Question

Prove that the conclusion, ∃xH(x), follows from the following premise(s): ∀x(G(x)→F(x))→∃x(J(x)∧H(x)) ∀x(G(x)→H(x))∧∀x(H(x)→F(x)) P.S.: Clearly state the name of each Inference Rule and/or Logical Equivalence applied at each step of your proof.

Answer 1

Answer and Step-by-step explanation:

To prove that the conclusion ∃xH(x) follows from the given premises, we will use logical equivalence and inference rules.

1. Premise: ∀x(G(x)→F(x)) → ∃x(J(x)∧H(x))

Inference Rule: Implication Introduction

2. Premise: ∀x(G(x)→H(x))∧∀x(H(x)→F(x))

Inference Rule: Conjunction Elimination

3. From premise 2, we have: ∀x(G(x)→H(x)) and ∀x(H(x)→F(x))

Inference Rule: Conjunction Elimination

4. From premise 3, we have: ∀x(G(x)→H(x))

Inference Rule: Universal Instantiation

5. From premises 1 and 4, we have: ∃x(J(x)∧H(x))

Inference Rule: Modus Ponens

6. From ∃x(J(x)∧H(x)), we can infer ∃xH(x) by existential instantiation.

Inference Rule: Existential Instantiation

Therefore, we have proven that the conclusion ∃xH(x) follows from the given premises using the inference rules of Implication Introduction, Conjunction Elimination, Universal Instantiation, Modus Ponens, and Existential Instantiation.