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.

A Thought Experiment against Humean Supervenience: Conceivability & Intuitions

One thought experiment that objectors against Humean Supervenience are quick to suggest is that of two possible worlds which both contain only one existing material object that always moves in a constant velocity. The thought experiment goes: in the first world, the particle is governed by the law that all particles move with constant velocity and in the second world, the particle is governed by the law that all particles obey Newton's Second Law; in other words, in the second (and not the first) world, if another particle had existed there, they may have had an effect on one another's velocity.

Is this a good counterexample via thought experiment? In a recent paper, Ned Hall tries to construe this counter-example as relying on the 'Conceivability entails Possibility' principle. And he therefore dismisses it. It is certainly true that the set-up of the thought experiment involves the (C-->P) principle, insofar as it must be committed to it being possible that there could be a world with only one particle and a world differing from another in only its modal facts.

But although I whole-heartedly dismiss the (C-->P) principle, I'm not so sure that a counterexample wherein two different objects in two distinct possible worlds are intrinsic duplicates and yet differ in their nomic-profiles is to endorse a possibility only on the basis of conceivability. Why is this such a leap in conceivability? I think the non-reductionist example is meant to (or at least, should aim to) exploit a seperate intuition; not the C-->P intuition. The intuition is: how a particular object (or kind of event) has been, is, or will be says nothing about ways it might have been.

It won't do to simply dismiss the non-reductionist thought experiment on the basis that it assumes that one world could differ in its laws while another one does not. For while this is assumed, and while it is an assumption that tips the scales, so to speak, it is not an altogether unwarranted assumption. And the onus is on the Humean to explicate the ways in which this thought example goes wrong. It won't be enough for the Humean to claim that the two worlds differing in their Laws is just impossible, any more than it will do for the non-reductionist to claim that it is possible. The non-reductionist position exploits an intuition, one that, for all I can tell, is fairly plausible, while the Humean position combats this intuition, but with no reason other than the fact that it does not accord with his theory.

Dawkins, Functional Complexity & Divine Ideas

I just returned from a great little talk at the Oxford Graduate Christian Union, being held at the Mitre Pub on High Street, where a Reverend Dr. Patrick Richmond gave a talk entitled "Swinburne vs. Dawkins: Is God Simple or Complex?" The talk was quite fascinating, as it covered Dawkins' argument against the probability of there being a God of any sort. The basic argument is this: any mind that could create the immense complexity in the world must itself be immensely complex. And we know from the world that any immensely complex thing cries out for an explanation as to how it was constructed, due to the fact that the more complex something is, the more improbable is its existence. Compare: the probability of complex molecules being created is quite low, due to the fact that millions and billions of elementary causal interactions between more fundamental particles could occur without such molecules being created.

In short, the argument is pretty bad. As Swinburne points out, God is not complex at all - He is actually infinitely ontologically simple. Not only this, but Dawkins' argument only extends to the statistical improbability of various material things coming together to create some materially complex thing. Any attempt to extend this to some sort of complexity in the Divine Mind (think: God's infinite knowledge of all possible worlds) yet seems rather ad hoc. As Richmond nicely pointed out in his talk, God's accumulated mass of ideas isn't a kind of complexity that seems to cry out for analysis - not, at least, by Dawkins' standards. According to Dawkins, only 'functional complexity' is in desperate need of explanation - things like watches and the human eye - but things that are complex but nevertheless functionally inert do not. Richmond's insight was to hold that the Divine Ideas are complex, but not for all of that forming a kind of system that cries out for explanation of their design; although, calling them 'non-functional' is probably quite the misnomer.

But I did ask a question of Richmond that I thought to be particularly sanguine to the discussion, one that may put the leverage back on the Dawkins side of things. To be clear, the original argument that Dawkins proposes is ghastly unacceptable. But if we explain things in the manner that Richmond and Swinburne opt for, we may yet be in a position that is open to more "But who designed that!?"-type criticisms. Consider, for example, the account that the Theist needs to give of God's knowledge of the possibilities of things. Presumably, this will come about in the classical form of God's knowledge of the essences of things - that is, God knows all of the possibilities of things in virtue of having a perfect knowledge of the essences of all things.

But although this is a seemingly acceptable answer, consider what kinds of things essences must be. They must be the objects of Divine Thoughts - or, alternatively, just the Divine Thoughts themselves. They must be simple things. But they must also somehow contain an exhaustive list of the entirety of the possible properties of the thing whose essence they are. For instance, God peers into the essence of me and notices that I could have been a fisherman or could have died in my infancy and the rest of the lot of potential world-careers I might have pursued or not pursued. Now all of this is fine, theologically speaking.

But there may yet be the further worry that although we have posited a simple object of Divine Knowledge, we have yet introduced another kind of complexity that itself cries out for explanation. For the essences of things are not only infinitely complex, but they also involve certain limits for possible values. Consider the essence of Adam - that is, the first man. Is it not part of the essence of Adam to eat of the fruit of the tree? Perhaps someone will deny this. But then, why do they deny it? It can only be denied on pain of a certain criteria for the essence of Adam. But then, we are still at the same problem - namely, the essences of things have certain innumerable facts about them that are somehow considered primitive; not just facts about possible futures, but also potential futures, possible pasts and -importantly - the limitations of all of these.

Essences, as objects of Divine Thought, on the face of it, seem to be both complex and functional in an attenuated sense - for they contain all kinds of specifications on which values are acceptable, which are not and which values will produce which values in which situation (cf. Molinistic 'Middle Knowledge'). But that kind of functionality is certainly functionality enough for the inquisitive atheist to question further: "What then, explains this precise (functional) complexity?"

An obvious move - and the one I eventually favour - is to opt for Primitivism. To ask why an essence is as it is is to ask a fundamentally misguided question - essences are themselves the things which explain why any contingent state of affairs is as it is, but they are no such state of affairs in need of explanation. However, there is the further nagging worry that this explanation is worth all the weight of Anslem offering the Fool the Dictionary entry for "God"; ie. arguing that "It's just in the very Definition of 'God' that He necessarily exists" is just not a very good argument for God's necessary existence. And if that's the best the Theist can do, he isn't doing very well at all.

But, as Quine once remarked, "there are primitives, and then there are primitives". Ontology must bottom-out in primitives, to be sure. But it's well to remember that the Theist still has the problem of arguing for primitives. Maybe he has argued away the primitiveness of Design in physical systems, but he hasn't, for all of that, done away with his need to find a better, more suitable primitive.

Compositional Vagueness in Material Objects

I recently attended a conference in Nottingham entitled 'New Directions in Metaphysics', hosted by the University of Nottingham. The venerable Dean Zimmerman was there, giving a talk on property dualism. His talk didn't excite me too much, but when he brought up the well-known problem of the vagueness of composition, it got me thinking again on these questions.

To refresh your memory, the problem of the vagueness of composition arises if we consider a reasonable definition of a material object - say, a human being. Suppose that a human being is defined as being that object that is composed of all of these material elements - say elements a1-an. This seems fairly straightforward, until we consider that it is very hard to, as it were, draw a line around those elements that compose us and those that do not. In fact, when we get a very detailed, close-up view of these elements, it becomes more and more obvious that there are many different elements that are just as entitled to be a part of an object as the ones that we at first chose to countenance. That is, there are many different sets of elements that are up to the task. Or consider another analogous problem: take all of the elements that one might say composes a particular human being (a1-an) and ask yourself whether or not it would be acceptable to define this human being by excluding merely one of these elements - say a3. Repeat and repeat the thought experiment until you reach absurdity.

This seems like a fairly important problem for defining material objects - for once the thought experiment is carried out, there seems to be an extremely large group of acceptable candidates, none of which seem to have any special claim to being the set that defines a particular material object.

An interesting response to this problem is the one given by Peter vanInwagen, who claims that the only material objects that exist are those that are biological - and they are individuated by being a particular group of elements all participating in the biological functioning of an object. Any of those fringe elements that made-up other candidates won't be acceptable then, as they are not aiding in the biological functioning of the object in question - they are, as it were, just idly floating about. Of course, a well known problem for this kind of view is that it entails that things like 'tables' and 'chairs' don't exist. It's a bullet that not many have particularly wanted to bite.

For my part, I think vanInwagen's response is on the right track in a certain respect, in that it individuates material objects using a certain dynamic criterion. It seems to me that the reason that the 'many candidate' problems all get started by assuming that the individuation of a material object must be a merely static - in the case of Lewisian examples, a particular set of elements. There is, of course, something to be said for wishing one's identity criterion remain stable - for instance, you don't want it to be the case that the identity criterion for a material object is constantly changing full stop. But the set-theoretic element identifications are inadequate, I think, because they fail to have any dynamic qualities at all.

I suppose the reason for this is mostly historical . Couple the fact that analytic philosophical methods have always wanted identity to be a stable, unchanging primitive fact with the further fact that the concepts of mathematics have always been considered the most unchanging facts and there's good reason to follow this kind of idea through. But it's hard for me to take any of this seriously in light of the current state of metaphysics, physics and the developments in quantum mechanics. As far as I can tell, all the evidence in quantum mechanical physics points towards defining physical objects as dynamical - the wavefunction is certainly, if anything at all, a dynamic representation of reality. If the wavefunction does represent something physical, it represents something that is not merely 'categorical', and certainly doesn't represent it as simply a set of particular elements; think here only of the fact of wave-particle duality! And metaphysics too, riding the crest of scientific discovery, is edging more and more towards an ontology of dispositions and propensities. So why should we be persuaded by thought experiments involving 'candidates' that are unchanging, static sets of elements?

There's every reason to believe that a better criterion of material object individuation can be given by something that is unchanging and yet has a dynamical aspect - perhaps dispositionally, via (Lorentz) invariant mass; cf. Leibniz, who once defined substance as 'a principle of action and passion'. Under this general criterion, we may define a material object by picking out its fundamental dispositional properties. This kind of analysis certainly has no concern for which elements 'belong' to to an object and which do not and makes the rival candidates of Lewis' thought experiments seem much less interesting.

It might still worry one that this kind of individuating technique leaves open the question of which elements of a set truly belong to a material object and which do not. But I'm not so sure that this is much of anything to truly be worried about. If a material object is defined by (and perhaps, is essentially) a set of fundamental dispositions, asking which material elements belong to it and which do not is not asking anything very interesting about it - unless, of course, one thinks that the contingencies of the moment-to-moment career of the object in question is of any real importance; it certainly isn't important for scientific inquiry at any rate.

Why Are You You?

I just attended a lecture by Stephen Priest, a lecturer here at Oxford, where he discussed what he took to be philosophical questions that needed theological answers. He had three such questions to discuss, the first of which was "Why are you you?".

Of course, he dismissed the fact that this was a silly question and defended that it has a non-trivial meaning. He then went on to tell a story about how what 'you' really are is a 'subjective viewpoint' or 'absolute interiority' (can you say Continentalist?) - in short, a unique phenomenological perspective. And so, being that this is the case, what 'you' really are isn't a physical body or a set of mental phenomena. He concluded therefore, that the question "Why are you you?" is an interesting one because of the disconnect between the 'subjective viewpoint' and a 'psycho-physical object'. Fair enough, perhaps.

As far as I can tell, either the question (a) expresses a tautology and so is useless or (b) can't be given any meaningful answer. (a) and (b) correspond to how one chooses to have the sentence refer. If the the word 'you' is a rigid designator for what 'you really are' (according to Preist), then the question "Why is this subjective viewpoint this subjective viewpoint?" is a pointless one: it expresses a tautology and has told us absolutely nothing.

If the word 'you' is not rigidly designating - and this is the option that Priest is opting for - then the real question is "Why is this subjective viewpoint inherent in this psycho-physical object?" If this is the question that Priest means to be asking, then it looks like this won't be such a trivial question; at least, not on the face - I suppose it could be questioned whether the two aren't interconnected in a deep way that constitutes a FAPP-identity.

But if that's the question Preist means to be asking, then that is the question to which Priest supposedly has an educated answer to; when I raised this concern during the Q&A, he responded that he did in fact have an answer: more or less "read my new book". But I am seriously skeptical of the very possibility of giving a meaningful answer here. What could one say about why this subjective viewpoint is associated with this psycho-physical object that was informative?

Now, Priest does invoke God in his discussion. But he is a Deist, not a Theist. And as this is the case, I can't see any move that would make such a connection (between a viewpoint and a psycho-physical object) explained. The Theist may say "this subjective viewpoint is with this psycho-physical object because God knew that having it arranged thusly would produce the most faithful and holy person possible". But this kind of answer certainly isn't available for Priest. And short of this kind of (seemingly ad hoc) answer, I don't see how one could answer the question meaningfully in such a way that the answer didn't turn out trivial in one sense or another.

The Status of 'Brute Facts'

Disregarding any theory that posited 'brute facts' - excepting 'identity' - used to be a common thread throughout my philosophical career. As any good Leibnizian, I am deeply committed to some form of the Principle of Sufficient Reason afterall. But as my thinking about metaphysics becomes more and more naturalized, I find myself not only denying this practice but also finding it just plain mistaken metaphysics.

Granted, it would be lovely if we could do away with 'brute facts'. But not only are there philosophical reasons for thinking this an impossible project - think: first mover - but there are good reasons for thinking that primitives ought to be had at a much 'higher level' of ontology. I for one, as a sortal essentialist, think that explanation stops at natural kind identities. So asking questions like "but why is this object this natural kind?" are not pointless, but metaphysically meaningless. You might as well be asking "but why is x identical to x?"

At the recent 'Philosophy of Cosmology' conference at St. Anne's College at the University of Oxford, Paul Davies gave an interesting talk. The gist was: we can always ask 'why' and physics may not always provide an answer. I approached him after his talk and asked him if he would ever be happy with primitives in physics, or philosophy for that matter. He insisted that the questions he raised were valid and important questions for physicists to ask themselves. Now, I agree with that, but I also recognize that there are primitives and then there are primitives. If we have a proposed primitive that is capable of theoretical reduction, then we have no primitive at all. But looking for contrastive reasons for every proposed primitive is, more often than not, a futile effort. In particular, Davies asked, "but why this law, rather than some other?"

I have recently been re-reading sections of Richard Swinburne's 'The Existence of God'. An amazing book to be sure, and there's no doubt that Professor Swinburne has done much to make theological belief intellectually credible. But in his chapter on 'The Nature of Explanation', he insists on these kinds of questions. In discussing the causal powers view of natural laws, he complains that '...the law does not explain why these substances have those powers'. But if these powers are the foundations for the natural laws - there just isn't any worth in asking such questions. If Swinburne is looking for an answer to "why negative charge repels negative charge, rather than attracting it", he is simply assuming that nature has available to it all of the content of the span of logically possible worlds. This is the only thing that can be going on in these types of questions: one must be a Humean about possibility.

If you think there's a meaningful answer to the question "why does negative charge repel negative charge, rather than attract it?", you must think that it is in fact possible that negative charge might have attracted other negative charges. But why think this? As far as I can tell, one would only think this by assuming that logical possibility is the possibility-space for the concrete world. But on a dispositional conception of possibility - to ask such a thing is to be asking a rather ad hoc question. For on this view, the causal powers themselves demarcate what is possible and what is not.

So there is simply no room to ask for any contrastive explanation as to why these powers are as they are. The short answer is: this is a brute fact about reality. And if you think that it is genuinely possible that like negative charges might have attracted, the onus is on you to explain why it is that purely logical possibilities should be countenanced as true de re possibilities for objects in the concrete world. And so you must make the dreaded conceivability entails possibility link.

Counterparts & Primtivism

Counterpart theorists argue that they can avoid an unwarranted primitivism about de re modality because they can explain why it is that objects have the de re modal properties they have. Other positions, such as Sortal Essentialism and/or Dispositional Actualism, must declare de re modal properties to be primitive facts. The Counterpart theorist however, can explain why x is possibly F: x has at least one counterpart that is F in some possible world.

L.A. Paul is concerned that the actualist who is not a Counterpart theorist must posit a spooky "Modal Force" inherent in objects that governs their de re modal properties.

But note that the Counterpart theorist has an even more disturbing primitive: Possible Worlds!; even more so, if - following the originator of Counterpart theory - you hold that these worlds are concrete entities.

I am more concerned about spooky "Other Worlds" that exist as a matter of primitive fact. Which primitive am I to prefer? I say the one that causes the least ripples throughout our current conceptual scheme and which is least capable of reduction. I therefore prefer the spooky Modal Force: it's adoption not only doesn't change the character of physics, but it is instead supported and perhaps suggested by modern physics; so it causes less ripples than the posit of an infinity of possible worlds. And such a concept does not, unlike possible worlds (abstract or concrete), cry out for conceptual reduction.