Necessity and Triviality

Abstract

In this paper I argue that there are some sentences whose truth makes no demands on the world, being trivially true in that their truth-conditions are trivially met. I argue that this does not amount to their truth-conditions being met necessarily: we need a non-modal understanding of the notion of the demands the truth of a sentence makes, lest we be blinded to certain conceptual possibilities. I defend the claim that the truths of pure mathematics and set theory are trivially true, and hence accepting their truth brings no ontological commitment; I further defend the claim that the truths of applied mathematics and set theory do not demand the existence of numbers or sets. While the notion of a demand must not be reduced to anything modal, I nonetheless argue that sentences that are trivially true must also be necessary, lest we violate a very weak version of the principle that truth depends on the world. I further argue that all necessary truths are trivially true, lest we admit unexplained necessities. I end by showing one important consequence of this: I argue that if there are truthmakers for intrinsic predications, they must be states of affairs rather than tropes.

Notes

¹That is, excluding only impossible objects: those transworld objects which are sums of objects from different worlds. Such things must have parts that are not spatio-temporally related to one another.

²Actually, to any possible worlds account of necessity.

³The error-theorist may admit the truth of ‘There is no largest prime number’, although for different reasons, obviously, than the Platonist.

⁴One of course could distinguish between soft and hard Platonism, hard Platonism being the doctrine that there really are numbers: i.e. that the existence of numbers is amongst the demands made on the world by the true theory of the world. The trivialist is not a hard Platonist but a soft Platonist: one who admits that there are numbers, but who thinks that the true theory of the world does not demand the existence of numbers.

⁵Rayo [forthcoming] also cites an epistemological advantage, but I will not go into that here. It should be fairly clear, however, why an account that does not have mathematical truths be about some realm of causally inert entities, or about the possibility of there being infinitely many concreta, should yield a more palatable epistemology than its rivals.

⁶If the demands of a sentence are the conditions its truth imposes on a world, the ontological commitments of the sentence are just those of its demands that are concerned with ontology: i.e. those of its demands that make a demand on what there is[Rayo 2007: 428].

⁷At least, she need not do so. It is no part of trivialism that any such paraphrase is in principle available.

⁸One would naturally want to see more details before signing on the trivialist's dotted line, but I do not have the space to do that here: see Rayo 2008; forthcoming] and Williams [forthcoming] for that.

⁹Again, see Rayo 2008 and Williams [forthcoming] for details, including exactly how to do all of this formally.

¹⁰If the stubborn modalist refuses to be moved, where do we stand? At the least, I don't think he can claim to have an advantage over me. The potential advantage would appear to be that he can offer a modal analysis of the demands made by a sentence, and hence needs only one primitive where I have two. But if what I argue in this paper is correct, paying attention to the demands the truth of a sentence makes opens the way for an analysis of necessity in terms of triviality; in which case, I only have one primitive as well, it's just a different one. So it looks like there's just a stand-off here.

¹¹By ‘trivial existence’ I mean the sense in which numbers exist according to the soft Platonism mentioned above: that numbers exist, but that the truth of this does not demand that the world contain numbers. The Xs non-trivially exist when ‘The Xs exist’ is true and the truth of that sentence demands that the Xs exist: that is, when the Xs are an ontological commitment of ‘The Xs exist’. Premise (2) is compatible with the claim that there are some necessary existents; it entails only that if ‘A exists’ is a necessary truth then the truth of that sentence does not demand the existence of A. That is, premise (2) says that A is not an ontological commitment of ‘A exists’ if A necessarily exists. The premise says that substantial existence is contingent, and that only trivial existence is necessary.

¹²Note that (ii) is weaker than the characteristic claim of truthmaker maximalists, which is that all the demands of a sentence are ontological demands: that is, that every demand made by a sentence is a demand on what there is, and never a demand on how things are.

¹³Many a presentist, for example, would admit that the truth of ‘There were dinosaurs’ makes a demand on the world but deny that it makes any demand on the ontology of the world. For the sentence to be true, she will claim, demands of the world that its past be a dinosaur-containing past—but the ontology of the world (which, for her, is what presently exists) can be whatever you like so far as the truth of that sentence is concerned.

¹⁴Or at least that there be simples arranged elephant-wise. In Cameron 2008 I argue that the truth of sentences proclaiming the existence of complex objects does not bring an ontological commitment to complex objects but only to their simple parts. Cf. Williams [forthcoming].

¹⁵Cf. Blackburn 1999. My argument here is hostage to fortune, of course, in that there may be some other explanation of necessity that will not lead to regress (and which meets the constraints laid out at the start of this paper). I have not argued against such a possibility; nor do I intend here to go through every possible explanation I can think of and argue that they don't work. But if you are sympathetic to the idea that there are some trivial necessities (such as the truths of mathematics), I think the benefits of having a unified account of necessity should give us at least pro tanto reason for holding that all explanations for the necessity of S are of the form: S is necessary because its truth makes no demands on the world. And in any case, even if it turns out to be false that all the necessary truths are trivial truths, it would still be an interesting thesis that there are trivial truths and that all the trivial truths must be necessarily true.

¹⁶Thanks to Elizabeth Barnes, Jason Turner, Robbie Williams and, especially, Agustín Rayo for helpful discussion. Thanks also to two anonymous referees for the AJP and to audiences at the University of Glasgow and the Centre for Metaphysics and Mind, University of Leeds.