If Descartes had tried to apply his Cogito principle objectively, he would have ended up saying 'I doubt I exist, but to doubt is to think, and to think one must be, so anyone who doubts my existence must accept that I exist, for they cannot doubt my existence without thinking of my existence'
I'm sure Inzababa would say something much more convoluted.
Of course there are, because there are different scopes.
In this, the actual world, it is not possible that JFK is the President of the US. This is a fact based on historical and political states of affairs.
In some other possible world, JFK was not assassinated, was extraordinarily fortunate, and still reigns to this day.
I like to distinguish these types of possibilia as 'localised possibilia' (things that are or are not possible given the way things are in the actual world) and 'modal possibilia' (things that are true in one or more possible worlds).
The actual world (unless you're an extreme modal realist) is this world. All other worlds are non-actual, yet possible.
If you are an extreme modal realist, then all worlds are actual, and we distinguish this world by simply saying 'this world'.
Then we can get into schemas, and worlds where certain maximal sets of propositions hold true, but that's irrelevant. The whole notion of 'different kinds of possible' is irrelevant, of course, because the fact that a bachelor is unmarried (and the other axioms we threw your way) is necessarily true under all scopes of standards of possibility.
A possible world, just to clear this up, is a maximal set of logically consistent propositions.
It's not so much an assumption as a reasoned inference.
Ax is a universal quantifier. It was created specifically for that purpose. To state 'Ax denotes that for all x or for any given x' is no more an assumption than to state 'The letter B is the letter B' or 'The colour red is not blue'.
But no, I'm tired of this, we'll ask the bloody questions.
Give us an axiom, or an argument, in predicate format, using modal logic if you wish, that demonstrates your grasp of logic. Use predicate format and the English vernacular to facilitate your demonstration.