Topics Logic

Question A question about logic. When symbolizing and making inferences in natural languages that contain such terms as "it is necessary that", "A ought to do X", "A knows X", and "it is always the case that", there are extensions of classical logic, respectively, modal, deontic, epistemic, and tense logic that attempt to deal with such natural language analogues. My question is: What about propositions that contain a mixture of all the above terms? For example, there are sentences in natural language of the form “It is necessary that John ought to always know that 2+2=4." Is there a logic that can effectively handle (i.e. symbolize and correctly infer) such propositions? If so, is this logic both sound and complete? If there is no such logic, what is a logician to do with such propositions? My intuition is that things get tricky when you mix these operators together and/or the classical quantifiers. Thanks kindly for your reply, A Concerned Thinker

Accepted:
February 25, 2006

Comments

Things get tricky anyway when you mix modal operators and the quantifiers, so it's best if we just leave it to the propositional case. And I'll add, just by the way, that it is quite controversial whether such "operator" treatments are correct for any of these cases, more so for tense, perhaps, than for the rest. Most semanticists nowadays, I believe, would take tense in natural language to be quantificational. And David Lewis, of course, held the same about "necessarily".

There are really two kinds of questions here: Can one write down some plausible logical principles governing (say) a language with two such operators? And then, can one develop a semantics for this language and prove soundness and completeness? As for the former question, the interesting issue is what principles should connect the two kinds of operators. Presumably, for example, we should have "If it is necessary that p, then it is always the case that p", but not conversely; and I suppose some people would have us assume "If it is necessary that p, then A knows that p". Most people would reject that claim, however, and it's not clear to me that there are any interesting relations between those two operators. (True, we have Kp → ◊p, but that's because we already had Kp → p.) In any event, once we have some such principles, then we can try to develop a semantics.

I don't know to what extent people have investigated logics with more than one modality, generally speaking (googling "bimodal logic" turned up a few things, but not a lot), but I know there is at least some work on this sort of topic. In provability logic, in particular, people have looked at "bimodal" logics. One has operators meaning "it is provable in PA that" and "it is omega-consistent with PA that". It turns out, indeed, that it is very difficult to set up a decent semantics for this logic, though there is a way around the problem, and one can show that the logic is sound and complete with respect to a certain semantics and also prove its "arithmetical completeness", which means that it really does give the logic of "it is provable in PA that" and "it is omega-consistent with PA that" (where we are talking about the logic of these expressions inside PA itself, not, say, inside ZFC). See Ch15 of Boolos's The Logic of Provability for the details (which I do not fully understand, let me quickly add).

So, in principle, yes, there can be problems when one mixes modalities. And I can well imagine there would be such problems if one tried tomix "normal" modalities, like "necessarily", with arguably non-normalones like "ought", since one might find it difficult to insulate thesemantics for the former from the impossible worlds needed for thelatter.

Until we know a bit more about what's possible here and what's not, we won't have to worry about what to do if there is no decent logic for such combinations.