Friday, April 28, 2017

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 (*).


Brandon said...

I'm not sure I follow the argument here. (Although that's perhaps not surprising, since I find I usually can't make any sense of the examples in the literature, either.)

(1) "There is a non-actual world" requires augmenting possible worlds language with actual worlds language (and, indeed, a particular version of actual worlds language), which is different; possible worlds language on its own can't say it.

(2) We could obviously say, Diamond(Reality is different). [I'm assuming that by 'box-diamond language' you just mean any propositional logic with box and diamond operators. If we allow applications to terms, it gets even simpler, something liek Diamond(This reality) & Diamond(Not this reality).] So is your claim that the "Reality is different" part has to be added in by hand? But formally there is nothing about possible worlds language that requires an interpretation of possible worlds as possible realities; to get from possible worlds talk to realities talk, this interpretation has to be added in by hand, as well.

So there are definitely assumptions here that I'm not understanding.

Alexander R Pruss said...

1: Good point. But without mentioning actual worlds, we can say this: There are at least two worlds.

2. What's the subject of "Reality is different" if not something like a world? What's the predicate if not "different from the actual world"?

Brandon said...

On (1), I'm a bit skeptical of the claim that "There are at least two worlds" is equivalent to "Reality can be different", although maybe there's some particular interpretation of the former that constrains it to the latter. (For instance, it looks like you might have to be ruling out any Lewisian interpretation. But Lewisian interpretations, despite their many problems, are not misusing the possible worlds language itself.)

On (2), if you take 'reality' to mean just 'the actual world', sure; but surely you are still using box-diamond language in saying Diamond(Reality is different). Box and Diamond as formal operators are perfectly open on terms or propositions; they apply to whatever you are allowing as terms or propositions, they don't prejudge what their meaning can be. And this is related to the point with my comment about 'adding the interpretation in by hand'. It doesn't look to me like anything that can be said about box-diamond language in this regard wouldn't also have be said, mutatis mutandis, about possible worlds language. Despite the name, 'possible worlds', formally speaking, are not worlds except exactly in that way you happen to be interpreting them to be. That's why we can use possible worlds language despite the fact that there is no general agreement on what our account of a possible world should be.

Alexander R Pruss said...

Regarding 2, you still have to say *what* reality could be different from.

Heath White said...


Let R be the One Big Truth, or the conjunction of all the little truths. Then

Diamond squiggle R.

Alexander R Pruss said...

Is R a sentence? An infinite one? Uncountably infinite?

Or do we say: Let P be the One Big Propositional Truth, and then stipulate that R expresses P.

But P is a candidate for being a world: indeed big propositions like that are a very plausible candidate for worlds.

Minor technicality: "Diamond squiggle R" doesn't say that reality could have been different. To say that, you need (M~R and MR).

Heath White said...

I was thinking that R was the One Big Propositional Truth, and R stipulatively expresses P. Yes, P would be a world. Why is that an issue?

IanS said...
This comment has been removed by the author.
Philippe Bélanger said...

Let R be the reality you and I are part of. Possibly, reality is different from R.

Alexander R Pruss said...

Sounds to me like R is a world.

Philippe Bélanger said...

It is, but I don't see why this is a problem. Why couldn't we talk about our world when we speak the language of diamonds and boxes?

Alexander R Pruss said...

Sure we can. But the point is that it's hard to get away from talking about possible worlds--or at least one of them.

Fabio Lampert said...

I know this is an old post, but here is an example of something that cannot be said in a box-diamond language and it's as simple as your example:

(1)"at the actual world, p."

If I were to formalize (1), I would use the actuality operator under its traditional semantics, i.e. M,w/=@p iff M,@/=p. If we define truth in a modal model by truth at every possible world in the model (as we see in any modal logic textbook), then (1) cannot be expressed in the basic modal language. If, on the other hand, we define truth in a model model as truth at the actual world of the model, then (1) can indeed be expressed.

Here is how a simple proof of the former claim can go: we build two bisimilar models for the basic (@-free) modal language with a single propositional symbol p, in which case the two models are equivalent with respect to the basic modal language while disagreeing about (1).

The first model, M=(W,0,R,V) is defined as follows: W={0,1,2}, 0 is the actual world of W, R={(0,1),(2,1)}, and V(p)={0,1}.
The second model, N=(U,a,S,T) is defined as follows: U={a,b,c}, a is the actual world of U, S={(a,b),(c,b)}, and V(p)={b,c}.

These can be illustrated as follows:

M: 0(p)--->1(p)<---2(-p) N: a(-p)--->b(p)<---c(p)

Now we just define the following relation Z between the two models: Z={(0,c),(1,b),(2,a)}. It's easy to check that Z is a bisimulation between M and N, whence the two models are equivalent for the basic modal language. Since, however, M satisfies the truth conditions for (1) at every world (for at every world in M it is true that p holds at the actual world) and N doesn't, the actuality enriched language is strictly more expressive than its actuality-free counterpart.

The beauty of this example is that it involves solely the propositional portion of the language, and not the first-order examples we see in the literature. I take it that you thought of a similar strategy with your example.