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.

## 10 comments:

The purpose would probably better be phrased slightly differently: I want to give reasons for rejecting metaphysics that contemporary Analytic metaphysicians themselves will at least find reasonable. This requires speaking their language -- the language of possible worlds and formal logic. My own personal reasons have to do with the fact that we cannot cognise the thing-in-itself, but I don't even want to try to translate this into the language of my colleagues.

Procrastinating a bit this beautiful Saturday morning, I came across a paper by Edward Zalta, `Logical and analytic truths that are not necessary' (J. Phil., 85:2 (Feb. 1988), pp. 57-74).

Zalta's principle claim: `by merely adding expressive power that allows us to talk about what is actually the case, the modal propositional calculus involves distinctions that undermine the traditional view that logical truths are necessary.'

And principal moral: `One should beware of the inference from logical truth to metaphysical necessity even in languages without rigid descriptions, for there may be an (implicit) actuality operator somewhere that invalidates the move.' (63, his emphasis)

"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."

You don't think that the only way one can be justified in believing that p is by having an *argument* for p, do you?

No, clearly not: Aristotelean axioms, at least, aren't justified by argument, and I completely allow you to take that view of axioms here. What third category would you like besides axioms and beliefs justified by arguments?

Give me a modal system with all the axioms assumed, and all I'll ever get out of it are modal theorems. But this is an uninteresting result.

Why can't I assume some premises too, and from these (in conjunction with the axioms and rules of inference) run my modal arguments? So far as I can tell, you've given no reason to suppose that one can't be justified in believing some modal truth through some means other than deductive argument. And if you haven't done this, then you really haven't crafted an interesting argument that's directed at anyone that actually does metaphysics, I'd think.

To answer your question directly, here are some examples of modal truths I take it I'm justified in believing in the basic way (roughly, not on the basis of any other belief, much less any *argument*). I employ these truths as premises in arguments--but they aren't the conclusions of any arguments (better: I don't need such arguments to have warrant for belief in these truths):

1. It's possible that determinism is true.

2. Possibly, I don't exist.

3. Necessarily, no prime minister is a prime number.

4. Necessarily, no red thing is non-colored.

You can certainly play around with any formal language you'd like. Take any P you like and show that, using S5, you can derive Q from P. I have no problems at all if you then go around asserting things like

S5 + P |- Q,

S5 |- P -> Q, and

S5 |- -P v Q.

But this seems just as interesting a result as things of the form

S5 |- R.

Now, maybe you think you have warrant for the axioms of S5. In this case, then you have warrant for such potentially interesting things as

P -> Q, and

R.

But why do you think you have warrant for the axioms of S5, when -- if you accept the argument of the post -- you don't even have warrant for the necessity of modus tollens, which is less formally complex than all of the axioms of S5 except, possibly, T?

Also, I don't see how your 1-4 are different from axioms. And I'd be interested to hear your account of how you have warrant for them.

Here's two other ways of putting the question: Why are you warranted in believing that the laws of sentence logic are true at every possible world? Or, Why should I believe that the laws of sentence logic are true at every possible world?

"Why are you warranted in believing that the laws of sentence logic are true at every possible world? Or, Why should I believe that the laws of sentence logic are true at every possible world?"

First, you ask two questions, and suggest that they're the same. They're not. The grounds of my warrant for p needn't be such that they also confer warrant for p for you.

Second, the warrant I've got for thinking that some logical truth is necessarily true comes in the same ways that I have warrant for other basic beliefs, I'd think. Truth-seeking cognitive faculties properly functioning in an appropriate epistemic environment, blah blah blah. I'll tell the same (or much the same) story here as I would tell for any other basic belief.

I was thinking about proper functionalism yesterday. But that's a topic for another post.

In any case, since I prefer an epistemology that allows me to at least fallibly check and see if I really do have warrant for my beliefs, you haven't really given an answer to the second version of the question. At least until someone invents the design-plan-o-scope. (Sorry for the snark, I've had too much caffeine this afternoon and I'm not really looking forward to a 2 1/2 hour seminar starting in 40 minutes.)

Following one of Solomon's better teaching strategies, I think I'll let you have the last word in this thread.

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