Wolfgang Schwarz

Notes on Asymmetric Modal Epistemology

If a statement p is impossible, then empirical information and a priori reasoning usually suffice to establish its impossibility. So if despite carrying out the relevant empirical investigations and a priori reasonings no impossibility shows up, this is a good reason to believe that p is possible. One might be tempted to say that our knowledge of possibility is always based on such a failure to detect the respective impossibility. This is what Bob Hale calls an asymmetric approach to modal epistemology. (See his "Knowledge of Possibility and of Necessity", Proceedings, 2003.)

Asymmetric approaches promise to reduce all modal knowledge to a single basic kind. In theory that sounds nice, but in practice I think it's wrong. Hale himself concedes that knowledge of actualised possibilities is often not based on failed refutations, but rather on knowledge of them being actualised. Once the single foundation is given up, I see little motivation not to accept other sources, like reasoning from combinatorial principles or imagination or even direct intuition. (Personally, I would prefer a modal epistemology that looks more like other branches of hyperintensional epistemology, probably drawing a lot on knowledge of semantics.)

There is another, more technical, reason why asymmetric approaches can't deliver a single and unified account: Suppose to figure out whether p is possible you always have to find out whether its negation is entailed by certain necessary truths N; if ~p is not so entailed, you're justified to believe that p is possible. Now let p be the statement that blue swans are impossible. p is not possible, so its negation should be entailed by N. But the negation of p is the statement that blue swans are possible. So N itself contains the information that blue swans are possible (and necessarily so). Hence failed refutation is not the only route to possibility.

Daniel Cohnitz argues (in German) that asymmetric approaches are wrong because they are inconsistent. His argument goes roughly as follows:

Let LC be a logic that results from first-order predicate logic by adding 'Nec' and 'Poss' operators, with the semantical rules

SN) 'Nec A' is true under a given interpretation iff A is true under all interpretations;
SP) 'Poss A' is true under a given interpretation iff A is true under some interpretation;

and the deduction rules

DN) From the provability of A one may infer the provability of 'Nec A';
DP) From the unprovability of A one may infer the provability of 'Poss ~A'.

It is easy to see that LC is incomplete: For any predicate logic sentence A, 'Poss ~A' is logically true in LC iff A is not a theorem of predicate logic. Thus if there was a mechanism that detects all LC-theorems of the form 'Poss ~A', this mechanism could also detect all non-theorems of predicate logic, contradicting Church's Theorem. Now let A be a logical truth of LC. Then ~A is true under no interpretation, and by (SP) neither is 'Poss ~A'. If LC is sound, 'Poss ~A' is therefore unprovable. But then A is provable -- as otherwise 'Poss ~A' follows from its unprovability by (DP) --, contradicting the incompleteness of LC we've just proven. Hence LC is not only incomplete but also unsound.

This shows that in the context of LC, (DP) is not an acceptable rule of inference. I would add that it is not an acceptable rule of inference at all: in a deductive system it should be effectively decidable whether a given step in a proof is a correct application of the rules or not. But for (DP), this would require an effective decision procedure for predicate logic. So (DP) should not be accepted as a rule in any case.

Is this a problem for asymmetric approaches? What we've learned is that if the asymmetric approacher holds that the necessary truths are exactly those statements provable in a certain formal system, she must not also claim that this system contains the 'rule' (DP). But that 'rule' doesn't represent the asymmetric approach. The asymmetric approach doesn't require that we may infer the provability of a possibility from the unprovability of a necessity. Instead, it says that we may infer a possibility from the corresponding failure to demonstrate a necessity.

Let NC be whatever condition picks out the necessary truths. The asymmetric approacher then says that we may infer 'Poss ~A' from '~NC(A)'. If NC is provability in a system L, this becomes

DP*) From the unprovability of A in L one may infer 'Poss A';

or, if for some reason you want to have it as a proper formal deduction rule:

DP**) From the provability of the unprovability of A in L one may infer the provability of 'Poss A'.

For suitable L, the rule can even be included into L itself:

DP***) From the provability in L of the unprovability of A in L one may infer the provability in L of 'Poss A'.

None of these give rise to any inconsistencies.

There is a small problem just around the corner, though: if NC is provability in some undecidable formal system then on the asymmetric approach there will be no effective method to decide for arbitrary p whether p is possible. But at least to me this doesn't sound too worrying.