What does it mean for something to be metaphysically necessary?

by Massimo Pigliucci

As I mentioned before, this semester I’m teaching a graduate level seminar on David Hume, and having lots of fun with it. During a recent discussion of sections 4 and 5 of the Enquiry Concerning Human Understanding (“Sceptical doubts concerning the operations of the understanding” and “Sceptic al solutions of these doubts”) the concept of metaphysical necessity came up.

As is well known, Hume wasn’t very keen on metaphysics in general. One of the most famous quotes by him (in section 12 of the very same Enquiry) says: “If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.” Ouch.

Anyway, back to metaphysical necessity. What might it mean for something to be metaphysically necessary, or — conversely — metaphysically impossible? Not surprisingly, there is a fairly large literature about this. The (far from comprehensive, but heavy on recent entries) section on metaphysical necessity of the PhilPapers archive lists 74 papers, with some of the most recent entries having titles like “Hume’s Dictum and Natural Modality: Counterfactuals”; “Radical Non-Dispositionalism and the Permutation Problem”; “Soames’s Deflationism About Modality”; and so forth.

But we’ll proceed here by looking briefly at the basics. First of all, metaphysical necessity/impossibility as opposed to what other kinds of necessity/impossibility? Two immediately come to mind: logical and physical. It is logically necessary that I either am me or am not-me, for instance [1]; it is also logically necessary, though for different reasons, that there is no such thing as a married bachelor. It is physically necessary that objects with mass attract each other; it is also physically impossible for me to both be here in New York and simultaneously in Rome [2]. And so forth.

Let’s see what we can glean from the above examples: in both instances concerning physical possibilities, and in one instance concerning logical possibility, the idea seems to be that there are certain “laws” that govern logic or physics, and that these laws are inviolable. Now, one could be skeptical about the a priori validity of the laws of logic (like W.V.O. Quine was), and one can even think of the laws of physics as simply empirical generalizations that could, in fact, admit of exceptions or have a limited domain of application (like, for instance, Nancy Cartwright does), but I won’t go there. As far as we are concerned, both logic and physics are solid enough, so to speak, to allow us to talk about things that are either possible or impossible given the respective sets of laws.

The remaining case (the impossibility of a married bachelor), of course, hinges on issues of definitions: since a bachelor is defined as an unmarried man, there simply cannot be any such thing as a married one, on penalty of (logical-semantic) contradiction. Definitions, of course, are tautological, and tautologies are often regarded with little interest in such discussions. But this is a mistake: think about the fact that mathematics (and much of logic itself) consists precisely in the working out of the tautological implications of certain axioms or premises.

So, where were we? Well, the discussion so far hints at one promising way to look for metaphysical necessity: search for laws of metaphysics. Unfortunately, that’s not at all a straightforward quest, because it is not clear what counts as a metaphysical law, as distinct from either a physical law or a law of logic — which of course doesn’t help our predicament at all.

Perhaps we should do what I’ve done above in the cases of logic and physics: look for examples first, then see what we can learn from them.

If you follow that route, one of the most commonly advanced examples of metaphysical necessity is… the existence of God! Since that is prima facie (I love it when I get to write that!) ludicrous — or it should be at the dawn of the 21st century — we will ignore it and proceed otherwise.

What else can be done? Well, there are some more intriguing examples of alleged metaphysical necessity, for instance “whatever is water is H2O” and “whatever is elemental gold has atomic number 79.”

Let’s look more closely: these are not examples of definitional necessity, like the bachelor. True, once we discovered that the molecular structure of water is H2O we could simply define water as that substance that has that chemical structure and be done with it. But this required an empirical discovery, it wasn’t true a priori from the get go, as is the fact that there cannot be a married bachelor. The reasoning is the same for gold being the element with atomic number 79.

Could it be that these two examples can be interpreted as instantiations of the laws of logic? Hard to see how. There is nothing logically contradictory in imagining a substance with the characteristics of water that is not made of two atoms of hydrogen and one of oxygen. But wouldn’t that contradict the laws of physics, at least? Ah, here things become tricky. Surely water behaves the way it is in our universe because the laws of physics are such that if a molecule has that structure then it will behave in that way. But it is hard to say which specific law of physics would be violated if something made of H2O actually behaved differently (say, it had a different freezing point at standard pressure).

Another way to think about this is to say that we can imagine a universe where the physics is (slightly) different and where, as a consequence, H2O doesn’t behave as our H2O. Of course, if that were the case, the H2O = water equation would not be a metaphysical necessity after all, but only a physical one. That’s because metaphysicians these days seem to make sense of the notion of metaphysical necessity by saying that something is metaphysically necessary if it is true in all possible worlds.

Talk of possible worlds is tightly connected with modal logic which, not surprisingly, is a set of logics that deal with expressions such as “necessarily,” “possibly,” etc. — which philosophers call modalities. There are a bunch of modal logics, including deontic (dealing with what is morally necessary or permissible), temporal, conditional and so forth. These have given origin to what is known as possible worlds semantics, the study of logical languages that make it possible for logicians to determine whether a given modal expression is inferentially valid or not (which, after all, is the whole point of any logic).

To return to our example: is it physically or metaphysically necessary that H2O = water? For this to be an example of metaphysical necessity, the equation would have to be valid in all possible worlds. But what makes a world possible to begin with? We could, again be talking about either logical or physical possibility (the former, should be clear, being much ampler than the latter). Let’s say we are talking about physical possibility: possible worlds are those worlds that could exist while instantiating a coherent set of physical laws.

Our world, obviously, realizes one of these possibilities. Worlds that, say, were different from ours only with respect to the gravitational constant would be our possible-neighbors, the closer to us as a function of how similar their gravitational constant is to ours.

One can easily extend this concept to a multidimensional landscape of fundamental physical constants, each varying within whatever range is physically possible for them to vary (e.g., although logically the gravitational constant could take any of an infinite number of values, it is perfectly possible that only a small subset of these values would yield a physically realizable universe).

If you smelled “multiverse” you are close. Despite some people’s reservations about the scientific status of the multiverse theory (reservations with which I sympathize), it does seem to make philosophical sense to deploy it within the context of this discussion. But if you don’t like that particular take, then think of possible worlds as the set of worlds that are mathematically realizable instead. While neither of these senses is the one normally used by philosophers who are interested in possible worlds semantics, I think they do help to get an intuitive grasp on the whole idea of “possible worlds,” because they give a fairly precise answer to the obvious question: possible in what sense?

So, again, water = H2O would be a metaphysical necessity just in case it had to be true in all possible worlds, say in the entire multiverse. My hunch is that this isn’t the case. It seems that some change in one physical constant or another would yield a pocket universe (within the multiverse) where a substance had the molecular structure H2O and yet had different physical characteristics from our water.

Still, there may be things that are metaphysically necessary in all possible instantiations of the multiverse. Perhaps the inter-conversion between matter and energy? Or the existence of fields from which matter emerges (like the Higgs)?

I am going to bet that most metaphysicians won’t like my analysis of metaphysical necessity as presented above, though. True, I have arrived at the conclusion that there can be such a thing as metaphysical necessity as smaller than logical necessity but ampler than physical necessity, thus legitimizing the concept. But I have also linked said concept operationally to either the multiverse as conceived by modern physics or its mathematical equivalent. If so, then discovering metaphysical necessities becomes either a matter for physics (because it is an empirical question) or for logicians-mathematicians (because it is a logical-mathematical thing). Which means that even our newfound way of thinking about metaphysical necessity either expands into logical necessity or collapses into physical necessity.

At the least, that’s the way I see it this week. Anyone out there have examples of metaphysical necessity that would rescue the concept from the Scylla or logic and the Charybdis of physics?

Postscript on the role of metaphysics

Interesting discussion so far. I wanted to add a few notes to further refine my thoughts about this issue. To begin with, I am leaning toward the conclusion that there is no such thing as metaphysical necessity. That’s in part because one cannot find metaphysical laws, and in part because I doubt there is such a thing as necessity, period. Nothing is physically or logically necessary - only possible or impossible.

True, once we establish certain constraints - for instance the laws of physics in our universe - then certain things necessarily happen. (Indeed, if you are a determinist, everything necessarily happens.) But there doesn’t seem to be a reason to think that the laws of physics themselves are necessary (multiverse and all that), so…

The same goes with logical necessity: once we pick certain axioms or premises, a number of things necessarily follow. But we could have picked different axioms or premises, so that those very same things wouldn’t follow at all.

Where, then, does that leave metaphysics? I still think it has a role to play, in the same sense that philosophy in general has a role to play. I have come to see philosophy as a type of critical inquiry that bridges logic (broadly construed) and science (and other sources of empirical knowledge), in the sense that it applies rigorous reasoning to whatever the issue at hand may be (e.g., ethics) while taking into account empirical input. This is nothing new: it is a restatement of Kan’t compromise between rationalism (the idea that one can derive a priori truths about the world) and empiricism (the idea that all truths derive from sense experience).

Similarly for metaphysics: I see it as a bridge between the Scylla of logic and the Charybdis of physics: the role of metaphysics is to make reasoned sense of what the natural sciences tell us about the world (in this I’m with people like Ladyman and Ross), as well as to elucidate how that knowledge fits with our understanding of abstract objects, such as mathematical and logical relations. But there are no laws of metaphysics, just like there are no laws of philosophy, so this endeavor is one of critically making sense of things, not of discovering or dictating how things are.


At least (again), this is what I think this week...

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[1] For the purposes of this discussion I will assume standard classical logic. The details would be different, but the general arguments the same, if we were using other kinds of logics.

[2] Non-locality does not apply to macroscopic objects of the size of a human being, for reasons that not even quantum physicists are particularly sure of.