Thursday, August 28, 2014

A very impossible world?

In a criticism of the Pearce-Pruss account of omnipotence, Scott Hill considers an interesting impossible situation:
  1. Every necessary truth is false.

While the criticism of the Pearce-Pruss account is interesting, I am more interested in a claim that Hill makes that illustrates an interesting fallacy in reasoning about impossible worlds. Hill takes it that a world at which (1) holds is a world very alien from ours, a world at which there are "infinitely many" "false necessary truths".
But that's a fallacious inference from:
  1. (∀p(Lp→(p is false))) is true at w
(where Lp says that p is necessary) to
  1. p(Lp→(p is false at w)).

Indeed, there is an impossible world w with the property that (1) is true at w and there is no necessary truth p such that p is false at w. Think of a world as an assignment of truth values to propositions. A possible world is an assignment that can be jointly satisfied—i.e., it is possible that the truth values are as assigned. An impossible world is an assignment that cannot be jointly satisfied. Well, let w0 be the actual world. Then for every proposition p other than (1), let w assign to p the same truth value as it has according to w0. And then let w assign truth to (1).


Mike Almeida said...

There are two options: in 'dead end worlds' (I think that's what K. Segerberg calls them) everything is necessary and nothing is possible and in non-normal worlds the reverse is true: everything is possible and nothing is necessary. In non-normal worlds there are infinitely many false necessary truths.

Alexander R Pruss said...

Note, though, that "everything is necessary at w" can have an internal and an external reading (((p)Lp) at w vs. (p)(Lp at w)), just as in the Scott Hill case (1) is an internal reading of "every necessary truth is false" and (2) is an external reading of it.

There is an impossible world w where everything is necessary in the internal sense, but there is no proposition p such that (Lp at w). So, internally, everything is necessary; externally, nothing is. For instance, take a world where the only true proposition is that everything is necessary. Every other proposition is non-true. Then in particular, no proposition of the form Lp will be true at that world.

Mike Almeida said...

Hi Alex,

I'm a little puzzled by your distinction. First, I don't recognize the syntax: neither of the sentences you discuss for external and internal readings is well-formed, at least, that I can tell. There is not only no propositional quantifier, there's nothing corresponding to 'in w' You're likely in a metalanguage you've chosen. But forgetting that, what are you quantifying over in the external case? Is it all propositions, period? All propositions in some world? Finally, is the distinction you're making the truth in a world/truth at a world distinction?

Alexander R Pruss said...

Yes, this is a case of the in/at distinction.

Maybe the simplest way to make sense of my syntax: I am assuming there is an "at w" sentential operator, and that we have a second order (or substitutional?) quantifier over sentences/propositions (tricky to get this exactly right).

In the external case, I can only quantify over the actual (true or false) propositions.

Scott Hill said...

Thanks for this, Alex. I see your point. It is well taken.