Thursday, October 7, 2010

Logic: Midterm Review

Form of the Midterm: First multiple choice questions about the lectures (ex: Aristotle thinks that skeptics like Heraclitus are no better than PLANTS, Kanada believes there are 6 categories, including KARMA/action, etc). Second short answers that demonstrate your knowledge of the forms that we have learned so far (ex: recreate the square of opposition, the forms of syllogisms, Nyaya proof). No essay.

Logic, Magic and Riddles: We read in Malinowski’s Rational Mastery by Man of His Surroundings that tribes use all of the grammatical forms and pieces of reasoning that we all use. We saw difference of practice and theory, constantly intertwined, in the difference between shallow water fishing and deep sea fishing. We saw in the next article that in baseball, the same sort of split occurs between hitting/pitching and fielding. In wisdom tales, understanding contradiction. Logic switched from the study of debate in early cultures to foundations of mathematics in modern times with the rise of algebra.

Kanada: Karma as action/energy. We saw the basic structure of IF-THEN hypothetical statements. The Modus Tollens transformation: (If p then q), means also (if not q then not p). Ex: If we know “If there are clouds then there is rain”, we know “If there is no rain then there are no clouds”.

Nyaya: We saw the Nyaya form of the two sided debate: (X is Y) vs. (X is not Y). We saw that there are four sources of knowledge: perception, inference, comparison, and testimony.
Inference shows us induction leading to deduction, gathering leading to rules/principles.

The Nyaya proof: Rule, Example, Reason, Conclusion
Wherever there is smoke, there is fire. As in a kitchen.
Because the hill is smoky, therefore the hill is on fire.
Whatever is produced is not eternal, as a pot. (If p, then n, pot is both)
Because it is produced, Sound is not eternal. (Because s is p, s is n)

Note the subtle attack on the older tradition of the Vedas in the last two examples.

While Nyaya scholars believe X is Y or not Y, exclusively, we see that Jains think it must be some Y and some not Y, neither ALL or None. Buddhists argue that X can sometimes and somewhere be Y but also sometimes and somewhere not be Y. Nagarjuna’s Catuskoti.

Aristotle: Genus and Species (ex: animal, human), substances can have contrary qualities but only at different times, “if someone admits contrary qualities, his contention is unsound”, skeptics are destroyers, plants.

Square of opposition, All X is Y, No X is Y, Some X is Y, Some X is not Y as four corners.
Top pair (universals) can’t both be true, bottom pair (particulars) can’t both be false.

If we know All X is Y is true, then we know two things are false: No X is Y & Some X is not Y.
If we know No X is Y, then we know All X is Y and Some X is Y are false.
If we know Some X is Y, then we know No X is Y is false.
If we know Some X is not Y, then we know All X is Y is false.

Exclusive and Inclusive OR:
Inclusive Or: X or Y and you can choose more than one.
Exclusive Or: X or Y but you can only choose one exclusively.

The Four Forms of the Perfect Syllogism:
First, the Positive Universal, ‘Barbara’:
All A are B, All B are C, therefore All A are C.
Example: If it’s a human, animal, living thing.
(Venn diagram of A within B, B within C, or partially/fully collapsed)

Second, the Negative Universal, ‘Celarent’
All A are B, No B are C, therefore No A are C.
Ex: All men are animals, No animals are stone, therefore No men are made of stone.

Third, the Positive Particular, ‘Darii’
Some A are B, All B are C, therefore Some A are C.
Ex: Some animals are humans, All humans are funny, therefore some animals are funny.

Fourth, the Negative Particular, ‘Ferio’:
Some A are B, No B are C, therefore Some A are not C.
Ex: Some animals are humans, No humans are reptiles, therefore Some H are not R.

Greek Skepticism: Heraclitus says do not trust experts, we are all a bunch of apes. Pyrrho argues anything against anybody, then the opposite. Sextus Empiricus says inference can be wrong.

Chinese Paradox: Lao Zi’s Wheel (positive, negative, both), Zhuang Zi’s perspective, sage’s this always has a that, Hui Shih’s left today arrived yesterday and greater and lesser similarity and difference, Kung-Sun Lung’s white horse is not a horse argument and two uses of “is”.