Question: In class we've saw the following fact:
$\phi$ is valid $\iff \phi^\forall $ is valid.
However on a test, the following prove/disprove question appeared:
$\Gamma$ is t-satisfialbe $\iff \Gamma^\forall $ is t-satisfiable where $\Gamma$ is a set of formulae.
The answer was that this is false using the example: $\lnot R(x) \wedge R(c)$
(t-satisfiable means that there is a structure and an assignment that satisfy the formula, whereas v-satisfiable is that there exists a structure that satisfies the formula for every assignment in it)
Thoughts
I was trying to understand how come these two "facts" live together, because I assumed that the first one can imply the second one. Would love for an explanation (with some examples preferrably).
The formula:
is obviously $t$-satisfiable in every structure with two elements; it is enough to consider a domain $D = \{ 1,2 \}$ and interpret the individual constant $c$ as $2$ and the predicate $R$ as "__ is Even".
Thus, the assignment $v(x)=1$ will satisfy the above formula.
But no structure can satisfy $\forall x \ (¬R(x) ∧ R(c))$ because at least the assignment $v'$ such that $v'(x)=c^D$, where $c^D$ is the element of $D$ "interpreting" $c$ will "falsify" the formula.