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.

Axiom T for physical possibility

Here is an argument for naturalism:

1. Only states that can be described by physics are physically possible.

2. Non-natural states cannot be described by physics.

3. Physical possibility satisfies Axiom T of modal logic: If something is true, then it’s physically possible.

4. So, non-natural states are physically impossible. (1 and 2)

5. So, non-natural states do not occur. (3 and 4)

I am inclined to think (1) is true, though it is something worth pushing back on. I think (2) is close to trivial.

That leaves me a choice: accept naturalism or deny that Axiom T applies to physical possibility.

I want to deny that Axiom T is a good axiom for physical possibility. The reason isn’t just that I think (as I do) that naturalism is actually false. The reason is that I think the axioms of physical possibility should hold as a matter of metaphysical necessity. But if Axiom T for physical possibility held as a matter of metaphysical necessity, then naturalism would be metaphysically necessary. And that is really implausible.

Yet Axiom T is very plausible. What should we do about it? Here is one potential move: Axiom T holds when we restrict our statements to ones formulated in the language of physics. This escapes the implausible conclusion that non-natural states are metaphysically impossible. But holding even this restricted axiom to be an axiom, and hence metaphysically necessary, still rules out the metaphysical possibility of certain kinds of miracles that I think should be metaphysically possible. So I think my best bet is to throw out Axiom T for physical possibility altogether. As a contingent matter of fact, it holds typically for statements formulated in the language of the correct physics. But that’s all.

A heuristic argument for the Brouwer axiom

Suppose that:

1. We cannot make sense of impossible worlds, but only of possible ones, so the only worlds there are are possible ones.

2. Necessarily, a possible worlds semantics for alethic modality is correct.

3. Worlds are necessary beings, and it is essential to them that they are worlds.

Now, suppose the Brouwer axiom, that if something is true then it’s necessarily possible, is not right. Then the following proposition is true at the actual world but not at all worlds:

4. Every world is possible.

(For if Brouwer is false at w₁, then there is a world w₂ such that w₂ is possible at w₁ but w₁ is not possible at w₂. Since w₁ is still a world at w₂, at w₂ it is the case that there is an impossible world.)

Say that the “extent of possibility” at a world w is the collection of all the worlds that are possible at w. Thus, given 1-3, if Brouwer fails, the actual world is a world that maximizes the extent of possibility, by making all the worlds be possible at it. But it seems intuitively unlikely that if worlds differ in the extent of possibility, the actual world should be so lucky as to be among the maximizers of the extent of possibility.

So, given 1-3, we have some reason to accept Brouwer.

Saying with possible worlds what can't be said with box and diamond

The literature contains a number of examples of a modal claim that can be made with possible worlds language but not in box-diamond language. Here is one that occurred to me that is simpler than any of the examples I’ve seen:

• Reality could have been different.

Very simple in possible worlds language: There is a non-actual world. (Note: This doesn’t work on the version of Lewis’s modal realism that allows for duplicate worlds. All the worse for that version.) But no box-diamond statement expresses (*). One can, of course, say that there aren’t any unicorns but could be, which implies (*), but that’s not the same as saying (*).

Narrowly logical necessity

My claim: Logical necessity understood narrowly
(a) violates the very-weak-Brouwer (VWB) axiom, or
(b) is not strong enough to make arithmetical facts be necessary, or
(c) makes every proposition necessarily true.

And of course a modality like that is clearly not what we mean by "necessity/possibility" or even "logical necessity/possibility". This post is an expansion of the brief argument here.

Now it's time to argue for my claim.

By VWB, I mean the following thesis:

• At least one proposition is necessarily possible.

According to the Brouwer axiom (and, a fortiori, given S5), every true proposition is necessarily possible. But VWB is much weaker than that. (In fact, it follows from Axioms T and Necessitation: given T, for any tautology p, Possibly(p) is a theorem; but by Necessitation, Necessarily(Possibly(p)) follows.) It is about as uncontroversial a modal axiom as one can get.

Now to prove my claim. 

A proposition p is narrowly logically possible provided that a contradiction cannot be proved from p while p is narrowly logically necessary provided that p can be proved (within the logical system that defines narrowly logical modality). 

Now suppose VWB is true for our narrow logical necessity. Then it is necessarily true that p is possible for some p. I.e., it can be proved that no contradiction can be proved from p. But if no contradiction can be proved from p, then our logical system is consistent: in an inconsistent classical system, a contradiction can be proved from every claim. And this conditional can be proved. 

Hence, given VWB, it follows that one can prove in the system that the system in question is consistent. It follows by Goedel's Second Incompleteness Theorem that either (i) the system is inconsistent or (ii) the system is not strong enough to contain arithmetic. In case (i), every proposition can be proved (this is classical logic, so we have explosion) and hence we have (c) and in case (ii) we have (b).  So if VWB is true, we have (b) or (c). Thus, in general, we have either (a) or (b) or (c).

Salmon's argument against S4

Start with:

1. If x originates from chunk α of matter and β is a non-overlapping chunk of matter, then x couldn't have originated from β.
2. If x originates from chunk α of matter and α' is a chunk of matter that almost completely overlaps α, then x could have originated from α'.

Iterating (2), and assuming a finite sequence of almost completely overlapping chunks between α and β, we conclude that an object x that originates from α possibly possibly ... possibly (with a finite number of possiblys) originates from β. By S4, we conclude that x could have originated from β, contrary to (1). Nathan Salmon uses this as an argument against S4.

But this is a mistaken line of thought. For (2) is not significantly more plausible than:

3. If x could have originated from chunk α of matter and α' is a chunk of matter that almost completely overlaps α, then x could have originated from α'.

Both (2) and (3) embody the same prima facie plausible small-variation intuition. If one thought (3) was false, one would have little reason to think (2) is true.

But given (3), Salmon's argument can be run without S4—all we need is T (what is actually true is possible). Iterating uses of (3) and modus ponens, we conclude that (1) is false. In other words, we cannot hold both (1) and (3). And since (2) has little plausibility apart from (3), we shouldn't hold both (1) and (2). Thus, Salmon's argument is not an argument against S4, but an argument against the conjunction of (1) and (2). And I say we should reject (2).

Two presentist ways of seeing worlds

If presentism is true, then right now, call it t₂, the proposition B that Bucephalus exists is false, but it once was true, namely at t₁. Now, at every time a token of the following sentence expresses a truth:

0. For all p, a proposition p is true if and only if it is true at the actual world.

Now, let's imagine ourselves at t₁. Then Bucephalus exists. Thus, B is true. Moreover, (0) expresses a truth, and so B is true at the actual world. So at t₁ the sentence

1. B is true at the actual world

expresses a truth. But now let's return to our time. B is false. But (0) expresses a truth, and so the sentence

2. B is not true at the actual world

does expresses a truth. Thus, (1) expresses a truth when said at t₁ but expresses a falsehood when said at t₂. This shows that either:

3. "The actual world" refers to different worlds at different times

or

4. The proposition that p is true at w can change in truth value, even if "p" and "w" refer rigidly to a proposition and a world, respectively.

Thus, the presentist has two ways of understanding possible worlds. Either possible worlds are tensed, so that at every time we inhabit a different possible world (that's option (3)) or else the "true at" relation is tensed, so that we inhabit the same world at different times, or when we say at t that p is true at w, we say something true if and only if p is true at t at w.

I think there is a problem for (4). Let p be the proposition that horses do or do not exist. Let t be the actual present time. Then p is true at every world, since it's a necessary truth. Now consider a world w where the time sequence does not include t. There are several options for this. Maybe in w, time comes to an end in 2011. Maybe time is discrete in w while in our world it is continuous, and so w either includes no times from our world or else w "skips over" t. Or maybe for some other reason the time sequence in w is radically different from our world's time sequence. Then p is true at w. But on (4), when we say that p is true at w, that is true if and only if p is true at t at w. But nothing is true at t at w, since t isn't a time at w.

Here's a slightly different way to see the point. When p is true at w, it is true either because there are or because there are not horses at w (this is an uncontroversial case of disjunctive grounding). Suppose it's true because there are not horses at w. But at which time are the horses not there at w? After all, w could have horses at some but not other times. Presumably, the relevant time is the present time. On proposal (3), every world comes along with its own present time, and this is fine. But on proposal (4), a world's relevant present time is our present time, and w doesn't have our world's present time.

One could try to solve this with counterpart theory for times. But one can suppose w won't have a counterpart to our time.

Here's a bolder move to defend (4) against our argument: The accessibility relation between worlds differs between times. The proposition p isn't true at all worlds, but only at all accessible worlds (this may or may not involve a denial of S5—S5 does not say that all worlds are accessible, but only that accessibility is an equivalence relation). And a world is only accessible if it includes the present time (or a counterpart to it?). This has the implausible consequence that what is metaphysically possible changes with time. For instance, if in w the time sequence comes to an end with 2011, then the proposition that w is actual was possible in 2011, but is no longer possible. But it's implausible that what is metaphysically possible changes with time.

If this is right, then the presentist should embrace (3). But is (3) plausible? Do we really live in different worlds at different times?

The presentist's other move is simply to abandon talking about worlds, and instead talk about, say, abstract times (in the Crisp sense).

First Order Logic and an ontological argument

[I also posted this on prosblogion.]

I want to give this argument in part to provoke a bit of discussion of the role of FOL in philosophy. I don't think the argument carries great weight, in large part because of Objection 2 (see the end).

1. (Premise) The inferences allowed by classical First Order Logic (FOL) combined with a modal logic that includes Necessitation are valid.
2. (Premise) If every being is contingent, then possibly nothing exists. (A material conditional)
3. Necessarily something exists. (By 1)
4. So, there is a necessary being. (By 2 and 3)

The proof of (3) is as follows. Classical logic allows (Ex)(x=x) to be inferred from (x)(x=x). Since (x)(x=x) is a theorem, so is (Ex)(x=x), and hence by the rule of Necessitation, we have: Necessarily (Ex)(x=x). And thus (3) follows. And of course Necessitation is a part of standard modal systems like M, S4 and S5.

I think (2) is intuitively plausible. Here is one way to try to argue for it:
5. (Premise for reductio) Premise (2) is false.
6. (Premise) The non-existence of non-unicorns does not necessitate the existence of unicorns.
7. Every being is contingent and it is necessary that at least one thing exists. (By 5)
8. Necessarily, if no non-unicorns exist, then at least one thing exists. (By 7)
9. Necessarily, if no non-unicorns exist, then at least one unicorn exists. (By 8) 
Since (9) contradicts (6), our reductio argument for premise (2) is complete.

(I am grateful to Josh Rasmussen for simplifying my original argument.)

Now, the weak point in the argument, I think, is premise 1, and specifically the assumption of classical FOL which allows the derivation of (Ex)F(x) from (x)F(x). In a free logic, this wouldn't happen.

But it is still an interesting fact, and a real cost to contingentism (the view that all beings are contingent), that it requires one to abandon classical logic or modify Necessitation. After all, there is some non-negligible prior probability that classical logic and Necessitation license only valid inferences.

Moreover, there is the question of why one should go for a free logic? If one's reason for going for a free logic is precisely that FOL licenses the derivation of (Ex)F(x) from (x)F(x), then one runs the danger of begging the question against the anti-contingentist, in that the derivation is valid (in the sense that necessarily if the premise is true, so is the conclusion) if there is a necessary being.

Objection 1: There is likewise a cost to the non-contingentist who is prevented from adopting those logics on which it is provable that possibly nothing exists.

Response: The non-contingentist who accepts such a logic can still make the move of distinguishing metaphysical and narrowly logical necessity. She can then say that the logic gives an account of narrowly logical necessity. Therefore, all that is shown in such a logic is that it is narrowly logically possible that nothing exists, but not that it is metaphysically possible that nothing exists. On the other hand, it is much harder for the contingentist to make the analogous move of saying that (3) is true in the case of "narrowly logical necessity". For it is widely accepted that if there is a distinction between metaphysical and narrowly logical necessity, the narrowly logical necessity is stronger of the two. Thus, if one accepts (3) with "narrowly logical necessity", one accepts (3) with metaphysical necessity, too.

Objection 2: There are other good reasons to accept free logic, besides the fact that FOL licenses the derivation of (Ex)F(x) from (x)F(x). Specifically, FOL+Necessitation implies that:
10. Necessarily (Ex)(x=a)
is true for every name a.

Response: This objection almost convinces me and is one of the main reasons why I think that while my argument lowers the probability of contingentism, it is not very powerful.

I do think there are two speculative responses to the objection, which is why I think my argument still has some weight.

i. The truth of (10) for every "name" a shows that FOL's "names" do not correspond in function to names in natural languages. In particular, they show that when translating natural language sentences into FOL, one can only employ FOL's "names" for necessary beings. This shows a significant limitation of FOL--namely, that FOL has no way of translating sentences like "Socrates is mortal." However, the fact that a logic has no way of translating a sentence does not mean that the logic's inferences are invalid. There is probably no standard formal logic that can translate all sentences of natural language.

ii. Another move in defense of FOL+Necessitation is that we should see the inclusion of non-dummy names in a language L as embodying existential assumptions about the referents of these names. Consequently, when we give the Tarskian semantics for a modal logic built on top of FOL, the recursive clauses for "Necessarily s" and "Possibly s" in a language L under an interpretation J should respectively read:
- Necessarily: If e(L,J), then s.
- Possibly: e(L,J) and s.
Here, e(J,L) is the conjunction of all metalanguage claims of the form "a* exists" where "a*" is a metalanguage name for the entity that the L-name "a" refers to under J, if L contains any names, and is any tautology otherwise. Then my initial argument needs to be run in a language with no names.

S5

In response to a question from a student, I explained S5 as the claim that modal truths don't vary between worlds—a modal truth, we might say, is a proposition of the form "Possibly p" or "Necessarily p" that is true. But actually, that's incorrect as an account of S5. S5 is compatible with there being modal truths in some worlds that do not obtain in others.

To see this, we need to make Robert Adams' distinction betwen truth at a world and truth in a world. On Adams' view, a proposition making de re reference to a particular only exists in a world where that particular exists. Consequently, propositions that reference contingent particulars de re are not necessary beings. Since there are modal truths involving contingent particulars, such as that possibly I will yell "Hurrah!" in five minutes, it follows that some modal truths exist in some worlds but not in others.

Adams' distinction, then, is basically this. We take as the primitive notion being true at a world. We might gloss this as follows: a proposition p is true at w provided that the state of affairs it reports obtains in w. We can then define being true in a world counterfactually: p is true in w iff it is the case that were w actual, p would be true. Since a proposition can only be true if it exists, it follows that p is true in w iff p is true at w and p exists in w. And if we like, we can say that p exists in w iff it is true at w that p exists.

On Adams' view, the proposition that Socrates either does not exist or is human is true at every world. But it is only true in those worlds in which Socrates exists.

Let us then follow Adams in allowing that some propositions exist in only some worlds. Then, in some worlds there will be modal truths that are not true in the actual world simply because they do not exist in the actual world. We now have two ways of defining modal operators. I'll just define possibility, M, since necessity is dual to it (Lp iff ~M~p). We say that M₁p iff there is a possible world w such that p is true at w. We say that M₂p iff there is a possible world w such that p is true in w. These give different results. For instance, if Adams is right about propositions about particulars, then M₁(Socrates never existed) is true but M₂(Socrates never existed) is false.

The modal logic we get with M₂ is no good. According to that modal logic, necessarily Socrates exists, but possibly there are no humans. So the modal logic we want is the one defined by M₁. But this modal logic is just fine for S5: it is basically just a restriction of a Plantingan modal logic with no scruples about non-actual particulars to those propositions that do not reference non-actual particulars de re.

Now we have a picture of modal logic that satisfies S5, even though there are modal truths in other worlds that are not modal truths in our world (because they do not exist as propositions in our world).