This means that A can be a part-whole system only if PA fulfils certain conditions - if it is acyclic (its transitive closure is antisymmetric) and if it fulfils some other more intricate 6 conditions, which we are not going to discuss here .
But given simply a notation, mathematical or otherwise, and indeed as thoroughly understood a notation as you please in point of the translation or truth conditions of its statements, who can say which of its true statements rank as postulates?
Now the notion of semantical rule is as sensible and meaningful as that of postulate, if conceived in a similarly relative spirit -- relative, this time, to one or another particular enterprise of schooling unconversant persons in sufficient conditions for truth of statements of some natural or artificial language L. But from this point of view no one signalization of a subclass of the truths of L is intrinsically more a semantical rule than another; and, if 'analytic' means 'true by semantical rules', no one truth of L is analytic to the exclusion of another.
Or, to change the figure, total science is like a field of force whose boundary conditions are experience.
But the total field is so undetermined by its boundary conditions, experience, that there is much latitude of choice as to what statements to re-evaluate in the light of any single contrary experience.
Specifically, because perceptions, intentions, beliefs, desires, and so on, are forms of intentionality, they carry within them the determination of conditions of success or failure.
We can generalize this point as follows: Any intentional state determines its conditions of satisfaction, and a normal animal that has intentional states must be able to recognize when the conditions of satisfaction are in fact satisfied.
If it is thirsty, it must be able to tell when it has drunk; if it is hungry, it must be able to tell when it has eaten; if it is trying to do something, it must know when it has done it, and so on. We can summarize this point by saying that when we supposed that our animals had intentional states we were already supposing that they had mental representations with propositional contents and conditions of satisfaction.
Rather, to have beliefs and desires, for example, is already to have something that determines conditions of satisfaction, and that implies the capacity to recognize success and failure.
I mean that there is something that sets the conditions of satisfaction; and because a condition is always a condition that such and such, it follows trivially that the conditions are propositional.