Logic for A.I. - Solutions

Ex. 17 Let us call the new system L, i.e. its axioms are all propositional tautologies (Axiom 1) plus the axioms 2> (Axiom 2) and 2φ∧2ψ → 2(φ∧ψ) (Axiom 3), and the rules modus ponens and φ→ ψ 2φ→ 2ψ We have to show that for all formulas φ `K φ ⇔ `L φ. ⇒: For this direction we have to show that L derives all axioms of K and all its rules. Axiom 1 of K is the same as Axiom 1 in L, thus we have nothing to prove. Axiom 2 of K is 2(φ→ ψ)→ (2φ→ 2ψ). We give a derivation of this formula in L: (φ→ ψ) ∧ φ→ ψ 2((φ→ ψ) ∧ φ)→ 2ψ (the rule from L) 2(φ→ ψ) ∧2φ→ 2ψ (axiom 3 of L and propositional logic) 2(φ→ ψ)→ (2φ→ 2ψ) (propositional logic)