To demonstrate the semantic equivalence between ~∃x[~Fx] and ∀x[Fx], use the truth-tree method:
Start with ~∃x[~Fx].
Apply negation to the existential quantifier, resulting in ∀x[~~Fx] (De Morgan's Law).
Simplify ~~Fx to Fx, yielding ∀x[Fx].
Comparing with ∀x[Fx] confirms their equivalence.
The truth-tree method confirms ~∃x[~Fx] as equivalent to ∀x[Fx].
To demonstrate the semantic equivalence of ~∃x[~Fx] and ∀x[Fx], use the truth-tree method step by step:
Negation of Existential Quantifier: Start with ~∃x[~Fx].
Apply the negation to the existential quantifier:
Transform ~∃x[~Fx] into ∀x[~~Fx] (by De Morgan's Law).
Simplify ~~Fx to Fx:
~∃x[~Fx] simplifies to ∀x[Fx].
Comparison with ∀x[Fx]:
Since step 3 leads to ∀x[Fx], both ~∃x[~Fx] and ∀x[Fx] are equivalent after simplification.
Conclusion:
The truth-tree method demonstrates that ~∃x[~Fx] is semantically equivalent to ∀x[Fx].
Ensure these steps are carried out accurately to show the logical equivalence of the given formulas using the truth-tree method.