Logic: Lecture 2, 09 August 2012
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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

Derived connectives

   and, implies (->), iff (<->), ..

   Truth tables and "completeness" of not and or

   Digression about intuitionism

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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