The Nature of Necessity: Triviality?

In a recent article by Ross Cameron in the Australasian Journal of Philosophy, it is argued that every necessary truth is necessary in virtue of it being trivially true. The discussion begins with the 'truths of mathematics' and attempts to show that these are truths that are trivially true in the following sense: their truth-conditions do not depend upon "the way the world is". The truth of <2+2=4> doesn't depend upon the way the world is, so it is a trivial truth in that, no matter which way the world happened to be (read: no matter which world turned out to be contingently actual), <2+2=4> would be true.

I've often had this kind of thought about truths of mathematics, logic and the like. I cannot find the claim that the truths of mathematics are contingent coherent simply because I would only count a truth contingent if there is some way the world might be such that the truth-value of a particular mathematical proposition would be made false (or true, if the proposition is @-false). In other words, I subscribe to the most limited and plausible form of the Truthmaker Theory. And because I have no understanding of what it would be like for the world being a certain way that would falsify mathematical truths (eg. <2+2=4>), I can't countenance mathematical truths as contingent. So far then, I am completely on board with the hypothesis that necessary truths are necessary in virtue of being trivially true.

Now, a natural way to understand what is going on in the case of mathematical truths being necessary in the aforementioned sense is that it's truth "places no demand on the world" - in other words, it's truth (or falsity) does not demand that the world be any which way at all; simply because, it's truth-value does not depend upon the world making it true in any particular way.

But things don't seem to be so simple when we begin to consider substantial metaphysical truths about concrete objects, dispositions, natural kinds, etc. When he comes to a discussion of these kinds of truths, the notion of a necessary truth placing "no demand on the world" takes a different form. Consider the stock example of a purportedly (a posteriori) necessary truth: ; Cameron rephrases this a bit as "If there is Water, there is H20". Surely here the notion that the proposition places no demand on the world seems a bit unsettling. Doesn't it, at the very least, make the claim that 'Water' and 'H20' are referentially co-extensive?; eg. that the set of all of the things picked out by 'Water' is in one-to-one correspondence with the set of all things picked out by 'H20'?

In discussing these substantial metaphysical truths, Cameron's new gloss on the notion of a necessary truth being trivially true in virtue of placing "no demand on the world" is that the two elements, say of an identity statement or a predication (as in <x is F>), make the "exact same demands" on the world. Cameron states:

"Since the truth of the antecedent places the same demands on the world as the truth of the consequent, the truth of the conditional itself places no demands on the world, and so this sentence is trivially true"

In other words, the "demand" that a particular world contain Water is exactly the same "demand" that that world contains H20. This seems, for the most part, right on target in explaining why a proposition is either necessarily true or false: is necessarily true because whenever you have Water, you have H20 and there is no way a world could be such that you have one but not the other.

But even so, I cannot agree with Cameron's conflation of "two terms placing the same demand upon the world" with "placing no demand upon the world". And I can't agree with equating the two notions simply because of the banal fact about extensional semantics that I pointed out above. It seems to me that mathematical propositions place "no demand on the world" in two ways: (1) in the aforementioned sense that there is no way the world could be such that it might alter the truth-value of the proposition and (2) in the sense that the proposition does not concern concrete existents. Now I am of the opinion that only concrete entities exist and that therefore, a world is to be identified with a possible configuration of concrete entities. But once you accept that assumption about ontology and (2), then it is rather easy to accept the truth of (1); in fact, (1) seemingly just follows from (2) with the 'concrete assumption' about world ontology.

But notice that substantial metaphysical truths, such as , do not place "no demand on the world" in the sense of (2), as mathematical truths do. I agree that it places "no demand on the world" in the sense of (1), simply because, as Cameron says, "when you have one, you have the other".

So perhaps it would be better to say that the sense of "placing no demand on the world" would be better characterized simply by (1) alone. Cameron begins his paper by dismissing the claim that necessary truths commit one to a particular ontology, so he would presumably not be happy with the claim that <2+2=4> is necessary because "when you have one, you have the other", as it plausibly commits one to at least having worlds where '2', '4', and the relation of addition all exist. I'm not pleased with the thought that mathematical truths commit us to an ontology either, but I don't think that the sense of "no demand" given by (1) needs to be interpreted existentially.

It seems to me that the "no demand" reading of triviality is to be understood as "when you have one, you have the other" without a necessary corollary to be interpreted existentially. In this way, we can claim that <2+2=4> is trivially true in the sense that, no matter which would happened to be actual, if you had 2 sets of 2 things there, you would have 4 things. Surely no possible world could alter the truth value of this claim - plausibly, not even a completely empty world could alter this truth. And, without being interpreted existentially, we can say also that is trivially true in the sense that, no matter which world happened to be actual, if you have Water there, you have H20 there.