An epistemological argument against metaphysics

Infamously -- at least, as infamously as I can make it -- I am radically indifferent to the concerns, problems, and techniques of metaphysics. I am thoroughgoing here: whether it's Shapiro, Quine, Heidegger, van Inwagen, Aristotle, Leibniz, or Aquinas, I think metaphysics is less interesting and important than the number of hairs on my head. This post is an attempt to justify that radical and thoroughgoing indifference to any metaphysicians that might happen to read it.

If the job of the sciences is to describe the way the world is, then, or so I've been told, the job of metaphysics is to describe the way the world must be. That is (well, this isn't exactly the same thing, but probably what's intended), metaphysics is in the business of discovering (metaphysically) necessary truths, truths of the form []p. This discovery often involves considering (metaphysically) possible truths, truths of the form <>p. I have problems with the idea that we can know such truths, if any such truths there be. That is, I don't think that beliefs that []p or beliefs that <>p are ever justified.

Let's consider an arbitrary necessary truth []p. How could we be justified in believing that []p? I want to consider this by asking what sort of argument could be given for []p. What sort of argument? Unless the metaphysician is prepared to say that we can come to know that []p by generalising empirical truths, presumably the argument should be deductive. More specifically, it should be logically valid and sound.

But what logic? The presence of the modal operator [] suggests that a modal logic. S5 is the most widely accepted, or so I've been told. Perhaps the reader prefers another modal logic. Or perhaps the reader thinks she can do her metaphysics with classical first-order logic. In any case, she will have a set of inferences rules for her logic, which can be stated as axioms, and these axioms will have to be necessary truths themselves. If they are not necessary, then her inferences will not apply across all possible worlds -- her logic will not be metaphysically sound. Furthermore, in order for our belief in the conclusions of the arguments built using these inferences to be justified, our belief in the necessity of these axioms must be justified itself.

In short, in order for metaphysical beliefs to be justified, our beliefs that logical axioms are necessarily true must be justified.

There, I think, only three meta-doxastic attitudes one can take towards axioms. First, on the Aristotelean or Classical attitude, axioms are contentual and must be self-evident. Alternatively, on the Hilbertian or Mathematical attitude, axioms are formal and simply stipulative. (Note that Hilbert himself was not a Hilbertian in this sense about all branches of mathematics -- he was an Aristotelean about finite arithmetic.) On the first attitude, axioms are the most basic and fundamental truths -- and probably this `fundamental' carries both metaphysical and epistemological tones. On the second attitude, axioms are simply laid down arbitrarily, and need not be true of anything. The first attitude has been prominent in Western thought since Antiquity. The second was a late nineteenth-century development out of non-Euclidean geometry and abstract algebra. The third or Contingentist attitude is to take axioms to be assumptions we make contingently, either implicitly or explicitly. Perhaps the axioms express reliable-but-not-necessary and contingent features of our cognition (a certain sort of neo-Humean might think this), or perhaps they express the theory of logic that best exemplifies the features we want a theory of logic to have (Quine thought this, Michael Friedman's `contingent a priori' is somewhat related, and my own view lies in the vicinity of both of these).

Illustrate these three with an example: modus tollens.

(((p -> -q) ^ q) -> -p)

The Aristotelean says that this is self-evident and fundamental in both metaphysical and epistemological senses. Bivalence and non-contradiction are deep features of reality and our knowledge of it. The Hilbertian says that this is just a certain rule for manipulating arrays of strings of symbols -- when you have a subproof that terminates in -q and a separate proof of q, then you can write down the negation of the premiss of the subproof. And the Contingentist says, perhaps, that this is a pragmatically useful posit -- it lets us create proofs by contradiction, which are very powerful indeed, and is only objectionable if you have a very strict understanding of the nature of mathematical reasoning or think truth means justified assertion.

If we take one of these attitudes explicitly, what sort of attitudes towards the axioms is justified? Are we justified in believing that they are necessarily true? For the Contingentist, we are at most justified in believing that axioms are true (that is, true of the actual world). Similarly, as there is no need on the Hilbertian attitude for the axioms to be true, we are at most justified in believing that axioms are true. To infer that axioms are necessarily true is to make a ridiculously hasty generalisation. We are justified, on these two attitudes, at best in believing them to be contingent truths. (Note that even this may not be justified: on certain sorts of Contingentism, such as the neo-Humean, we are not even justified in believing that they are true!) Finally, for the Aristotelean, if self-evidence is indeed justification, then it seems that we are justified in believing that the axioms are true. But self-evidence doesn't imply necessity. Indeed, for classical epistemological foundationalists, sensory beliefs are self-evident, but are assumed not to be necessary at all. If self-evidence is not justification, then it doesn't seem that we are even justified in believing that the axioms are true.

The problem is just this: Any story we tell about being justified in believing that the axioms of logic are true is consistent with them not being necessarily true. Indeed, any story we tell is consistent with the axioms of logic only being true of the actual world, and false of every other possible world.

Something entirely parallel happens if we prefer an externalist account of knowledge, replacing justification with warrant. Every non-ad hoc story the externalist epistemologist-metaphysician tells us about how our belief that the axioms of logic are true is warranted is consistent with them not being necessarily true, and indeed with them only being true of the actual world. The only way out that I can see is to propose that belief that []p is warranted just in the case that []p is true, or something equivalent, which is clearly ad hoc.

To summarise: In order to be justified in our belief that []p, a paradigm example of an important claim of metaphysics, we must first be justified in believing that the axioms of logic are necessarily true. But on no reasonable account of the status of axioms are they necessarily true. Hence our belief that the axioms are necessarily true is not justified. Hence we are justified in believing that []p.