Question

Solve using Syntactic Methods

├ (A → C) → [(A & B) → C]

Only use Rules of Inference for &, v, Rules of Inference for →, Rules of Replacement for ~, &, v, Rules of Replacement for ~, &, v, Axioms, Theorems, and Central Syntactic Concepts, Conditional Proof, Indirect Proof, and Advanced Indirect Proof i.e. Simp, Disjunctive Syllogism, Conjunction, Idempotence, Modus Ponens, Modus Tollens, Hypothetical Syllogism, Constructive Dilemma, Double Negation, Commutation, DeMorgans Law, Association, Distribution, Implication, Transportation, Exportation and Equivalence.

Answer 1

To solve the expression ├ (A → C) → [(A & B) → C] using syntactic methods, we apply the rules of inference and replacement step by step.

Step-by-step explanation:

To solve the expression ├ (A → C) → [(A & B) → C] using syntactic methods, we will apply various rules of inference and replacement. Let's break it down step by step:

1. Start with the expression ├ (A → C) → [(A & B) → C].
2. Apply the rule of implication to convert the first conditional: ├ ~(A → C) v [(A & B) → C].
3. Use the rule of implication again to convert the second conditional: ├ ~(A → C) v (~(A & B) v C).
4. Use DeMorgan's Law to distribute the negation: ├ (~(A → C) v ~(A & B)) v C.
5. Apply the rule of implication to convert ~(A & B): ├ (~(A → C) v (~A v ~B)) v C.
6. Apply DeMorgan's Law again: ├ (~(A → C) v ~A v ~B) v C.
7. Combine the two disjunctions: ├ (~(A → C) v ~A v ~B v C).

Thus, the final solution using syntactic methods is: ~(A → C) v ~A v ~B v C.