Tuesday, September 10, 2013

Impossible worlds

Suppose w is an impossible world. Then impossible things may be possible at w. For instance, w might be a world where square circles are possible. But an impossible world need not be such that impossible things are possible at it. After all, an impossible world w might have the same modal truths as our world does and violations of them. Thus, there will be two impossible worlds: One where there are square circles and square circles are possible, and one where there are square circles despite their impossibility. Moreover, there will be an impossible world that is just like ours except for some or all modal truths. Imagine a world just like ours except that every proposition is possible and another just like ours except that no proposition is possible.

When I say these things, I seem to be near the boundary of coherence—and maybe on the wrong side of it. But one can give precise descriptions of such worlds by saying precisely which propositions are true at them. For instance, consider a world w1 such that a proposition p is true at w1 if and only if p is actually true or p is a proposition expressing the possibility of a proposition q (for any q), and all other propositions are false.

Heath White said...

Well, standard Kripke semantics would clarify this.

Whether some impossibility p is possible at an impossible world w is just a matter of whether 'actually p' implies 'possibly p', which is to say it is a matter of whether the T axiom is true. But whether that axiom is true is not a feature of the world but of the frame.

Likewise, whether other modal truths are true at this world is a feature of the accessibility relations between worlds, not a truth local to the world. (That is not exactly right but you see what I mean.)

Alexander R Pruss said...

I doubt that standard Kripke semantics do justice to the full range of impossibilia.

Mike Almeida said...

It is hard to make good sense of impossible things being possible at w. It can be true that, say, and impossible state of affairs p obtains at w, but that would not make p possible. For an impossible p to be possible at w, there must be a possible world w' at which p obtains. But of course that's not true for any impossible p. I think what is getting conflated is obtaining at an impossible world and being possible at an impossible world. On the other hand, there are lots of possible states of affairs that obtain at impossible worlds, but that's maybe too obvious to mention.

Dagmara Lizlovs said...

As for square circles remember the formula "Pie are squared", or is it "Pie 'r' squared", or "Pi 'r' squared", or "Pi r**2". See, skool learned me good, or is it skule lerned me gud, or is it skool teached me good? Anyway, English is really my second language. Latvian is my first. Sometimes when it comes to coherence, I hit my own boundaries where I know the Latvian word for something and just can't think of the English word for it. Anyways most coherence problems at 8:30 AM disappear with a cup of strong coffee. The times when I've been on the boundary of coherence or maybe on the wrong side of it, where when I had been out all day goose hunting in freezing rain, or out deer hunting all day in wintry mix and a good wind. In these cases what restores me to coherence is a cup of tea and a shot of Drambuie. Oh by the way, deer archery season has just started. :-)

Alexander R Pruss said...
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Alexander R Pruss said...

Mike:

Surely for any impossible proposition p, there is some impossible world w at which p holds. Otherwise, some impossible propositions are more impossible than others: there are those that are impossible, but not too impossible to be true at some impossible world, and there are those that are so impossible that they aren't true even at an impossible world.

It seems to me that once one admits impossible worlds into one's ontology, one should not put further constraints on them. After all, why do we want impossible worlds? We want them to do things like modeling inconsistent doxastic commitments. But inconsistent doxastic commitments can be of all sorts. One can just as easily be committed to the claim that some impossible things are true as one can to the claim that some circles are square. So we want a world at which it is true that some impossible things are true and a world at which it is true that some circles are square.

A handy model for worlds that allows for impossible worlds is that a world is a function from propositions to truth values (including, if need be, truth values like "neither true nor false"). Those functions f that are such that possibly (p)(p has the truth value f(p)) are possible worlds.

All other functions define impossible worlds. But then any truth value assignment will yield a world. There will be a world which has the same truth value assignment as our world, except that at it the proposition <it's impossible that 2+2=4> is assigned TRUE (though <2+2=4> is assigned TRUE, just as at our world).

thirdmillennialtemplar said...

Why not simply say that there are no logically impossible worlds? A proposition P is impossible if and only if there is no possible world in which P is true. Since all 'worlds' are possible, and we only adopt the convention of talking about 'impossible worlds' as a facon de parler, we can say that there is no impossible world w1 in which some logically false proposition P1 obtains as true. That just looks to me like the bastardization of our common modal language game.

Construe Logically possible worlds as possible states of affairs. To say that some proposition or set of propositions is impossible is just to say there is no logically possible world exemplifying the stipulated state of affairs. That proposition or set of propositions does not 'obtain in a logically impossible world', but rather fails to obtain at all possible worlds. What we mean by the expression "P obtains in an impossible world" is that P does not obtain in any possible world. Why not simply say that no 'world' (understood as a stipulated state of affairs) is impossible - only the incoherent state of affairs is impossible, and this, being incoherent, isn't anything at all, it is no world at all.