GEB class notes jan 18

1
GEB class notes jan 18

• A Logic Puzzle
• Classification Problems,
• Equivalence Relations,
• and Disjoint Unions
• Logic and Set Theory

2
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?

3

• 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 (Im
  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?
• HINT Try a statement of the form You will not
  give me

4
Classification Problems

• Statements can be self-referential, well-formed,
  true, provable, meaningful
• The Game of 20 Questions
• The three of these things are kind of the
  same Game

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Set Complements
The Universe
A
not A
6
A Partition of the Universe
animal
vegetable
other
mineral
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A Partition by Overlapping Sets
A
B

• How many disjoint subsets in this partition?

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What does kind of the same mean? 3 essential
properties

• For every A, A is kind of the same as A.
• If A is kind of the same as B, then B is
  kind of the same as A.
• 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

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RESULT

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

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DeMorgans Laws

• Sets not(A U B) not A I not B
• In Logic, Not(A OR B) (Not A) AND (Not B)

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Summary at a very basic level,

• reality seems to be imitated by sets,
• set operations seem to correspond to logical
  operations,
• logical operations seem to correspond to thought
  processes,
• thought processes seem to be well-expressed
  through the language of AND, OR, and NOT.

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A Set Operation for If Then
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Which Box Contains the Gold?

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

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Whats Going On?

• 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.
• Experimentally, we saw that the logically
  consistent interpretation was not consistent with
  reality --- who said it had to be?