Truth Pluralism and Mathematics

When I started out as a graduate student writing a thesis defending mathematical nominalism, my naïve view was that mathematical claims (the correct ones) were true in a different way to empirical claims (the correct ones). Before long though, under the influence of Tarski, the basic model theory I was teaching in Logic 1, etc. I came to reject that view. Truth had to do with satisfaction in the Tarskian sense which required a domain of objects to do the satisfying. Recently though I came across this passage from Huw Price:

[W]e need to distinguish the notion of keeping track as something that we do within the assertoric language game – a notion constituted, within the game, by the fact that the normative structures always hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player – from a notion that we might employ from outside the game, in saying that in at least some of its versions its function is to aid the players in keeping track of their physical environment. … [T]here’s a temptation to call both kinds of external constraint ‘truth’, but we shouldn’t make the mistake of thinking we’re dealing with two aspects of sub-species of a single notion of truth. Both notions may be useful, for various theoretical purposes, but we shouldn’t confuse them’ (Expressivism, Pragmatism and Representationalism, 191)

I’m beginning to think that my naïve view might just be the right one. Any assertoric discourse [Could there be a non-assertoric discourse? I’m not sure.] with standards of correctness and incorrectness will need a truth predicate for the sorts of expressive purposes deflationists get excited about. (We need a truth predicate to express commitments without having to state them explicitly. E.g. if a theory \(\Gamma\) entails infinitely many things I can say ‘Everything \(\Gamma\) entails is true’ but I can’t possibly explicitly assert everything entailed by \(\Gamma\)  Hence truth predicates are indispensable for certain expressive purposes.) Mathematical discourse seems like a good candidate for a discourse that’s governed by internal rather than external standards. In empirical matters the world gets to answer back: we bump up against the world, probe it, test it experimentally. But mathematical investigations don’t involve interactions with mathematical objects, they involve proofs. Of course the results of these investigations can surprise us or be counterintuitive. But none of that requires an external world of mathematical objects; only that the normative structures constituted within the game ‘hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player’. Objectivity doesn’t require objects. Moreover, as I’ve argued before there’s good reason to think that the norms of correctness and incorrectness of mathematical discourse are internal, rather than external, in the senses above.

Wright on Deflationism

I’m reading Crispin Wright’s Truth and Objectivity for a reading group at the moment, where he sums up an argument against deflationism about truth in the following way (I quote at length):

The deflationist holds that “true”, although gramatically a predicate, denotes no substantial quality of statements, or thoughts, but is merely a device of assertoric endorsement, of use to us only because we sometimes wish so to endorse a single statement, referred to in a way which doesn’t specify it’s content, or batches of statements all at once. Apart from applications of those two kinds, it is, for the deflationist, a complete explanation of the truth predicate that it satisfies the Disquotational Schema. It is a consequence of this general conception of the role of the truth predicate that it can register no norm governing assertoric discourse distinct from warranted assertibility. Yet the central place assigned to the Disquotational Schema—and thereby to the Negation Equivalence—actually clashes with that consequence, for it follows that, while normative of assertoric discourse, and indeed coincident in (positive prescriptive) normative force with warranted assertibility, “true” is nevertheless potentially extensionally divergent from warranted assertibility—and hence has to be accounted as registering a distinct such norm. Since it’s compliance or non-compliance with a norm distinct from assertoric warrant can hardly be an “insubstantial”property of a statement, and since a uniform account is possible of what it is for any particular statement so to comply, deflationism collapses. (pp. 71–2)

For reference, the Disquotational Schema is:

(DS) “P” is T is and only if P

One can’t, without engaging in a kind of doublethink, say or believe things like ‘P and it is not warrantedly assertible that P’ for some proposition P, since you can rationally assert P if and only if it is warrantedly assertible (for you) that P. But truth can be used to contrast with warranted assertibility. Take (DS) and substitute ‘It is not the case that P’ for P:

(i) ‘It is not the case that P’ is T is and only if it is not the case that P.

From (i) and (DS) one can infer:

(ii) It is not the case that P if and only if it is not the case that ‘P’ is T.

And from (i) and (ii) you get:

(iii) ‘It is not the case that P’ is T if and only if it is not the case that ‘P’ is T.

But (iii) cannot be right if T means warrantedly assertible, so ‘true’ registers a norm distinct from warranted assertibility. Deflationism is the view that ‘true’ just is a devise of assertoric endorsement, and Wright thinks that a mere devise of assertoric endorsement couldn’t register a norm distinct from warranted assertibility, so deflationism must be false.

But there is a way to register the truth norm without using the truth predicate. One can say ‘P and it is not warrantedly assertible for S that P’ or ‘¬P and it is warrantedly assertible for S that P’ where S is some person other than yourself, and in doing so can register the truth norm without using the truth predicate. What is required is that one contrasts one’s own perspective with that of another. Registering the truth norm requires some kind of I-Thou contrast. If that’s the case then ‘true’ is not what, fundamentally, allows one to register the truth norm contrasting with the norm of warranted assertibility, and deflationism is off the hook.

I think this might tap into something deep about objectivity—more specifically, our ability to see the world as being objective or to conceptualise there being objective facts that outstrip our ability to know them—as it coheres with something Robert Brandom says about objectivity. Brandom (I won’t spell out the details here) also argues that conceptualising objectivity requires I-Thou relationships. This is made explicit in paradigmatically referential of-statements like ‘He believes of this criminal that he is an innocent man.’ Understanding “of” requires navigating between one’s own perspective and that of another. Since “of” is how we refer objectively to the world, talking (and hence thinking) about the world objectively requires navigating between one’s own perspective and that of another.

Softening the Blow of Truth Fictionalism

In the last post I said that mathematical fictionalism has consequences that sound terrible, but really aren't worrisome at all, when you think about what they actually entail. Something similar could be said about alethic fictionalism, or fictionalism about truth. Consider the truth predicate.  It’s a widely held view that the purpose of having a truth predicate is to allow people to undertake commitments to certain claims without having to explicitly state those claims.  So, if a theory $\Gamma$ entails infinitely many claims that you want to endorse, you can say ‘Everything entailed by $\Gamma$ is true', rather than explicitly state every claim entailed by $\Gamma$, which would be impossible.  So the job of the truth predicate isn’t to ascribe a special property, TRUTH, to things, but to allow us to undertake commitments without having to articulate those commitments explicitly.

Nevertheless, you could consistently hold that, even though what explains why we have a truth predicate has nothing to do with ascribing the property TRUTH to things, the meaning of a predicate is always to ascribe a property to something.  And if, in addition, you held that there is no such property as TRUTH, then you would end up being a fictionalist about truth discourse.  Any claim of the form ' $\phi$ is true’ would be false, since nothing has the property of truth.  But this wouldn’t really matter, since the truth predicate would still allow us to do what it was designed to do.  (It would, however, sound like a really bad result.)

Dialetheism for cheap?

It’s easy to “make” a new truth.  I can define the term busy* thusly:

For any x, x is busy* iff it contains more than five items.

Given this definition it is true that the room I am currently in is busy*.  I can define another term busy** thusly:

For any x,

(1) x is busy** if it contains five items or more, and

(2) it is not the case that x is busy** if it contains seven items or fewer.

Now consider a room containing six items; it is both true and false that the room is busy**.  Clearly the concept of busyness** is inconsistent, yet the sentence ‘This room is busy**’ seems to express a proposition—inconsistent claims are not unintelligible in virtue of their inconsistency.  Since the claim expresses a proposition it has a truth value, and in cases where ‘this room’ designates a room containing six items, the claim will be both true and false.

Now, we might not be too worried about inconsistencies of this sort, since they involve no worldly contradiction—there is nothing inconsistent or incoherent about a room containing six items—only the deployment of inconsistent concepts.  Yet, so long as some sentence or proposition is both true and false—regardless of whether this involve a worldly contradiction)—then, in classical logic, by the misnomed (yes, that is a word) ex falso quodlibet, it follows that every sentence or proposition is true, which is absurd.  As such, cheap dialetheism of this sort is sufficient to show that we must reject classical logic in favour of a relevance logic.

It seems to me something must be wrong with this argument, but I’m not sure what.

The Inverse Indispensability Argument

Most philosophers take it that the truth term plays the role of a predicate.  Since predicates denote properties this provides prima facie reason to think that truth is a property; many claims that we take to be correct appear to be ascriptions of a truth property to sentences or propositions.  However, as Quine famously pointed out, we would require the truth predicate for certain expressive functions—viz. undertaking commitments without the need to express them explicitly—whether or not there is a property of truth, and this fact constitutes an undercutting defeater for the prima facie reason to think that truth is a property.  In other words, the indispensability of a truth predicate (for purposes other than attributing a property of truth to sentences) undercuts the reason to think that ‘is true’ denotes a property.

Mathematical talk refers to quantifies over abstract mathematical objects.  Since referring terms denotes objects, mathematical talk provides prima facie reason to think that mathematical objects exist.; many claims that we take to be correct appear to be descriptions of an abstract realm of mathematical objects.   However, we would require reference (or apparent reference) to mathematical objects in order to describe concrete systems (in order to model physical phenomena mathematically) whether or not there are mathematical objects, and this fact constitutes an undercutting defeater for the prima facie reason to think that there are mathematical objects.  In other words, the indispensability of mathematics (for purposes other than describing a realm of abstract mathematical objects) undercuts the reason to think that mathematical terms denote extant abstract mathematical objects.