Paradoxes in Logic, Mathematics and Computer Science

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Lecture 5

• Logic is a Game II

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Limitations of Propositional Logic

• How does propositional logic express a statement
  like All men are mortal.
• If there are finitely many of those men, then we
  can say
• (Man 1 is mortal) ? (Man 2 is mortal) ?
• Notes
• The above sentence could be very long.
• What if there are infinitely many men?

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Predicate (First Order) Logic

• For the statement All men are mortal.
• We use the predicates
• Man(x), which means x is a man.
• Mortal(x), which means x is mortal.
• We then write (?x)(Man(x) ? Mortal(x)) to mean
  For every x, if x is a man, then x is mortal
• Note (A ? B) is an abbreviation for (?A ? B),
  i.e. the above sentence reads
• For every x, either x is not man, or (if x is
  indeed a man) x is mortal

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Syntax of Predicate Logic

• Formed from atomic formulas like
• Man(x), Mortal(x) unary predicates,
• Friend(x,y) a binary predicate, which means x
  is a friend of y.
• Addition(x,y,z) a ternary predicate, which
  means x y z.
• Using the propositional connectives ?,?,?
• And the quantifiers (?x) and (?x)

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Syntax of Predicate Logic (cont.)

• Sometimes we can also use names for
• Constants, e.g. socrates, 0, ?, etc..
• Functions, e.g. mother(x), which means the
  mother of x, or S(x), which means the successor
  of x.
• We can then say the following terms
• mother(socrates) the mother of Socrates
• S(S(S(0))) This is the definition of 3.

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Semantics of Predicate Logic

• We need a nonempty set U (the universe) which
  includes all entities we are talking about.
• Each unary predicate symbol, e.g. Man(x), is
  interpreted by a subset of U, e.g. the subset of
  all men.
• Each binary predicate symbol, e.g. Friend(x,y),
  is interpreted by a binary relation over U, e.g.
  the set of all pairs (x,y), for which x is a
  friend of y. Etc..

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Semantics of Predicate Logic (cont.)

• Each constant symbol, e.g. socrates, is
  interpreted by a particular element of U, e.g.
  the person Socrates.
• Each function symbol, e.g. S(x), is interpreted
  by a particular function on U, e.g. the function
  f(x) x 1. Etc..
• U together with all interpretations forms a
  structure, e.g. U (U, Man, Friend,),
• or N (N, 0, S).

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Semantics of Predicate Logic (cont.)

• As before
• ? means not, ? means and, ? means or.
• Also
• (?x) means for all x in U (a big AND)
• (?x) means there exists an x in U (a big OR)
• Notes We also have the short notations
• (A ? B) (?A ? B) means A implies B
• (A ? B) (A ? B)?(B ? A) means A iff B

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Example

• Consider the sentence
• A (?x)(?y)(x lt y) ? (?x)(?y)?(y lt x)
• Note We are using the notation (x lt y) instead
  of lt(x,y).
• Questions 1) What does A say?
• 2) Is A true about the set of natural numbers
  with its natural order?
• 3) Is A true about the set of real numbers with
  its natural order?

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A Truth Game on the Sentence A

• Initial Position The sentence (?x)(?y)(x
  lt y) ? (?x)(?y)?(y lt x)
• with a structure (U,lt).
• Two Players Verifier (V) and Falsifier (F).
• Rules 1,2,3) If A (B ? C), or (B ? C), or ?B,
  the rules are as before (see Lecture 4).
• 4) If A (?x)B(x) (or (?x)B(x) ) F (or V)
  chooses an element c?U and the players play on
  B(c).
• 5) If A has the form c lt d, V wins iff c lt d is
  true in (U,lt).

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When does Verifier win?

• Theorem For all predicate sentences A, and all
  structures U,
• 1) V can win from a position A iff A is true in
  U.
• 2) F can win from a position A iff A is false in
  U.
• Proof By induction on the complexity (the
  length) of A.
• Base is clear. We then consider the cases
• Cases 1,2,3 A (B ? C) or (B ? C) or ?B (as
  before)
• Case 4,5 A (?x)B(x) or (?x)B(x)

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Example

• Consider the game on the sentence B (?x)(?y)((x
  lt y) ? (?z)(?w)
• ((y z?w ? (z 1 ? w 1)) ? (y 2
  z?w ? (z 1 ? w 1))))
• With the structure N (N,1,2,,?).
• Questions
• 1) What does B say?
• 2) Who wins in this game (V or F)?
• Note This is the Twin Prime Conjecture.

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• Thank you for listening.
• Wafik