To determine the validity of the given statement, let's break it down and analyze it step by step using logical equivalences.
1) ∃x (S(x) ∧ P(x)) ⇔ ¬∀x (S(x) → ¬P(x))
First, let's simplify the right side of the equivalence using the negation of the implication:
2) ∃x (S(x) ∧ P(x)) ⇔ ¬∀x (¬S(x) ∨ ¬¬P(x))
Next, applying De Morgan's Law to the negation of the universal quantifier:
3) ∃x (S(x) ∧ P(x)) ⇔ ¬(∀x ¬S(x) ∨ ∀x ¬¬P(x))
Since the quantifiers are over the same variable, we can rewrite it as:
4) ∃x (S(x) ∧ P(x)) ⇔ ¬(∀x ¬S(x) ∨ ∃x P(x))
Now, using the negation of the disjunction:
5) ∃x (S(x) ∧ P(x)) ⇔ ¬(∀x ¬S(x)) ∧ ¬(∃x P(x))
Applying the negation to the universal quantifier:
6) ∃x (S(x) ∧ P(x)) ⇔ ∃x S(x) ∧ ¬(∃x P(x))
Finally, using De Morgan's Law again:
7) ∃x (S(x) ∧ P(x)) ⇔ ∃x S(x) ∧ ∀x ¬P(x)
The statement is now simplified. From this, we can observe that the left side of the equivalence states that there exists an x such that S(x) is true and P(x) is true. Meanwhile, the right side of the equivalence states that there exists an x such that S(x) is true and for all x, P(x) is false.
Therefore, the original statement is not valid.