Examples of Proofs I

 

 

Example 2.7 (a). For any well-formed formula A, (A → A).

Proof. We shall construct proof in L.

1. (A → ((A → A) → A)) → ((A → (A → A)) → (A → A)) Instance of Axiom A2.
2. (A → ((A → A) → A)) Instance of Axiom A1.
3. ((A → (A → A)) → (A → A)) From 1 and 2 by modus ponens.
4. (A → (A → A)) Instance of Axiom A1.
5. (A → A) From 3 and 4 by modus ponens.

 

 

 

Exercise 2d. Show that {(A → (B → C))} (B → (A→ C)) holds for any well-formed formulas A, B and C of L.

Proof.

1. A → (B → C) Hypothesis.
2. (A → (B → C)) → ((A → B) → (A → C)) Instance of Axiom A2.
3. (A → B) → (A → C) From 1 and 2 by modus ponens.
4. (B → (A → B) Instance of Axiom A1.
5. (B → (A → C) From 3 and 4 by hypothetical syllogism.