharist): Paul condemns the man with his father’s wife as something even pagans know is terrible; neither Paul nor Thomas mentions Oedipus.
• Human (acquired) virtues are in one way more stable than infused virtues, which one mortal sin destroys; without them there is instability; Paul: the man who neglects his family is worse than a pagan. One cannot neglect the natural and assume the supernatural.
• Theology as belief seeking understanding fits wonder being natural. Psalm 99: “know that the Lord is God, he made us, his we are, the flock he tends” matches the three books of the Summa Contra Gentiles (God in himself, God as maker, God as end and providence). “Enter his gates with thanksgiving, his courts with praise”: asking, then thanksgiving (thinking of what was received, not seeking more), then praise (wholly outside oneself) — the gate to the court, and how heaven will be.

His words #

• set of all / set of none: the evident first-figure principles to which valid forms are reduced [ed.: dictum de omni et nullo]
• square of opposition: used to move from denying necessity of a conclusion to the possibility of its contradictory
• law of incompatibility: a proposition incompatible with true premises must be false
• “the pause that helps you to avoid mistakes”: checking that premises are true with chosen examples
• power (of a figure): ability to get a universal conclusion
• natural hearing [ed.: Physics]
• fifth book of wisdom [ed.: Metaphysics V]
• distribution: textbook term for a predicate spread over all or some; not the same as universal
• obsequium rationabile: Paul’s “reasonable service” of faith
• belief seeking understanding: the old definition of theology

Texts #

Read in class

• Plato, Phaedo: soul not the harmony of the body (three arguments, Simmias)
• Shakespeare, As You Like It: “natural bond of sisters”; the Duke’s speech in the forest
• Psalm 99 (his “correct numbering”): opening verses, “Enter his gates with thanksgiving”

Mentioned

• Aristotle, Prior Analytics, conversion and second figure (inferred)
• Aristotle, Metaphysics V, senses of “beginning”, with Thomas’s commentary
• Aristotle, Physics I: what changes is composed
• Thomas Aquinas, an article on the necessity of faith (locus not given)
• Thomas Aquinas, Summa Contra Gentiles I-III; Summa Theologiae
• John Paul II, Fides et Ratio
• Augustine, a letter on faith and reason (via Father Boulay)
• Paul, 1 Corinthians (man with his father’s wife; Thomas’s division by sacraments); an epistle on the man who neglects his family
• Sophocles, Oedipus the King
• Greek commentators on the Metaphysics; Cajetan
• Thérèse of Lisieux, her life; Heisenberg on never beginning at the beginning of reality

His questions #

• Are examples enough to show something is a syllogism? One example shows something is not always so; no number of examples shows it always and necessarily so.
• Why can you show by examples that something is not a syllogism but not that it is? Necessity requires always; a counter-example destroys “always”; validity is seen only from the set of all and none or the square of opposition.
• Can you find examples for A, B, C satisfying both conditions (some A is not B, no C is B)? Animal, dog; cat and stone.
• How many valid forms are there in the second figure? None among particulars, two mixed, two universal — four.
• What do you see about the premises in the second figure? The major is always universal; the minor can be any of the four.
• Why is the second figure less powerful than the first? Power is a universal conclusion; the second gives only universal negatives, the first also universal affirmatives.
• Why did our Lord choose brothers as the first apostles? At least to show harmony between the natural and supernatural bond; perhaps that nature disposes for grace — left partly open.
• How should one take “trust completely in God, distrust completely yourself”? “You got to be careful the way you state that”; addressed only through the harmony of nature and grace — left open.