Logic: Lecture 2, 06 August 2015
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Review of syntax, valuations, validity, satisfiability, truth tables

Notation:

  v |= a for v(a) = tt
    |= a for a is valid, i.e. v |= a for all v

Axiomatizations

  System H

  (A1) A -> (B -> A)
  (A2) (A -> (B -> C)) -> ((A -> B) -> (A -> C))
  (A3) (~B -> ~A) -> ((~B -> A) -> B)

       A, A -> B 
  (MP) ---------
           B

  Presentation: 

    Axiom schemes vs concrete axioms and substitutions
    A,B will denote schemes, a,b will denote concrete formulas

  Definition: "Derivation" or "proof" of a in system H

    Finite sequence of formulas ending with a such that each
    formula is an instance of an axiom or is obtained from a
    pair of earlier formulas in the sequence by applying MP.

  Notation:

   |- a for a is derivable (or "a has a proof", or "a is a theorem")

  Example: Derivation of p -> p

  |- (p -> ((p -> p) -> p)) -> ((p -> (p -> p)) -> (p -> p))  (A2)
  |- p -> ((p -> p) -> p)                                     (A1)
  |- (p -> (p -> p)) -> (p -> p)                              (MP)
  |- p -> (p -> p)                                            (A1)
  |- p -> p                                                   (MP)


Theorem to prove:  |- a iff |= a

  Soundness    : If |- a then |= a
  Completeness : If |= a then |- a

Proof of soundness

  By induction on the length of deriviation

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Adding constraints

  May constrain valuations to reflect connections between
  propositions.

  e.g. p = "Sky is blue", q = "It is raining", insist that p -> ~q
  always holds

  Under this assumption, ~~q -> ~p (contrapositive), or q -> ~p
  will also always hold.  How can we derive this formally?

  Assume X, prove a
  Formulas in X are concrete, not axiom schemes 

  X |- a

  Redefine deriviations: 

  A sequence where each formula is an instance of an axiom,
  obtained using MP, or a member of X.

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