GEB class notes jan 18 A Logic Puzzle Classification Problems, Equivalence Relations, and Disjoint Unions Logic and Set Theory

Basic Question What hope is there that a certain “portion of reality can be imitated in its behavior by a set of meaningless symbols governed by formal rules”?

n I offer two prizes --- Prize 1 and Prize 2. You are to make a statement. If the statement is true, then I will give you one of the prizes (I’m not saying which one). If your statement is false, then you get no prize. Obviously you can be sure of winning a prize by saying, e.g., “Two plus two is four,” but suppose you have your heart set on Prize 1; what statement could you make that would guarantee that you will get Prize 1? n HINT: Try a statement of the form “You will not give me …”

Classification Problems n Statements can be self-referential, well-formed, true, provable, meaningful … n The Game of 20 Questions n The “… three of these things are kind of the same” Game

Set Complements A not A The Universe

A Partition of the Universe animal vegetable mineral other

A Partition by Overlapping Sets n How many disjoint subsets in this partition? A B

What does “kind of the same” mean? 3 essential properties n For every A, A “is kind of the same as” A. n If A “is kind of the same as” B, then B “is kind of the same as” A. n If A “is kind of the same as” B and B “is kind of the same as” C, then A “is kind of the same as” C

RESULT: n If a relation is symmetric, reflexive, and transitive, n Then the collection of sets of things that are “kind of the same” forms a partition of the Universe into a disjoint union.

DeMorgan’s Laws Sets: not(A  B) = not A  not B n In Logic, Not(A OR B) =(Not A) AND (Not B)

Summary: at a very basic level, n reality seems to be imitated by sets, n set operations seem to correspond to logical operations, n logical operations seem to correspond to thought processes, n thought processes seem to be well- expressed through the language of AND, OR, and NOT.

A Set Operation for If … Then …

Which Box Contains the Gold? n Two boxes are labeled "A" and "B". n A sign on box A says "The sign on box B is true and the gold is in box A". n A sign on box B says "The sign on box A is false and the gold is in box A". n Assuming there is gold in one of the boxes, which box contains the gold?

What’s Going On? n Your reasoning was sound. If each sign is either true or false, then the only interpretation that is logically consistent with those signs is “the gold is in Box B.” n Experimentally, we saw that the logically consistent interpretation was not consistent with reality --- who said it had to be?