Monday, February 5, 2018

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:

  1. 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.


Wesley C. said...

What are your thoughts on the objection that Barry Miller's property view of existence, which seems essential for Thomism given it accepts the Real Distinction between essence and existence and is also essential for the Existential Argument, is in contradiction to axiom S5 or Brouwer's axiom?

Angra Mainyu said...


Is 1. true?
As I interpret it, it seems to be false, though I might be misreading. But let's consider the following scenario:

S1: Scientists are mistaken. Water is not composed of H2O, but in fact, water is simple.

It seems to me that there is no possible world at which S1 is true, but we can surely make sense of S1 (and thus, of a world containing S1).
So, there are impossible worlds one can make sense of (I'd say there are plenty more examples).
Granted, if one distinguishes between worlds that are possible as actual vs. as counterfactual, then S1 is possible as actual. But if one includes worlds that are possible as actual (but not as counterfactual) in 1., then the argument does not seem to be about the kind of possibility you want to focus on.

Alexander R Pruss said...

Some worries have to do with explosion. I know relevance logic can limit that, but I am still suspicious.

Angra Mainyu said...

You mean worries about impossible worlds?
I'm not so familiar with relevance logic, but as far as I know, the explosion worries involve worlds that have contradictions (either only considering logical symbols including quantifiers, or also the (internal) meaning of the words), but impossible worlds that do not involve those things do not appear to be a problem (as far as I know). In any case, we can make sense of many of them. In addition to a world in which water is not H2O, we can make sense of cases involving proper names, like:

w3: Hespherus is not Phosphorus.
w4: Jack the Ripper was Montague John Druitt.
w5: Jack the Ripper was Seweryn KÅ‚osowski.
w6: Jack the Ripper was Aaron Kosminski.

w3 is not possible, and at most one of the others is, it seems to me. Whether we can make sense of worlds involving contradictions (of one sort or another) is more difficult, but even if we can't, impossible worlds like the previous ones seem to be a problem for premise 1.

Or do you think there is some (correct) theory of proper names that makes all of those worlds possible? In any case, there is still the problem of water and H2O, or similar ones.

Side note: we can consider the assertion "Jesus behaved immorally more than once". I believe that's possible, but Christians (probably, if I'm getting this right) hold it's not, yet it's also something we can make sense of.

Perhaps, you can modify 1.' to make just the claim that only possible worlds exist, without making the claim that we cannot make sense of impossible ones.

Alexander R Pruss said...

Sure, you can make sense of individual impossible statements. But a world needs to have determinate answers to many other questions. In the world where the Morning Star is other than the Evening Star, what makes the two of them be the same as our Venus and yet different from each other?

Maybe my intuitions here were guided by the wrong way of thinking about possible worlds, though, something like what Kripke criticizes when he talks of the telescope view--you look at another world as if through a telescope and you try to see what's there. But maybe the better view is something like Kripke's view that we just stipulate facts about possible worlds.

In that vein, I suppose one could think of a possible world as just a collection of propositions, and then impossible worlds will be those collections of propositions that couldn't be all true together without other propositions being true as well. On that view of worlds, my argument is unsound.

Angra Mainyu said...

In re: what makes them different, I would say that in many scenarios in which the Morning Star is not the Evening Star (maybe not all; that's more complicated), there is no Venus. People who claim Venus exist are either mistaken or lying. The Morning Star and the Evening Star exist there, but are two different objects, one that we see in the morning, the other in the evening. What makes them different from each other depends on the sub-scenarios.

Granted, that does not provide conditions fully determining a world. However, that lack of determinacy is just as present in the case of run of the mill possible worlds. For example, let w8 be a possible world in which Clinton defeated Trump and is now the POTUS. What makes it the case that she won? Well, that depends on the sub-variant of the scenario we use to describe the world. There are plenty of different ways in which she possible wins, just as there are plenty of understandable differences (even if possible only as actual, not as counterfactual) between the Morning Star and the Evening Star.

So, I don't think indeterminacy is a problem for making sense of impossible worlds (at least, when they're possible as actual), assuming it's not a problem for making sense of possible ones.

That said, I'm not claiming impossible worlds exist. I take no stance on that (I think the statement would be unclear without a lot of clarification, and with clarification, the question would dissolve so to speak). In any case, I'm only arguing that we can sometimes make sense of impossible worlds, and if you changed your first premise for something like

"1'. The only worlds there are are possible ones.",

then the objection I'm raising would no longer work, though one might raise a weaker point regarding the warrant of 1'.
On the telescope view of worlds (if I get it right), you could argue in support of 1' without having to defend the stronger claims in 1.