I want to determine the following formula is logically valid or not?
I know that,
|I| = ω,
a^ I = 10,
P^ I (n) = t, if n is prime, f, otherwise,
Q^I (m, n) = t, if m < n, f, otherwise
How can i find if the below formula is logically valid or not?
P(x) ⊃ ∃yP(y)
If you mean $\forall P(x) \rightarrow \exists P(y)$, then that is valid: since every object in the (non-empty!) domain has property $P$, there is of course going to be at least one thing in the domain that has property $P$.
But if you really meant for that formula to be $P(x) \rightarrow \exists y P(y)$, then that formula is not a sentence, and so it has no truth-value, and so it is not necssarily true either, which is what we typically mean by being 'logically valid'.
Then again, we could define a 'logically valid formula' as one that will be satisfied by every object of the domain, and then yes, it would be a logically valid formula, since given any object, if it has property $P$, then there is something with property $P$, and if it does not have property $P$, then the antecedent of the conditional would be false, and hence the conditional be true (to be exact, for any object $d \in D$, and where constant $c$ were to denote $d$ under the extended interpretation $I'$, the sentence $P(c) \rightarrow \exists y P(y)$ is always true). Which also means that $\forall x(P(x) \rightarrow \exists y P(y))$ is logically valid in the normal sense of the term.