Back when I was a long-haired fresher, I was taught logic from a book by Wilfrid Hodges called Logic. I’ve since lent the book to someone from whom I don’t expect to get it back, but if I remember correctly, at the end the author wonders what expressions might one day have a logic developed for them. He wonders about ‘obviously’. He says he thinks that P follows from Obviously P, and that it’s unclear whether or not Obviously obviously P follows from Obviously P. I don’t remember him saying so in the book, but these principles correspond to the modal axioms T and S4. The S5 axiom would say that Obviously it’s not obvious that P follows from it’s not obvious that P, though I don’t think he mentioned that one at all.
I don’t have a logic for ‘obviously’.
I do however have an idea for an analysis of ‘obviously’, which I’ll tell you about now. I think that the fundamental facts about obviousness – or at least as fundamental as you can get while still using the concept of obviousness – are facts about what is obvious to a person. Facts of the form P is obvious to S. My first pass at an analysis of that is this:
If S considered whether or not P, S would be disposed to know that P.
This will admit some false positives, such as it usually being obvious to everyone that they are considering something, even when they’re not. We can patch that up easily enough:
P, and if S considered whether or not P, S would be disposed to know that P.
I expect that that’s still going to have some false positives like someone is considering something, and probably false negatives like S isn’t considering anything. (If not that one, then others.) This is the same kind of problem you get with counterfactual analyses of anything, but that needn’t make them useless. I’d hope you could deal with the false negatives by being careful about what you held fixed, perhaps making it a counterpossible conditional if necessary. And I’d quite like to deal with the false positives by saying something like this:
There’s a fact F that P, and if S considered whether or not P, S would be disposed to know F.
The idea is that for each G thing there is a different fact that there are G things, and if considering whether or not there were G things would create another G thing, that’d be another fact, and knowing that wouldn’t mean you knew one of the pre-existing facts. That involves a weird and unexplored (by me) ontology of facts though, so I’ll just hitch my wagon to those of the people already working on the false positives problem for counterfactual analyses.
False positives and negatives aside, I think the basic idea is pretty good. It allows that a priori facts can be obvious, which they can, without making them all obvious, which they’re not. It allows different things to be obvious to different people, which they are. It also allows people to miss obvious things, which they do, and it holds that this will usually but not always be things they haven’t considered, which also seems right to me. When you say that something is obvious simpliciter, your intentions or the context will supply some actual or hypothetical person or people (and an actual or hypothetical situation for them to be in) to relativize it to. This will also go for the sentential adverb obviously, except when you’re using it as a meaningless filler. (Not that there’s anything wrong with that.)
The T axiom obviously follows this analysis. I don’t think it decides the other modal axioms one way or the other, although I haven’t thought very hard about it. I suppose it could help us make some progress on the other axioms though, because it’d mean they had some consequences for knowledge or dispositions, which you might already have views on. Progress!