Reference no: EM132381164

Axiomatic Derivations Problems -

Problem 1 - Show that the following hold by exhibiting derivations from the axioms:

1. (φ Λ ψ) → (ψ Λ φ)

2. ((φ Λ ψ) → χ) → (φ → (ψ → χ))

3. ¬(φ ∨ ψ) → ¬φ

Problem 2 - Prove Proposition: Γ is inconsistent iff Γ|- φ for every φ.

Problem 3 - Prove Proposition: 1. |- (φ → ψ) → ((ψ→ χ) → (φ → χ);

2. if Γ U {¬φ} |- ¬ψ then Γ U {ψ} |- φ (Contraposition);

3. {φ, ¬φ} |- ψ (Ex Falso Quodlibet, Explosion);

4. {¬¬ φ} |- φ (Double Negative Elimination);

5. If Γ |- ¬¬ φ then Γ|- φ;

Problem 4 - Prove that Γ |- ¬φ iff Γ U {φ} is inconsistent.

Problem 5 - Prove Proposition: If Γ U {φ} and Γ U {¬φ} are both inconsistent, then Γ is inconsistent.