User:Lord Farin/Sandbox/Completeness

Theorem
Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:


 * $\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$

Proof

 * 1) Every tautology is equivalent to a tautology in CNF
 * 2) CNF formula is tautology iff it is a conjunction of tautologies
 * 3) A disjunction is a tautology iff it contains both a variable and the negation of that variable
 * 4) Every tautological disjunction is derivable in H2
 * 5) Every tautological CNF is derivable in H2
 * 6) Replacement theorem for tautologies
 * 7) Replacement theorem for proofs in H2
 * 8) Formulae whose CNF is derivable are derivable in H2

Tautology is equivalent to tautology in CNF
Existence of Conjunctive Normal Form of Statement

CNF formula is tautology iff it is conjunction of tautologies
Immediate from definition of conjunction

Disjunction of literals tautology iff contains both variable and negation
Finite amount of work

Tautological disjunction derivable in H2
From previous + rule of addition + association + commutation

Tautological CNF derivable in H2
From previous + rule of adjunction

Replacement Theorems

 * User:Lord Farin/Sandbox/Completeness/Replacement Tautology
 * User:Lord Farin/Sandbox/Completeness/Replacement H2