# Semantics- predicate calculus and quantifiers

I have a sentence of "No A is B" (No child is sad) I been given 2 formulas: ¬∃x[C(x) ⋀ S(x)]
∀x[C(x) →¬ S(x)]

I needed to show that they are the same by deriving the truth conditions. I got that: "Iff for every d ∈ Dm, d ∉ Fm(C) or d ∉ Fm(S)"

Now, I have a new formula that is suggested "∀x[¬[C(x) ⋀ S(x)]]", I need to say if it is equivalent to the other two formulas. I think it is not, but I cant find an explanation.
Can someone help me out here?

• They are equivalent by simple manipulation. It comes from the first one by moving the negation inside the quantifier (and reversing the quantifier). You get the second one by expanding " A implies B" to "(not A) or B" and moving the negation outside the alternation. Jan 9, 2023 at 12:21
• Right. The first one is due to DeMorgan's Laws, which deal with negatives and quantifiers, and the second one is Reduction of Implication, which says that "A implies B" is equivalent to "not-A or B". These are explained (and more theorems are given) in the Logic Study Guide. Jan 10, 2023 at 16:55