Explaining the A Priori in Terms of Meaning and Essence

UPDATE (Nov 2019): A descendent of this post, 'Apriority and Essential Truth', has been published in Metaphysica.

It wasn't just the positivists who thought there was a tight connection between meaning and truth in the case of a priori propositions:

However, it seems to me that nevertheless one ingredient of this wrong theory of mathematical truth [i.e. conventionalism] is perfectly correct and really discloses the true nature of mathematics. Namely, it is correct that a mathematical proposition says nothing about the physical or psychical reality existing in space and time, because it is true already owing to the meaning of the terms occurring in it, irrespectively of the world of real things. What is wrong, however, is that the meaning of the terms (that is, the concepts they denote) is asserted to be something man-made and consisting merely in semantical conventions. (Gödel (1951/1995), p. 320.)

Perhaps we should try to recover some insight from the idea, nowadays highly unfashionable within philosophy (but alive and well in the broader intellectual culture, I think), that a priori truths like those of mathematics are in some sense true owing to their meanings. Philosophers often used to express this by calling such propositions 'necessarily true', but since Kripke that sort of usage has been crowded out by another.
  
Noteworthy in this connection is that Kripke was not altogether gung ho about his severance of necessity from apriority:

The case of fixing the reference of ‘one meter’ is a very clear example in which someone, just because he fixed the reference in this way, can in some sense know a priori that the length of this stick is a meter without regarding it as a necessary truth. Maybe the thesis about a prioricity implying necessity can be modified. It does appear to state some insight which might be important, and true, about epistemology. In a way an example like this may seem like a trivial counterexample which is not really the point of what some people think when they think that only necessary truths can be known a priori. Well, if the thesis that all a priori truth is necessary is to be immune from this sort of counterexample, it needs to be modified in some way. [...] And I myself have no idea it should be modified or restated, or if such a modification or restatement is possible. (Kripke (1980), p. 63.)

This may make it sound like the required modification would consist in somehow ruling out the problematic contingent a priori truths from the class of truths whose epistemic status is to be explained. But Chalmers' idea of the tyranny of the subjunctive suggests another route: try instead to find a different notion of necessity - indicative, as opposed to subjunctive, necessity; truth in all worlds considered as actual, rather than truth in all worlds considered as counterfactual - better suited to the explanation of apriority.

Now, in Chalmers' epistemic two-dimensionalist framework, indicative necessity is itself explained in epistemic terms. But if we try for a more full-bloodedly semantic conception of it, we may get something more explanatory of the special epistemic status of a priori truths. The notion we are after is something like: a proposition is indicatively necessary iff, given its meaning, it cannot but be true. And the modality here is not supposed to be epistemic.

But what aspect of its meaning? Sometimes 'meaning' covers relationships to things out in the world, and even the things out there themselves. What we are interested in is internal meaning. Putnam's Twin Earth thought experiment - though this is not how he used it - lets us see the distinction we need here. We want to talk about meaning in the sense in which Earth/Twin Earth pairs of propositions mean the same. This can be articulated using the middle-Wittgenstein idea of the role an expression plays in the system it belongs to (see Wittgenstein (1974, Part I)).

So, what if we say that a proposition is indicatively necessary iff any proposition with its internal meaning must, in a non-epistemic sense, be true? Can indicative necessity in this sense be used to explain apriority?

Maybe not, since there are indicatively necessary truths which are indicatively necessary only because their instantiation requires their truth. Example: language exists. (Language is here understood as a spatiotemporal phenomenon.) This is indicatively necessary, because any proposition with its internal meaning must be true, if only because the very existence of that proposition requires it to be true. Its truth comes about from the preconditions for its utterance, but - you might think - not from the internal meaning itself. It is interesting to note that it is indicatively necessary, but it lacks the special character of a priori propositions whereby they, in some sense, don't place specific requirements on the world.

This situation pattern-matches with Fine's celebrated (1994) distinction between necessary and essential properties. Socrates is necessarily a member of the set {Socrates}, but that membership is not part of his essence, since it doesn't have enough to do with Socrates as he is in himself. Likewise, he is necessarily distinct from the Eiffel Tower, but this is no part of his essence. So let us throw away the ladder of indicative necessity and instead hone in on the notion of essential truth. A proposition is essentially true iff it is of its internal meaning's essence to be true (i.e. to be the internal meaning of a true proposition).

Thus, with encouragement from Gödel and Kripke, we can develop ideas from Chalmers, Putnam, Wittgenstein, and Fine, to yield:

To say that a proposition is a priori is to say that it can, in some sense, be known independent of experience. (You may need experience to get the concepts you need to understand the proposition, but you don't need any particular further experience to know that the proposition is true.) What is distinctive about these propositions which explains their being knowable in that peculiar way? It is that their internal meanings - their roles in language - are, of their very essence, the internal meanings of true propositions; any proposition with that internal meaning must be true, and not for transcendental reasons relating to the pre-conditions of the instantiation of the proposition, but as a result of that internal meaning in itself.

So we can have an account of apriority which explains it in terms of a tight connection between meaning and truth, freed of its accidental associations with conventionalist and deflationary views about meaning, modality and essence.

This is not to say that a priori propositions' truth is to be explained in a case by case way by considerations about meaning and essence. That would be to crowd out the real mathematical justifications of non-trivial mathematical truths. But explaining apriority in general in this way wards off misunderstandings which come from treating a priori truths too much like empirical truths. And that is what makes it an explanation.

References

Chalmers, David J. (1998). The tyranny of the subjunctive. (unpublished)

Fine, Kit (1994). Essence and modality. Philosophical Perspectives 8:1-16.

Gödel, Kurt (1951/1995). Some basic theorems on the foundations of mathematics and their implications. In Solomon Feferman (ed.), Kurt Gödel, Collected Works. Oxford University Press 290-304. (Originally delivered on 26 December 1951 as the 25th annual Josiah Willard Gibbs Lecture at Brown University.) 

Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.

Putnam, Hilary (1973). Meaning and reference. Journal of Philosophy 70 (19):699-711. 

Wittgenstein, Ludwig (1974). Philosophical Grammar. University of California Press.