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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