Paradoxes in Logic, Mathematics and Computer Science

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Lecture 4

• Logic is a Game

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What is a Logic?

• The word Logic has many meanings, e.g. check
  http//en.wikipedia.org/wiki/Logic.
• In our context, a logic is a set of sentences,
  written in a formal language, and has
• A Syntax Rules for defining legal sentences.
• A Semantics Rules for giving meaning to those
  sentences.

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Propositional (Boolean) Logic

• Syntax Formed from propositional variables X, Y,
  Z, etc. using the operations
• ?A, (A ? B), (A ? B).
• A different notation A?, AB, AB
• Semantics Each propositional variable says
  something that is true or false about our world.
• ?A means not A.
• (A ? B) means A and B.
• (A ? B) means A or B (or both).

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Example

• Consider the sentence
• A ((?X ? Y) ? ?Z) ? ?((?Y ? Z) ? X).
• Let X and Y be true, and Z be false.
• Question Is A true or false?
• Notes
• We build the truth values of the parts of A
  bottom-up.
• We can visualize A as a tree or as a Boolean
  circuit.

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A Truth Game on the Sentence A

• Initial Position The sentence A ((?X ?
  Y) ? ?Z) ? ?((?Y ? Z) ? X),
• with X Y 1 (true), and Z 0 (false)
• Two Players Verifier (V) and Falsifier (F).
• Rules From the position A
• 1) If A (B ? C), F moves to B or C.
• 2) If A (B ? C), V moves to B or C.
• 3) If A ?B, the players switch roles and play
  on B.
• If A is a propositional variable, V wins iff A1.
• Note Due to Rule 3, V may end up being F at the
  end.

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When does Verifier win?

• Theorem For all propositional sentences A, and
  all truth assignments to its propositional
  variables,
• 1) V can win from a position A iff A is true.
• 2) F can win from a position A iff A is false.
• Proof By induction on the complexity (the
  length) of A.
• Base is clear. For the step, we have the cases
• Case 1 A (B ? C)
• Case 2 A (B ? C)
• Case 3 A ?B

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Back to the sentence A

• Question Starting from the position A ((?X
  ? Y) ? ?Z) ? ?((?Y ? Z) ? X),
• Can V win the truth game on A for every truth
  assignment to X,Y, and Z?
• Answer No. Using two methods
• 1) Construct a Truth Table for A and get a 0 in
  the column of A.
• 2) Use the Reduction Method to show that there is
  a truth assignment for which F wins.

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Valid Sentences

• Definition A propositional sentence A is valid
  (tautology) iff it evaluates to true for all
  truth assignments to variables occurring in it.
• OR (iff V can win for all truth assignments.)
• Examples
• 1) A ((?X ? Y) ? ?Z) ? ?((?Y ? Z) ? X) is NOT
  valid.
• 2) B ((?X ? Y) ? ?Z) ? ((?Y ? X) ? Z) is is
  valid.

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A Validity Game on Propositional Sentences

• Initial Position A propositional sentence, e.g.
  B ((?X ? Y) ? ?Z) ? ((?Y ? X) ? Z)
• Two Players Validator (V) and Falsifier (F).
• Rules
• 1) F chooses a truth assignment for X,Y, and Z,
  e.g. X Y 1, and Z 0.
• 2) V and F play the truth game on B.
• Theorem For all propositional sentences B, V
  can win the validity game on B iff B is valid.

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• Thank you for listening.
• Wafik