Suppose narrowly logical necessity LL is provability from some recursive consistent set of axioms and narrowly logical possibility ML is consistency with that set of axioms. Then Goedel’s Second Incompleteness Theorem implies the following weird anti-S5 axiom:
- ∼LLMLp for every statement p.
In particular, the S5 axiom MLp → LLMLp holds only in the trivial case where MLp is false.
For suppose we have LLMLp. Then MLp has a proof. But MLp is equivalent to ∼LLp. However, we can show that ∼LLp implies the consistency of the axioms: for if the axioms are not consistent, then by explosion they prove p and hence LLp holds. Thus, if LL∼LLp, then ∼LLp can be proved, and hence consistency can be proved, contrary to Second Incompleteness.
The anti-S5 axiom is equivalent to the axiom:
- MLLLp.
In particular, every absurdity—even 0≠0—could be necessary.
I wonder if there is any other modality satisfying anti-S5.
6 comments:
I've been thinking about this half of the day. The weirdness will carry over to any logic that is similar enough to an alethic modal logic; the temporal interpretations are, if anything, more obviously absurd. Deontic and epistemic interpretations, farther out, are less immediately absurd, but not really less weird. The epistemic interpretation implies that it itself is not known to be true, and that, in fact, nothing can be known to be unknown by someone. The deontic interpretation seems to give the result that absolute anything can be prohibited or required.
There are modal logics that don't take propositions, so maybe some of them would be far enough away from the alethic interpretation not to inherit the weirdness? I'm not familiar enough with those that take programs, but I'd imagine they are still too close. You can do a unary mereological interpretation of modal operators, which take things that can overlap something or be part of something; if L is then 'is part of something' and M is 'overlaps something', then MLa would be 'Every a is a part of something that overlaps something'. And it itself doesn't sound all that weird. Perhaps it helps that there is no obvious mereological analogue to an operator applying to an absurd proposition.
Isn't everything a part of something (namely itself) and doesn't everything overlap itself?
Here's maybe a context where we have anti-S5. Suppose parliament is omnipotent in the following way: there are no limits on what laws they can pass, except insofar as they have passed a law limiting what laws they can pass. And suppose, further, that parliament right now has passed no such laws. Then interpret:
Lp = It is legally required for parliament to act so that p.
Mp = It is legally permitted for parliament to act so that p.
Then, MLp is correct for all p: parliament is permitted to make anything legally required. (Even absurdities! Of course, it would be silly to outlaw 2+2=4, but they could, or so we suppose.)
But note an interesting difference. In the narrowly logical modality case, it's not just that MLp happens to be true for all p, but it is a theorem, and hence it's *metaphysically* necessarily always true that MLp.
But parliamentary-legal-anti-S5 is not metaphysically necessarily true, because the very omnipotence of parliament means that they *metaphysically could* forbid making certain things legally required.
Yes, that's a good point; under the usual interpretations MLa would be trivially true. But, of course, ~LMa looks like we get the same weirdness to begin with, since it says that a overlaps something that is not a part of anything; but everything is a part of itself. (The unary mereology interpretation breaks the usual translations between L and M, not something I had really thought about before, which is why the two are not equivalent.) So we do actually have an analogue of allowing absurdities here. Merely restricting parthood wouldn't help any, so mereological interpretations of modal operators also inherit the oddness.
It's possible to have legal fictions and even entirely reasonable legal fictions that would be metaphysically absurd -- so for instance, you could legally fix pi at 3.1416 for some legal purpose that doesn't require greater precision and for which greater precision might be a waste of time. It's a much more robust claim to think parliament could make any absurdity at all a required legal fiction for some purpose, but I suppose it's at least consistent with some accounts of legal fiction.
Your point about the difference is, I think, an important one. People sometimes talk as if all modalities are reducible to metaphysical ones, but this really seems to depend on how metaphysical they were to begin with.
Here's a very different candidate. Suppose that Mp indicates that p is thinkable, in the sense that one can think about p,and Lp indicates that one cannot think about ~p. Then MLp says that one can think about not being able to think about ~p.
Another modal logic that satisfies anti-S5 is this one:
Lp is p&~p
Mp is ~(p&~p).
This modal logic has T but not Necessitation, since nothing is necessary on it (and Mp is a theorem for all p, so we have anti-S5).
That's not gerrymandered, but it is trivial.
Note that anti-S5 cannot allow for both T and Necessitation. For if p is a tautology, then by Necessitation, Lp is a theorem, so by T, Mp is a theorem, so by Necessitation LMp is a theorem. In fact, it cannot allow for what one might call weak-T and Necessitation, where weak-T says that if Lp, then Mp. (While T says that accessibility is reflexive, weak-T says that every possible world has something accessible from it--not necessarily itself.)
That makes sense; it explains why combining it with anything that seems remotely like an ordinary alethic modal logic gives strange results, and also why the deontic and provability cases leave more room. Anything with the so-called Brouwer axiom on its own, if p then LMp, would obviously be out too, since together they imply that everything is false. Since the S5 system can be built using B, too, the anti-S5-ness of anti-S5 seems very well assured.
Taking inspiration from that, another candidate, this time doxastic: Let Lp be 'p is believed by believer A' and Mp '~p is not believed by believer A'. Then ~LMp is "A does not believe that he does not believe that not-p". MLp is "A does not believe that he does not believe that p". While A doesn't seem very intelligent, A also doesn't seem impossible -- if A has beliefs but no beliefs about his beliefs, for instance.
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