Let φ(x) and ψ be first order formulas so that x is not free in ψ.
Show why this sentence is valid:
( (∀xφ(x) )→ ψ) ↔ (∃x( φ(x) → ψ) )
I hope someone can give me a simple semantical explanation
(no need to show a natural deduction)
Let φ(x) and ψ be first order formulas so that x is not free in ψ.
Show why this sentence is valid:
( (∀xφ(x) )→ ψ) ↔ (∃x( φ(x) → ψ) )
I hope someone can give me a simple semantical explanation
(no need to show a natural deduction)
The semantics of first-order logic is designed to reflect our meta-level concepts of "implies," "for all," etc. Do you see why this formula is true when the connectives are interpreted as these natural-language concepts? Hint: for the left-to-right direction, consider the cases when ∀xφ(x) is true and when it is false.