Prove the following by deductive system（Using First-order Logic）: If x does not occur in ψ, then ∃x : S • (φ(x) ⇒ ψ) ⊨ (∀x : S • φ(x)) ⇒ ψ.

1. Assume that x does not occur in ψ.
2. Let a be an arbitrary element of S such that φ(a) ⇒ ψ.
3. Assume ∀x : S • φ(x).
4. By Universal Instantiation, φ(a).
5. Using Modus Ponens with (2) and (4), we can conclude that ψ.
6. Therefore, (∀x : S • φ(x)) ⇒ ψ, as desired.

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Prove the following by deductive system:

As an AI language model, I am not able to present a deductive proof without a specific statement or theorem to prove. Please provide me with a statement or theorem that needs to be proven, and I would be happy to assist you with a deductive proof.

Prove the following by deductive system: ⊨ φ ∨ ¬φ (i.e., ∅ ⊨ φ ∨ ¬φ)

Proof:

1. Assume ¬(φ ∨ ¬φ) for the sake of contradiction.
2. By De Morgan's Law, ¬(φ ∨ ¬φ) is equivalent to ¬φ ∧ ¬¬φ.
3. By Double Negation, ¬¬φ is equivalent to φ.
4. Therefore, ¬φ ∧ φ.
5. By the principle of Non-Contradiction, a statement and its negation cannot both be true.
6. Therefore, ¬(φ ∨ ¬φ) leads to a contradiction.
7. By the principle of Explosion, a contradiction implies anything.
8. Therefore, φ ∨ ¬φ must be true, as assuming its negation leads to a contradiction.
9. Therefore, ⊨ φ ∨ ¬φ, as it is true under all interpretations.

Note: This proof uses the principle of Non-Contradiction and the principle of Explosion, which are two fundamental principles of classical logic.