So having read some stuff on modal logic (basically just what the Stanford Encyclopedia of Philosophy has to say) I was trying to make up a modal logic of my own. So what follows is just brainstorming. It might not make any sense at all if you try to work out the details. But as I was stuck in trying to figure out those details I thought I'd mention it here.
Usually □ is used to mean things like "it is necessarily the case that", "I know that" or "it ought to be the case that" but I thought I'd see if it could be used to mean "it is almost certainly the case that". Here's the intuition I'm trying to capture: □p would be roughly like saying p holds with probability 1-O(ε). So □□p would mean that we expect it holds with probability (1-O(ε))² which is the same as 1-O(ε). So we'd have □□p⇒□p. I intend "almost certainty" to mean so close to certainty that if A almost certainly tells the truth, and so do B, C, D up to Z, then if Z tells you that Y says that X claimed....that A said p then you could still be nearly certain that p is true.
Other examples of things that I'd expect to hold would be "continuity" conditions like
and I'm nearly certain that
□A⇒◊A (where ◊A=¬□¬A by definition).
Anyway, it seems like a perfectly good modal logic to me. But when I tried to work out the possible world semantics I got nowhere, mainly because p⇒□p upsets things. I was also toying with the idea that "almost certain" is a bit like the strict mathematical sense of "almost always" and that there might be some kind of semantics that involves sets dense in some topology. But I couldn't work out the details of that either.
So can anyone who is knowledgable about these things tell me if what I'm trying to do is sensible? If I fill in more details can I use possible worlds to prove completeness and soundness?
The really wacky idea I had (and remember, I'm just brainstorming) is that you might be able to set up a differential calculus of logic. Just as you compute derivatives of a function w.r.t. x by perturbing x slightly and looking at how f(x) varies, □A might be seen as a slight deformation of A, hence my use of the term "continuity" above. I'd then expect to see a version of the Leibniz rule appear somewhere in the theorems of this modal logic.
Anyway, this might be going nowhere, so brainstorming over.
Ordinary propositional calculus can be viewed as being set theory in disguise. Pick a set X. Now translate a logical proposition to a statement of set theory by 'atomic' propositions to subsets of X, ∧ to ∩, ∨ to ∪, ⊤ to X; ⊥ to ∅ and ¬ to the complementation in X operator. A proposition is true if and only if its translation always equals X. For example ¬p∨p translates to P∪(X\P) which is clearly X.
For my proposed modal logic put a topology on X. Then □ translates to the Cl operator where Cl(A) is the smallest closed set containing A. The things I want to be true again correspond to propositions whose traslations are all of X.
I think this modal logic captures exactly what I want. You can get a feel for it by thinking about propositions like x=3 on the real line. This is false "almost everywhere" (ie. on a dense subset of the real line, ie. a subset whose closure is the entire real line) so it makes some kind of sense to want to say □¬(x=3).
With some keywords to search on I was now able to find this paper. Page 29 (following the document's own page numbering) essentially describes what I have except that they have swapped □ and ◊. Essentially they're saying that this is the standard modal logic known as S4. This topological interpretation of S4 goes back to work by Tarski and McKinsey in 1944.
And now I can reason about almost-certainty with certainty.