if p v r is satisfiable then p is satisfiable

Question

When checking if a propositional logic is true or not, do you need to consider all possible models of it? For example in this question:
if p v r is satisfiable then p is satisfiable

If you only consider the model p = false and r = true then p isn't satifisable. But if you consider all possible models of p v r, p is satisfiable, as there the exists a model p = true and r = false.
Which is the correct method to think about this?
What if p v r is valid?

Thank you in advance

Answer 1

The former way is the correct way: just because $p \lor r$ is satisfiable does not mean that $p$ is satisfiable, unless $r$ implies $p$.

Answer 2

It depends on whether you consider $p$ and $r$ to be propositional variables or placeholders for possibly more complex formulas.

If they are two fixed propositional variables, then "$p \lor r$ is satisfiable" and "$p$ is satisfiable" are simply both true, and "if true then true" is a true claim.

However if $p$ and $r$ are placeholders, then "$p\lor r$ is satsifiable" doesn't get a truth value before you choose a concrete wff for $p$ to stand for. And in that case, it is natural to understand your claim as

For all wffs $p$ and $r$ it holds that if $p\lor r$ is satisfiable, then $p$ is satisfiable.

And that is manifestly not the case. One counterexample is to let $p$ be $(A\land \neg A)$ and $q$ be $A$, where $A$ is a propositional variable.