When we claim that something is a priori we are claiming that it can be known to be true independently of experience. Of course we may be mistaken in judging that something is a priori (Kant famously thought that we could know space to be Euclidean a priori), but it seems self evident that there are at least some simple truths, such as 5 = 5, or “all bachelors are unmarried” that simply must be a priori.
Let us first turn then to the logical truths (and thus mathematical truths), which can supposedly be known a priori. We might reason that since there is only one way to correctly reason logically then this way could be discovered without reference to the external world, since the other possibilities would be in some way incoherent. However, this conclusion rests on a flawed premise, namely that there is only one coherent way to reason logically. Numerous other logical systems have been developed in recent years, sometimes proving useful and sometimes being nothing more than curiosities. Even if we were to start with the same set of axioms, in some systems we would be able to deduce that 5 = 5, in some that 5 = 4, in some both, and in some neither. How then can we claim that 5 = 5 can be known a priori? We might think that we could relativize our claim to the logical system we deduced it from (i.e. 5 = 5 under classical logic), but this isn’t real progress because every conclusion can be deduced under some logical system (some systems allow one to deduce all possible statements). So we would still be forced to conclude that every statement could be known a priori, which is unacceptable. What we really want is some way to pick out, in an a priori fashion, which logical system is the “right” one, and thus be able to discard the conclusions derived from the others. Unfortunately the way we decide which logical system is “right” is by observing which one most closely reflects the way truth is preserved in our experience. For example, we know that a person can’t be in two places at once, so any logical system that allowed us to deduce that I could both be in the store and not be in the store at the same time (from some basic set of axioms about objects and locations) would be a bad one. But we can’t make this move without some appeal to experience, or common sense derived from that experience. And thus to say that some statements are a priori because they are logically necessary is to say that every statement can is a priori. Thus if the a priori is to be a meaningful category statements cannot be said to be a priori because of their logical status.
So then, how else can we pick out what is a priori? Another possibility is that whatever can be known to be true based solely on the definitions of words (or the meaning of words) is an a priori truth. Now obviously there might be some issues with assuming that the definitions of words can be known outside of experience, but let us just assume for a moment that they can. The problem with this is that either some of the statements that are thought to be a priori under this criterion will be false or the words aren’t guaranteed to actually refer to existing objects. To see why consider a person who knows the meaning of every word shortly after the discovery of the neutrino. When neutrinos were discovered it was thought that they couldn’t interact with matter, so the definition of a neutrino was something along the lines of “a particle produced by reactions: … that does not interact with matter”. And so such a person could know sentences like “all bachelors are unmarried” and “neutrinos don’t interact with matter” to be true a priori. But neutrinos do interact with matter, it just happens very rarely. And thus at least some of the things this person knows to be true a priori are actually false. Or if we don’t want to say that they are false we could say that their use of “neutrino” didn’t refer to real neutrinos, and hence the sentence they thought to be true a priori was really meaningless. Both responses may be acceptable, the problem really is that the person can’t pick out which words don’t refer to real objects, or which sentences are false, a priori. Again, this would make the a priori essentially useless, because even if we can put forward some statement as something that can be known a priori we have no guarantee that the statement is either true or meaningful.
There is perhaps one exception to this: sentences dealing with words that refer to things constructed by the words themselves, such as “bachelor”. Unlike neutrons there is nothing more to being a bachelor than what we define it as. Even with this exception in mind, we can conclude from the above arguments that claiming something to be a priori is to say little. It might have some force when speaking about things we have created through our definitions, but there is no need to investigate such things, and to show that some claim about them was a priori would be to tell us only what we already knew. Of course, what comes to mind when I think of claims that are put forward as a priori is bad metaphysics. There are some philosophers who, though some conceptual analysis, will produce claims that they claim are a priori, and thus supposedly true and unquestionable. But as we have just shown these claims have little force, and can be safely ignored (since their subjects are never things that we have constructed, but rather things such as consciousness, causation, nature, ect).