It is at least tempting to answer that PIJ is an analytic truth.
That's because he thought that the adequacy of a definition itself is a matter of analytic truth and that what makes a truth a logical truth is that it is an analytic truth concerning certain symbols.
In keeping with current orthodoxy, we assume that being metaphysically necessary (a property of propositions or sentences) is not to be identified with being a logical truth, being an analytic truth, being a conceptual truth, or being an a priori truth.
Now perhaps this is an analytic truth, in which case we could grant it to Pascal without further discussion — perhaps it is constitutive of rationality to maximize expectation, as some might say.
If, for example, it is an analytic truth that a judgment does not count as moral unless it is appropriately motivating then it trivially follows that anyone who accepts a moral judgment is appropriately motivated or is practically irrational simply because it follows that anyone who accepts such a judgment is appropriately motivated.
Thus, Hartshorne says, "That God exists is one with his essence and is an analytic truth … but how, or in what actual state of experience or knowledge or will, he exists is contingent in the same sense as is our own existence" (Hartshorne 1948, 87).
A similar picture of logic takes consequence to be a matter of what is analytically true, and it is not an analytic truth that water is H2O.
But he argued that when the terms flanking the sign of identity were what he called rigid designators, an identity statement, if true at all, had to be necessarily true, but need not be knowable a priori, as an analytic truth would be.
The first question arises because Hume's Principle doesn't seem to be a purely analytic truth of logic; if neither Hume's Principle nor the existential claim that numbers exist is analytically true, by what faculty do we come to know (the truth of) the existential claim?
The main objection with which the HP-er needs to contend is that Hume's Principle is neither a logical nor an analytic truth.
