First off, my experience is only with some applications of logic (in linguistics and knowledge representation), as opposed to formal logical analysis. But, I'm looking for a formal answer to my question above namely: When we make a logical assertion--say, $F(a)$ as in $\text{mortal(Socrates)}$--are we justified in making further assertions related to that assertion?
I know that in first-order logic, when we assert $F(a)$, we are committed to the existence of $a$. In other words, there exists $x$ such that $F(x)$ where $x=a$. This consequence is a theorem of first-order logic. But given that we've asserted that $F(a)$ is $\text{true}$, is it also a theorem in second-order logic that there exists $y$ such that $y(a)$ where $y=F$. It makes sense to me to say that $F$ also exists, since it is being instantiated by $a$, which we already know exists. But is it a second-order theorem? (My scientific background argues that if a property is instantiated, then that property exists.)
In like manner, does the whole proposition, or state-of-affairs, $F(a)$ exist in some higher-order logic, if the components $a$ and $F$ exist? Is it justified or consistent, using some brand of formal logic, to think that it could, or at least be so stated? Is there a name for such a logic that takes propositions as variables so you can talk about whether a proposition exists or not, using the existential quantifier? (ps: I don't know whether such a question makes sense theoretically, since the existence of a proposition seems tantamount to its truth--a vexed topic.)