Deduction Theorem Counterexample Relational Logic

Counterexample: Γ = {} φ = p(x) ψ = ∀x p(x) Now, applying Deduction Theorem, from prop. logic, substituting above Γ |= (φ ⇒ ψ) iff Γ ∪ {φ} |= ψ yields: ({} |= (p(x) ⇒ ∀x p(x)) ⇔ ({p(x)} |= ∀x p(x)) The LHS of this equivalence is the logical entailment {} |= (p(x) [...]

Prop. Logic Metatheorems

Deduction Theorem: Δ |- (φ ⇒ ψ) if and only if Δ∪{φ} |- ψ. Substitution Theorem: Δ |- (φ ⇔ ψ) and Δ |- χ, then it is the case that Δ |- χφ←ψ. ChainingTheorem: If Δ|-(φ⇒ψ)andΔ|-(ψ⇒χ), then Δ |- (φ ⇒ χ).