Why do $\exists x\in S: p(x)$ and $\forall x\in S: p(x)$ have different logical structures?
89808
When to use implication and when to use conjunction?
19209
Why is compactness in logic called compactness?
31212
The existence of the empty set is an axiom of ZFC or not?
20556
Isomorphism between two models of Hilbert's Axioms with different notions of congruence
62936
Proving equivalencies of two versions of the Axiom of Choice
79367
Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$.
105455
Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent.
35115
Questions on how to prove that a set of connectives is NOT functionally complete
9158
Proving that a propositional theory of any cardinality has an independent set of axioms
87928
Proof of logical equivalence of biconditional and other proposition
88281
Mixing and Distributing Qualifiers ($\forall x$, $\exists x$)
48614
Why is "$P \Rightarrow Q$" equivalent to "$\neg Q \Rightarrow \neg P$"?
93557
Intuitively, why does (B → S) ≡ (¬B ∨ S)?
578
In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?
42150
Prove that $\forall x \neg P(x) \iff \neg \exists x P(x)$
101163
Does there exist a universal formula of first-order logic that is satisfiable only by structures with infinite domains?
45481
Prove or disprove validity: $(\forall x \exists y (P(x) \supset Q(y))) \supset(\exists y \forall x (P (x) \supset Q(y)))$