Modal details in Unger's argument against his existence

Unger famously argues that he doesn’t exist, by claiming a contradiction between three claims (I am quoting (1) and (2) verbatim, but simplifying (3)):

1. I exist.

2. If I exist, then I consist of many cells, but a finite number.

3. If I exist and I consist of many but a finite number of cells, then removal of the least important cell does not affect whether I exist.

Unger then says:

these three propositions form an inconsistent set. They have it that I am still here with no cells at all, even while my existence depends on cells. … One cell, more or less, will not make any difference between my being there and not. So, take one away, and I am still there. Take another away: again, no problem. But after a while there are no cells at all.

But taken literally this is logically invalid. Premise (2) says that I consist of many but a finite number of cells. But to continue applying premise (3), Unger needs that premise (2) would still be true no matter how many cells were taken away. But premise (2) does not say anything about hypothetical situations. It says that either I don’t exist, or I consist of a large but finite number of cells. In particular, there are no modal operators in (2).

Now, no doubt this is an uncharitable objection. Presumably (2) is not just supposed to be true in the actual situation, but in the hypothetical situations that come from repeated cell-removals. At the same time, we don’t want (2) to be ad hoc designed for this argument. So, probably, what is going on is that there is an implied necessity operator in (2), so that we have:

4. Necessarily, if I exist, then I consist of many cells, but a finite number.

The same issue applies to (3), since (3) needs to be applied over and over in hypothetical situations. Another issue with (3) is that to apply it over and over, we need to be told that removal of the cell is possible. So now we should say:

5. Necessarily, if I exist and I consist of many but a finite number of cells, then removal of the least important cell is possible and does not affect whether I exist.

Now, I guess, we can have a valid argument in S4.

Is this a merely technical issue here? I am not sure. I think that once we’ve inserted “Necessarily” into (4) and (5), our intuitions may start to shift. While (2) is very plausible if we grant the implied materialism, (4) makes us wonder whether there couldn’t be weird situations where I exist but don’t consist of many but a finite number of cells. First, it’s not obviously metaphysically impossible for me to grow an infinitely long tail? That, however, is a red herring. The argument can be retooled only to suppose that I necessarily have many cells and I actually have a finite number. But, second, and more seriously, is it really true that there is no possible world where I exist with only a few cells? In fact, perhaps, I once did exist with only a few cells in this world!

Similarly about (5). It’s clear that right now I can survive the loss of my least important cell. But it is far from clear that this is a necessary truth. It could well be metaphysically possible that I be reduced to some state of non-redundancy where every cell is necessary for my existence, where removal of any cell severs an organic pathway essential to life. I would be in a very different state in such a case than I am right now. But it’s far from clear that this is impossible.

Perhaps, though, the modality here isn’t metaphysical modality, but something like nomic modality. Maybe it’s nomically impossible for me to be in a state where every cell is non-redundant. Maybe, but even that’s not clear. And it’s also harder to say that the removal of the least important cell has to (in the nomic necessity sense) be nomically possible. Couldn’t it be that nomically the only way the least important cell could be removed would be by cutting into me in ways that would kill me?

Furthermore, once we’ve made our modal complications to the argument, it becomes clear that of the three contradictory premises (1), (4) and (5), premise (1) is by far the most probable. Premise (1) is a claim about my own existence, which seems pretty evident to me, and is only a claim about how things actually are now. Premises (4) and (5) depend on difficult modal details, on how things are in other worlds, and on metaphysical intuitions that are surely more fraught than those in the cogito.

(One of the things I’ve discovered by teaching metaphysics to undergraduates, with a focus on formulating logically valid arguments, is that sometimes numbered arguments in published work by smart people are actually quite some distance from validity, and it’s hard to see exactly how to make them valid without modal logic.)

Necessity and the open future

Suppose the future is open. Then it is not true that tomorrow Jones will freely mow the lawn. Moreover, it is necessarily not true that Jones will freely mow the lawn, since on open future views it is impossible for an open claim about future free actions to be true. But what is necessarily not true is impossible. Hence it is impossible that Jones will freely mow the lawn. But that seems precisely the kind of thing the open futurist wishes to avoid saying.

What I think is wrong with the proof of the Barcan formula

The Barcan formula says:

0. ∀x□ϕ → □∀xϕ.

The Barcan formula is dubious. Suppose, for instance, that the only things in existence are a, b and c, and let ϕ(x) say that x = a ∨ x = b ∨ x = c. Then the left-hand-side of (0) is true, since necessarily a = a, b = b and c = c. However the right-hand-side is not true, since it’s false that necessarily everything is one of a, b and c: even if there are only three things in existence, there could be more.

The Barcan formula can be proved in the Simplest Quantified Modal Logic (SQML) with S5.

Recently, a correspondent asked what I do about the fact that I accept S5 and yet presumably reject the Barcan formula. This gnawed at me for a bit, and I thought about the proof of the Barcan formula as presented by Menzel. I think I now have a pretty firm idea of where I get off the boat in the proof, and it has nothing to do with S5.

The first two steps of the proof are:

1. ∀x□ϕ → □ϕ (quantifier axiom)

2. □(∀x□ϕ→□ϕ) (from (1) by Necessitation).

Claim (1) is hard to dispute. But claim (2) isn’t right. Let ϕ be the formula D(x), where D(x) says that x is divine. Then (2) says:

2. □(∀x□D(x)→□D(x)).

By Generalization, which I think is hard to dispute, we get:

3. ∀x□(∀x□D(x)→□D(x)).

But (3) is false. For let a be me. Then (3) says the following about me:

3. □(∀x□D(x)→□D(a)),

i.e., that in the possible worlds where everything is necessarily divine, I am necessarily divine. But that’s just false. For I don’t exist in possible worlds where everything is necessarily divine. Only God exists in those worlds.

So I think the problem lies with Necessitation, which is the rule that says that theorems are necessary and yields (2) from (1). Here is my story as to what the problem with Necessitation is. Some logics have presuppositions. We can, for instance, imagine a theological logic that presupposes the existence of God. If a logic has presuppositions, then unless we have established that the presuppositions are themselves necessary truths, we are not entitled to assume that the theorems of that logic are themselves necessary. Instead, all that we are entitled to assume that the theorems of that logic necessarily follow from the presuppositions.

Now, infamously, classical logic has an existential presupposition: all the names and terms are names and terms for existing things. Because it has an existential presupposition, unless we have established the necessity of the existential presupposition, all we can say about theorems is that they necessarily follow from the existential presupposition, not that they are actually necessary.

Assuming we have a name for me in the language, it is indeed a theorem of classical logic that if everything is necessarily divine, then I am necessarily divine. But we cannot conclude that it is necessary that if everything is necessarily divine, then I am necessarily divine. For that would imply that in the world where only God exists, I would exist as well and be God. Rather, all we can conclude is that:

4. It is necessary that: if I and all the other things whose existence is presupposed exist, then if everything is necessarily divine, I am necessarily divine.

And that is trivially true, because in the worlds where everything is necessarily divine, I don’t exist.

A posteriori necessities

The usual examples of a posteriori necessities are identities between kinds and objects under two descriptions, at least one of which involves a contingent mode of presentation, such as water (presented as “the stuff in this pond”, say) and H₂O.

Such a posteriori necessities are certainly interesting. But we should not assume that these exhaust the scope of all a posteriori necessities.

For instance, Thomas Aquinas was committed to the existence of God being an a posteriori necessity: he held that necessarily God existed, but that all a priori arguments for the existence of God failed, while some a posteriori ones, like the Five Ways, succeeded.

For another theistic example, let p be an unprovable mathematical truth. Then p is, presumably, not a priori knowable. But God could reveal the truth of p, in which case we would know it a posteriori, via observation of God’s revelation. And, plausibly, mathematical truths are necessary.

For a third example, we could imagine a world where there is an odd law of nature: if anyone asserts a false mathematical statement, they immediately acquire hideous warts. In that world, all mathematical truths, including the unprovable ones, would be knowable a posteriori.

A necessary truth that explains a contingent one

Van Inwagen’s famous argument against the Principle of Sufficient Reason rests on the principle:

1. A necessary truth cannot explain a contingent one.

For a discussion of the argument, see here.

I just found a nice little counterexample to (1).

Consider the contingent proposition, p, that it is not the case that my next ten tosses of a fair coin will be all heads, and suppose that p is true (if it is false, replace “heads” with “tails”). The explanation of this contingent truth can be given entirely in terms of necessary truths:

2. Either it is or is not the case that I will ever engage in ten tosses of a fair coin.

3. If it is not the case that I will, then p is true.

4. If I will, then by the laws of probability, the probability of my next ten tosses of a fair coin being all heads is 1/2¹⁰ = 1/1024, which is pretty small.

My explanation here used only necessary truths, namely the law of excluded middle, and the laws of probability as applied to a fair coin, and so if we conjoin the explanatory claims, we get a counterexample to 1.

It is, of course, a contingent question whether I will ever engage in ten tosses of a fair coin. I have never, after all, done so in the past (no real-life coin is literally fair). But my explanation does not require that contingent question to be decided.

This counterexample reminds me of Hawthorne’s work on a priori probabilistic knowledge of contingent truths.

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

• anti-realism about set theory

• Hamkins’ set-theoretic multiverse

• contingent mathematical truth.

Closure views of modality

Logical-closure views of modality have this form:

1. There is a collection C of special truths.

2. A proposition is necessary if and only if it is provable from C.

For instance, C could be truths directly grounded in the essences of things.

By Goedel Second Incompleteness considerations like those here, we can show that the only way a view of modality like this could work is if C includes at least one truth that provably entails an undecidable statement of arithmetic.

This is not a problem if C includes all mathematical truths, as it does on Sider’s view.

Logical closure accounts of necessity

A family of views of necessity (e.g., Peacocke, Sider, Swinburne, and maybe Chalmers) identifies a family F of special true statements that get counted as necessary—say, statements giving the facts about the constitution of natural kinds, the axioms of mathematics, etc.—and then says that a statement is necessary if and only if it can be proved from F. Call these “logical closure accounts of necessity”. There are two importantly different variants: on one “F” is a definite description of the family and on the other “F” is a name for the family.

Here is a problem. Consider:

1. Statement (1) cannot be proved from F.

If you are worried about the explicit self-reference in (1), I should be able to get rid of it by a technique similar to the diagonal lemma in Goedel’s incompleteness theorem. Now, either (1) is true or it’s false. If it’s false, then it can be proved from F. Since F is a family of truths, it follows that a falsehood can be proved from truths, and that would be the end of the world. So it’s true. Thus it cannot be proved from F. But if it cannot be proved from F, then it is contingently true.

Thus (1) is true but there is a possible world w where (1) is false. In that world, (1) can be proved from F, and hence in that world (1) is necessary. Hence, (1) is false but possibly necessary, in violation of the Brouwer Axiom of modal logic (and hence of S5). Thus:

2. Logical closure accounts of necessity require the denial of the Brouwer Axiom and S5.

But things get even worse for logical closure accounts. For an account of necessity had better itself not be a contingent truth. Thus, a logical closure account of necessity if true in the actual world will also be true in w. Now in w run the earlier argument showing that (1) is true. Thus, (1) is true in w. But (1) was false in w. Contradiction! So:

3. Logical closure accounts of necessity can at best be contingently true.

Objection: This is basically the Liar Paradox.

Response: This is indeed my main worry about the argument. I am hoping, however, that it is more like Goedel’s Incompleteness Theorems than like the Liar Paradox.

Here's how I think the hope can be satisfied. The Liar Paradox and its relatives arise from unbounded application of semantic predicates like “is (not) true”. By “unbounded”, I mean that one is free to apply the semantic predicates to any sentence one wishes. Now, if F is a name for a family of statements, then it seems that (1) (or its definite description variant akin to that produced by the diagonal lemma) has no semantic vocabulary in it at all. If F is a description of a family of statements, there might be some semantic predicates there. For instance, it could be that F is explicitly said to include “all true mathematical claims” (Chalmers will do that). But then it seems that the semantic predicates are bounded—they need only be applied in the special kinds of cases that come up within F. It is a central feature of logical closure accounts of necessity that the statements in F be a limited class of statements.

Well, not quite. There is still a possible hitch. It may be that there is semantic vocabulary built into “proved”. Perhaps there are rules of proof that involve semantic vocabulary, such as Tarski’s T-schema, and perhaps these rules involve unbounded application of a semantic predicate. But if so, then the notion of “proof” involved in the account is a pretty problematic one and liable to license Liar Paradoxes.

One might also worry that my argument that (1) is true explicitly used semantic vocabulary. Yes: but that argument is in the metalanguage.

Optimalism about necessity

There are many set-theoretic claims that are undecidable from the basic axioms of set theory. Plausibly, the truths of set theory hold of necessity. But it seems to be arbitrary which undecidable set-theoretic claims are true. And if we say that the claims are contingent, then it will be arbitrary which claims are contingent. We don’t want there to be any of the “arbitrary” in the realm of necessity. Or so I say. But can we find a working theory of necessity that eliminates the arbitrary?

Here are two that have a hope. The first is a variant on Leslie-Rescher optimalism. While Leslie and Rescher think that the best (narrowly logically) scenario must obtain, and hence endorse an optimalism about truth, we could instead affirm an optimalism about necessity:

1. Among the collections of propositions, that collection of propositions that would make for the best collection of all the necessary truths is in fact the collection of all the necessary truths.

And just as it arguably follows from Leslie-Rescher optimalism that there is a God, since it is best that there be one, it arguably follows from this optimalism about necessity that there necessarily is a God, since it is best that there necessarily be a God. (By the way, when I once talked with Rescher about free will, he speculatively offered me something that might be close to optimalism about necessity.)

Would that solve the problem? Maybe: maybe the best possible—both practically and aesthetically—set theory is the one that holds of necessary truth.

I am not proposing this theory as a theory of what necessity is, but only of what is in fact necessary. Though, I suppose, one could take the theory to be a theory of what necessity is, too.

Alternately, we could have an optimalist theory about necessity that is theistic from the beginning:

2. A maximally great being is the ground of all necessity.

And among the great-making properties of a maximally great being there are properties like “grounding a beautiful set theory”.

I suspect that (1) and (2) are equivalent.

Brute necessities and supervenience

There is something very unappealing about unexplained, i.e., brute, metaphysical necessities that are “arbitrary”. For instance, suppose that someone said that some constant in a law of nature had the precise value it does by metaphysical necessity. If that contant were 1 or π or something like that, we could maybe buy that. But if the constant couldn’t be put in any neat way, could not be derived from deeper metaphysical necessities, but just happened necessarily to be exactly 1.847192019... (in a natural unit system) for some infinite string of digits? Nah! It would be much more satisfactory to posit a theory on which that constant has that value contingently. “Arbitrariness” of this sort is evidence of contingency, though it is a hard question exactly why.

Here is an application of this epistemic principle. It seems very likely that any view on which mental properties supervene of metaphysical necessity on physical ones will involve brute metaphysical necessities that are “arbitrary”.

For instance, consider a continuum of physical arrangements, starting with a paradigmatic healthy adult human and ending with a rock of the same mass. The adult human has conscious mental properties. The rock does not. Given metaphysically necessary supervenience, there must be a necessary truth as to where on the continuum the transition from consciousness to lack of consciousness occurs or, if there is vagueness in the transition, then there must be a necessary truth as to how the physical continuum maps to a vagueness profile. But it is very likely that any such transition point will be “arbitrary” rather than “natural”.

Or consider this. The best naturalist views make mental properties depend on computational function. But now consider how to define the computational function of something, say of a device that has two numerical inputs and one numerical output. We might say that if 99.999% of the time when given two numbers the device produces the sum of the numbers, and there is no simple formula that gives a higher degree of fit, then the computational function of the device is addition. But just how often does the device need to produce the sum of the numbers to count as an adder? Will 99.99% suffice? What about 99.9%? The reliability cut-off in defining computational function seems entirely arbitrary.

It may be that there is some supervenience theory that doesn’t involve arbitrary maps, arbitrary cut-offs, etc. But I suspect we have no idea how such a theory would go. It’s just pie in the sky.

If supervenience theories appear to require “arbitrary” stuff, then it is reasonable to infer that any supervenience is metaphysically contingent—perhaps it is only nomic supervenience.

This line of argument is plausible, but to make it strong one would need to say more about the notion of the “arbitrary” that it involves.

Aesthetic reasoning about necessary truths

We prefer more elegant theories to uglier ones. Why should we think this preference leads to truth?

This is a classic question in the philosophy of science. But I want to raise the question in connection with philosophical theories about fundamental metaphysics, fundamental ethics, philosophy of mathematics and other areas where our interest is necessary rather than contingent truth. Why should we think that the realm of necessity has the kind of aesthetic properties that would make more beautiful theories more likely to be true?

Here are two stories. The first story is that we are so constructed that we tend to find beauty in those philosophical theories that are true. It is difficult to explain why there would be such a coincidence if we are the product of naturalistic evolution, since it is unlikely that such a connection played a role in the survival of our species tens of thousands of years ago. If God exists, we can give an explanation: God gave us aesthetic preferences that guide us to truth.

The second story is that fundamental necessary reality is itself innately beautiful, and beautiful theories exhibit the beauty of their subject matter. And we recognize this beauty. It is puzzling, though: Why should fundamental necessary reality be beautiful? The best explanation of that which I can think of is again theistic: God is beauty itself, and all necessary truths are grounded in God.

Of course, one might simply reject the claim that our aesthetic preferences between theories lead to truth. But I think that would be the end of much of philosophy.

I think that in the order of knowing, aesthetics and ethics come first or close to first.

Is a necessary being inconceivable?

Consider this argument:

1. Obviously necessarily, if N is a necessary being that exists, it is impossible that N doesn't exist.
2. It is conceivable that N doesn't exist.
3. So it is inconceivable that N exists.

For this argument to work, we need to be able to make the inference from:

4. Obviously necessarily, if p, then necessarily q.
5. Conceivably not q.
6. So, not conceivably p.

Suppose that p just is the statement that necessarily q. Then (4) is uncontroversial. If the above argument form is good, then so is this one:

7. Conceivably not q.
8. So, not conceivably necessarily q.

But why can't we conceive both of not q and of necessarily q? Why should the ability to conceive of one thing, viz., the necessity of q, preclude the ability to conceive of another, viz., not q?

The principle that conceivability is defeasible evidence of possibility may seem relevant, but I don't think it establishes the point. That I can conceive of necessarily q is evidence of the necessity of q. That I can conceive of not q is evidence if the possibility of not q. So, if both, then I have evidence for two contradictory statements. Nothing particularly surprising there: quite a common phenomenon, in fact!

Suppose A and B are contradictory statements. It may be that evidence for A is evidence against B. But is evidence for A evidence against there being evidence for B? If it is, it is very weak evidence. Likewise, even given the principle that conceivability is evidence for possibility, the argument from (7) to (8) is very weak, much weaker than the inferential strength of this principle.

To summarize: The strength of the inference from (1) and (2) to (3) in the original argument is about equal to the strength of evidence that the existence of evidence for A provides against the existence of evidence against A. But the existence of evidence of A provides very little evidence against the existence of evidence against A. So the original argument is a very weak one. It would be improved if the conclusion were weakened to the claim that it is impossible that N exists, and then I would focus my attack on (2).

A quick route from mathematics to metaphysical necessity

The Peano Axioms are consistent. If not, mathematics (and the science resting on it) is overthrown. Moreover, it is absurd to suppose that they are merely contingently consistent: that in some other possible world a contradiction follows logically from them, but in the actual world no contradiction follows from them. So the Peano Axioms are necessarily consistent. But they aren't logically necessarily consistent: the consistency of the Peano Axioms cannot be proved (according to Goedel's second incompleteness theorem, not even if one helps oneself to the Peano Axioms in the proof, at least assuming they really are consistent). So we must suppose a necessity that isn't logical necessity, but is nonetheless very, very strong. We call it metaphysical necessity.

More against neo-conventionalism about necessity

Assume the background here. So, there is a privileged set N of true sentences from some language L, and N includes, among other things, all mathematical truths. There is also a provability-closure operator C on sets of L-sentences. And, according to our neo-conventionalist, a sentence p of L is necessarily true just in case p∈C(N).

Moreover, this is supposed to be an account of necessity. Thus, N cannot contain sentences with necessity operators and C must have the property that applying C to a set of sentences without necessity operators does not yield any sentence of the form Lp, where L is the necessity operator (It may be OK to yield tautologies like "Lp or ~Lp" or conjunctions of tautologies like that with sentences in the input set, etc.) If these conditions are not met, then we have an account of necessity that presupposes a prior understanding of necessity.

Now consider an objection. Then not only is L(1=1) true, but it is necessarily true. But now we have a problem. For C(N) by the conditions in the previous paragraph contains no Lp sentences. Hence it doesn't contain the sentence "L(1=1)".

But this was far too quick. For the neo-conventionalist can say that "L(1=1)" is short for something like "'1=1'∈C(N)". And the constraints on absence of necessity operators is compatible with the sentence "'1=1'∈C(N)" itself being a member of C(N).

This means that the language L must contain a name for N, say "N", or some more complex rigidly designating term for it (say a term expressing the union of some sets). Let's suppose that "N" is in L, then. Now, sentences are mathematical objects—finite sequences of symbols in some alphabet. (Or at least that seems the best way to model them for formal purposes.) We can then show (cf. this) that there is a mathematically definable predicate D such that D(y) holds if and only if y is the following sentence:

• "For all x, if D(x), then ~(x∈N)."

But if y is this sentence, then y is a mathematical claim. If this mathematical claim isn't true, then y is a member of N. But then y is true. On the other hand, if y is true, then being a mathematical claim it is a member of N, and hence y is false. (This is, of course, structurally like the Liar. But it is legitimate to deploy a version of the Liar against a formal theory whose assumptions enable that deployment. That's what Goedel's incompleteness theorems do.)

To recap. We have an initial difficulty with neo-conventionalism in that no sentences with a necessity operator ends up necessary. That difficulty can be overcome by replacing sentences with a necessity operator with their neo-conventionalist analyses. But doing that gets us into contradiction.

(It's perhaps formally a bit nicer to formulate the above in terms of Goedel numbers. Then we replace Lp with n∈C*(N*) where n is the Goedel number of p, and C* and N* are the Goedel-number analogues of C and N. Diagonalization then yields a contradiction.)

One place where I imagine pushback is my assumption that C doesn't generate Lp sentences. One might think that C embodies the rule of necessitation, and hence in particular it yields Lp for any theorem p. But I think necessitation presupposes necessity, and so it is illegitimate to use rules that include necessitation to definite necessity. However, this is a part of the argument that I am not deeply confident of.

An argument against neo-conventionalism in modality

The neo-conventionalist account of necessity holds that necessity is just a messy property accidentally created by our conventions. We historically happened to distinguish a certain family N of true sentences. For instance, N might include the mathematical truths, the truths about the identities of natural kinds (e.g., "water = H₂O"), the truths about the scope of composition, etc. Then we said that a sentence is necessarily true if and only if it is a member of the closure C(N) of N under some logical deduction rules. (Alternately, one might do this in terms of propositions.)

Here is a criterion of adequacy for a theory of modality. That theory must yield the following obvious, uncontroversial and innocuous-looking fact:

1. Necessarily, some sentence is not necessary.

Some things just have to be possible. (Note: In System T, if p is any tautology, then necessarily ~p is not necessary.)

A neo-conventionalist proposal consists of a family N of true sentences and a closure operator C. For any neo-conventionalist proposal, we then can raise the question whether it satisfies condition (1). Formulating this condition precisely within neo-conventionalism takes a bit of work, but basically it'll say something like this:

2. "Some sentence is not a member of C(N)" is a member of C(N).

There is a more intuitive way of thinking about the above condition. A family A of sentences is such that C(A) is all sentences if and only if the family A is C-inconsistent, i.e., inconsistent with respect to the rules defining C. (This is actually a fairly normal way to define inconsistency in a wide range of logics.) So (2) basically says:

3. "N is C-consistent" is C-provable from N.

Put that way, we see that our innocuously weak assumption (1) is actually a pretty strong condition on a neo-conventionalist proposal. It is certainly not guaranteed to be satisfied. For instance, a neo-conventionalist proposal where N is a finite set of axioms and C is a formal system (with the axioms and formal system sufficient for the operations in C) will fail to satisfy (3) by Goedel's Second Incompleteness Theorem.

This last observation shows that the question of whether a neo-conventionalist proposal satisfies (3) can be far from trivial. Now, in practice nobody espouses a neo-conventionalist proposal with a finite set of axioms. All the proposals in the literature that I've seen just throw all mathematical truths in, so Goedel's Second Incompleteness Theorem is not applicable.

But even if it's not applicable, it shows that the question is far from trivial. And that is unsatisfactory. For (1) is obviously true. Yet on a neo-conventionalist proposal it becomes a very difficult question. That by itself is a reason to be suspicious of neo-conventionalism. In fact, we might say: We know (1) to be true; but if neo-conventionalism is true, we do not know (1) to be true; hence, neo-conventionalism is not true.

Now, one can probably craft neo-conventionalist proposals that satisfy our constraint. For instance, if N is just the set of mathematical truths (considered broadly enough to include truths about what sentences are C-provable from what) then "N is C-consistent" will be true, and hence a member of N, and hence C-provable from N. But of course that's just another proposal that nobody endorses: there are more necessities than the mathematical ones.

And here's the nub. The neo-conventionalist isn't just trying to craft some proposal or other that satisfies (1). She is proposing to let N be those truths that we have conventionally distinguished (she may not be making an analogous move about C; she could let C be closure under provability in the One True Logic). But we did not historically craft our choice of distinguished truths so as to ensure (3). Consider the following curious definition of an even number:

4. A number is even if and only if it has the same parity as the number of words in my previous blog post.

This account might in fact get things right—if we are lucky enough that the number of words in my previous post is divisible in two. But I did not choose my wording in that post with that divisibility in mind. I chose the wording for completely different reasons. We don't have reason to think, without actually counting, that (4) is correct. And even if it is correct, it is only my luck that I happened to choose an even number of words, and we don't want a theory to rest on luck like that.

Necessary coincidences

On standard naturalist views, neither the objective facts of mathematics and morality nor their grounds (e.g., Platonic entities, etc.) have any influence on how matter behaves and hence on how we think. This seems to imply that if our mathematical or moral beliefs happen to be true, that's just a coincidence. But merely coincidentally true belief isn't knowledge (maybe it's Gettiered knowledge). Now consider a response on which:

1. Of biological necessity, we have evolved through unguided natural selection to have mathematical or moral beliefs of type N.
2. Of metaphysical necessity, most mathematical or moral beliefs of type N are true.
3. Therefore, there is no coincidence here and nothing that calls out for further explanation.

(Erik Wielenberg offered a response of roughly this sort last week here.) The response presupposes that there cannot be coincidences between necessary truths that call out for further explanation.

But there can. There are two real numbers, x and y, between 0 and 1 with the following property. If you write them out in binary, divide up the bits into groups of eight, and then put the bits into ASCII code, then you actually find a lot of comprehensible text in each. In particular:

4. In x, there are infinitely many occurrences of "Consider the following proposition:", and each of them is followed by a well-formed arithmetical sentence (say, written in TeX) and a period. In fact, all possible arithmetical sentence thus occur in x.
5. In y, at exactly the same point as each "Consider the following proposition:" string occurs, there instead occurs "That's true" or "That's false."
6. Moreover, "That's true" occurs in y precisely when the proposition given in that place in x is true, and "That's false" occurs when the proposition is false.

But of course, it is necessary that x and y have the binary expansion they do.

Now, if we're given two such numbers x and y, the above is an apparent coincidence that calls out for explanation. And maybe an explanation can be given, say in terms of a selection effect: Perhaps the reason we're considering these two numbers is because a logically omniscient being exhibited them to us, and the being chose the two numbers for these remarkable properties. No surprise then!

But what if turned out that x=π and y=e satisfy (4)-(6)? Then we would consider the above coincidence truly remarkable. We would search for some deep mathematical reason for it. But suppose this search fizzled out and we came to conclude that although, of course, it is necessary that π and e have the properties (4)-(6), e.g., it being necessary that Fermat's Last Theorem occur at location 12848994949494888 in π (I assume it doesn't) and "That's true" in e at the same location and so on, mathematically this is just an incredibly unlikely coincidence. That would be a highly intellectually unsatisfying position. So unsatisfying that we would reach for a metaphysical explanation like Descartes' story about God having designed mathematics or a science fictional one like Carl Sagan's novel about aliens having embedded a message in π. We would have good reason to accept such an explanation if it were offered, and if we rejected there being such an explanation, we would have to say we have just a coincidence.

Thus, we can imagine cases of agreement between necessary mathematical facts which genuinely call out for explanation. And we can imagine concluding that although they call out for explanation, there is none, and hence we have a coincidence. Thus we can imagine a coincidence in the realm of necessary truth.

Necessity is not provability

A plausible account of necessity is that p is necessary provided that p can be proved in the correct logical system K and p is possible provided that its negation cannot be proved. Assuming K is axiomatizable and proves enough of the axioms of arithmetic, this account can be shown to be incorrect.

Fix any sentence s in K. It follows from Goedel's Second Incompleteness theorem that there is no K-proof of s's being K-unprovable (for if there were such a proof, then it would follow that there is a K-proof of K's consistency, since if K is inconsistent, then every sentence, including s, can be proved in K). But on the account of modality under consideration, this means that it is possible that s is K-provable, i.e., it is possible that s is necessary.

In other words, this account of modality implies that every sentence is possibly necessary. But it is absurd to think that 0=1 is possibly necessary!

I think much the same reasoning can be used to disprove Swinburne's account of necessity, since where we are not dealing with directly referential rigid designators, Swinburne's account agrees with the provability account.

I am skirting over distinctions between s and its Goedel number, but I think that's a mere technicality to work out in greater precision.

This makes for a nice way to see a relationship between the two incompleteness theorems. The first one tells us that not everything true is provable. From the second we learn that not even everything necessary is provable.

An Aristotelian argument from a necessary being to a necessary concrete being

Suppose that none of the participants in World War II had ever existed. Then it would have been impossible for World War II to occur. Why? Because World War II's existence is solely grounded in the existence, activities, properties and relations of the participants, and

1. If an entity x's existence is solely grounded in the existence, activities, properties and/or relations of the Fs, then it is impossible for x to exist without at least one of the Fs existing.

Now add this Aristotelian axiom:

2. If x is abstract, then x's existence is solely grounded in the existence, activities, properties and/or relations of concreta.

Finally, add this:

3. Every being is either concrete or abstract.
4. There exists a necessary being.
5. There is a world where no one of the contingent concrete beings of our world exists.

One might try to give the number three as an example of a necessary being to support (4).

Now, let N be the necessary being of (4). If N is essentially concrete, we get to conclude that there is a concrete necessary being. If N is essentially abstract, then N is grounded in the existence, activities, properties and/or relations of concreta. If some concreta are necessary, we conclude that there is a concrete necessary being. So suppose all concreta are contingent. Then the beings that N is grounded in don't exist at the world mentioned in (5), which violates the conjunction of (1), (2) and the necessity and abstractness of N. So, no matter what, it follows from (1)-(5) that:

6. There is a necessary concrete being.

A sufficient condition for a subjunctive conditional

Start with the idea of grades of necessity. At the bottom, say[note 1], lie ordinary empirical claims like that I am typing now, which have no necessity. Higher up lie basic structural claims about the world, such as that, say, there are four dimensions and that there is matter. Perhaps higher, or at the same level, there are nomic claims, like that opposite charges attract. Higher than that lie metaphysical necessities, like that nothing is its own cause or that water is partly composed of hydrogen atoms. Perhaps even higher than that lie definitional necessities, and higher than that the theorems of first order logic. This gives us a relation: p<q if and only if p is less necessary than q.

Let → indicate subjunctive conditionals. Thus "p→q" says that were it that p, it would be that q. Let ⊃ be the material conditional. Thus "p⊃q basically says that p is false or q is true or both. Then, the following seems plausible:

1. If ~p<(p⊃q), then p→q.

I.e., if the material conditional has more necessity than the denial of its antecedent, the corresponding subjunctive conditional holds.

Suppose it's a law of nature that dropped objects fall. Then the material conditional that if this glass is dropped, then it falls is nomic and hence more necessary than the claim that this glass is not dropped, and the subjunctive holds: were the glass dropped, it would fall.

Moreover, the subjunctives that (1) can yield hold non-trivially, if there are grades of necessity beyond metaphysical necessity (on my view, those are somewhat gerrymandered necessities), and this yields non-trivial per impossibile conditionals. Let p be the proposition that water is H₃O, and let q be the proposition that a water molecule has four atoms. Then ~p<(p⊃q), because p⊃q is a definitional truth while ~p is a merely metaphysical necessity. Hence were p to hold, q would hold: were water to be H₃O, a water molecule would have four atoms.

I wonder if the left-hand-side of (1) is necessary for the non-trivial holding of its right-hand-side.

Leibniz and the necessity of optimality

Leibniz famously holds that:

1. God creates the best logically possible world.

In order to resist logical fatalism, Leibniz denies that (1) is a logically necessary truth.

But this causes a problem for him that I am not sure he recognizes. Either (1) does or does not logically follow from God's perfection. If it does follow, then (1) will be logically necessary, since Leibniz thinks it is logically necessary that there be a perfect God because of the ontological argument.

But if (1) does not logically follow from God's perfection, then how do we know that (1) is in fact true? Leibniz is not so blind to the evils of the world as to think we can conclude (1) from an optimistic appraisal of the world around us. His theodical work insists on our not knowing much about what the infinite universe is like, and our thus being unable to form a justified judgment that the universe is non-best. If we could see that the universe is best, he wouldn't have to go to that trouble.

Leibniz does have a backup plan. In one piece, he notes that even if (1) were true, it wouldn't logically follow that it is logically necessary that this world is best. For Leibniz, a proposition is logically necessary provided it has a finite proof, whereas contingently true propositions have only an infinite proof. Thus, Leibniz insists—and very plausibly so—that even if our world is best, that fact cannot be finitely proved. So Leibniz could simply affirm that (1) is necessary, but that fatalism does not follow. He doesn't want to do that, though.

To make it harder for Leibniz to resist the logical necessity of (1), consider the following little argument:

2. Logically necessarily, a being that fails to create the best possible world is imperfect (in power, knowledge or morality).
3. Logically necessarily, there is a perfect being, and God is that perfect being. (By the ontological argument.)
4. So, logically necessarily, God creates the best possible world.

Leibniz defends the idea that a perfect being couldn't create less than the best, so he seems committed to (2). And he is definitely committed to (3).

I think, though, there is a neat way out for Leibniz. I do not know if he ever takes this way out—it would be interesting to search the texts carefully to see. The neat way out is to deny (2). Instead, recall Leibniz's controversial but insistent claim that if a perfect being were faced with a choice between two equally good worlds, that perfect being would not create anything. More generally, it is plausible that Leibniz would say:

5. Logically necessarily, if x is a perfect being and there is no best possible world, x creates nothing.

Moreover, on Leibnizian grounds, the following is very plausible:

6. Logically necessarily, if there is a perfect being and a best possible world, the perfect being creates the best possible world.

Putting (5) and (6) together, we get a way to both deny the necessity of (1) and a way to know that (1) is true. First, there is no finite proof that there is a best world. Any proof would require comparisons between infinitely many worlds and would, plausibly, be an infinite proof. So it is not logically necessary that there is a best world or that God creates the best on Leibniz's finite-proof understanding of necessity. Second, because we can know with certainty that God created something (argument: necessarily, anything other than God is created by God; I exist and am not God because I lack many perfections; hence, something is created by God), by (5) we conclude that there is a best possible world, and by (6) that God created it.

I do not know if this line of thought is in Leibniz's texts, but I think every step in the story is one that he should endorse given his other views.