Thursday, October 15, 2009

Midterm Review for Logic

Form of the Midterm: First, there will be 10 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, there will be several short answers that demonstrate your knowledge of the forms that we have learned so far. No essays, just the central forms on which we concentrated (ex: recreate the square of opposition, the difference between inclusive and exclusive or, etc).

Logic, Magic and Riddles (first week): 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.

As a discipline, Logic switched from focusing on the practice of debate in early cultures to foundations of mathematics in modernity.

Kanada: Six objects of knowledge: (subst, attrib, act/karma, genr, part, inher, void).
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).
For example, 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:
Positive:
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)
Negative:
Whatever is eternal is not produced, as in the soul (If not n, then not p, soul is neither)
Because it is produced, Sound is not eternal.

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 and think it must be some Y and some not Y, neither ALL or None.

Aristotle: We saw that Aristotle believes in 10 categories, Genus and Species (ex: animal, human), that substances can have contrary qualities but only at different times.
Ar. says “if someone admits contrary qualities, his contention is unsound”, skeptics like 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.

Principle of Non-Contradiction: X must be either true or false, but not both.
Principle of Bivalence: X must be either true or false, but not neither (it must be one of the two).

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.