Continuing my rough thoughts on Fitch's Paradox. Up to this point I have suggested that, contrary to first appearances, the paradox is not generated by KP but by a peculiar interpretation. The argument is that, when you keep track of who knows p and under what conditions, the paradox fails to go through, unless you assume that the knower in question is such that it is possible, no matter what the circumstances, that that knower knows the truth in question. And on that assumption it is perhaps rather plausible that this is only true of an omniscient person; or, rather, that it is a knower that either knows no truths (and therefore is not really a knower) or knows every truth, i.e., of a person of whom it can be said that, for any truth you please, they are able to know it, and for whom the knowledge of any truth has exactly the same conditions of knowing. If such a person knows any truth, his knowing of any other truth requires no change in circumstances, or, to put it in other words: such a person doesn't have to learn truths (knowing any truth, that person is in a position to know any truth) and therefore is not dependent on conditions for passing from ignorance to knowledge.
But so far I haven't really looked very closely at this notion of "possibility of knowing." So I'd like to look more closely at KP:
(KP) p → ◊K{∃x|∃c}p
And, in particular, at what this "◊K" means, or can mean. To put it more colloquially, what can people consistently mean when they say, "Every truth is knowable," or "Every truth can be known by somebody"?
I already noted that we have to be careful here, because 'knowable' very often hides a temporal operator: If I ask if it's knowable whether such-and-such is true, I will very often mean "Is it currently possible to come to know that such-and-such is true?" Or, in other words, "Is it possible for such-and-such to come to be known?" This is very different from the following two interpretations:
Is it possible that such-and-such is known now?
Is it possible that such-and-such is known (at some time, past, present, or future)?
Now I think it's fairly clear that the most natural reading of sentences in modal logic is to read it in one of these two ways. Thus, the most natural interpretation of a sentence like KP, just in general, is "If p is true, it is possible that p is known," where only the context will determine the ampliation (this time or any time at all). However, KP is supposed to be a sentence that is very plausible to a lot of people, and is the sort of thing they might appeal to in discussion. But the only people to whom it will be plausible that "If p is true, it is possible that p is currently known" are those who believe that there currently exists an omniscient being (although strictly speaking if someone believed that the community of all epistemic agents -- and this would have to be a community immensely larger than the human race -- is collectively omniscient, they could hold this, too -- but who believes that?); and even they wouldn't usually mean a sentence like KP in this sense.
So of the interpretations canvassed so far, the only one that is even remotely plausible as an interpretation of KP is:
If p is true, it is possible that it is known at some time (by someone under some circumstances).
And even this is a very strong claim. Take, for instance, the location of a molecule fourteen billion years ago. Someone who held the above claim would be committed to saying either that there existed epistemic agents at the time for whom knowing it was possible, or that, at some point afterward, epistemic agents could arise who could discover it, or that, in fact, there is no truth to be known. And so on with the features of every molecule in all the universe, at any time. I very much doubt that people really intend this; it's only a plausible claim to make if you assume the existence either of an omniscient being or of a collectively omniscient community.
I think it's also possible that we are seeing a conflation of propositional and predicate modality, although how far it messes things up is an open question. As we saw when we were looking at Sommers-Englebretsen Term Logic, a great deal of sense can be made of a position that distinguishes modalities applying to part of the predicate from modalities applying to the whole proposition. And it seems to be a good idea in cases like this, because 'p could be known by S' is compatible with 'It is possible that p is known by S' -- 'p could be known by S' is true when certain conditions for S's having the ability to know p are met, whereas 'It is possible that p is known by S' requires that conditions for actually knowing p are not ruled out (whether in fact p is actually known or not). It is possible for S to have the ability to know p, but to be prevented from knowing p by some accident or incidental condition, for instance.
None of this is particularly surprising in itself, but it highlights the layers and layers of complexity hidden in these apparently simple statements. And it is relevant to this paradox. Michael Fara has an interesting paper called the Paradox of Believability (PDF) which notes a parallel sort of paradox (for believability and 'superagents') that is resolvable by closer attention to modal issues similar to these. And that suggests that we can't be lax about them here.
Again, all this is sketchy. Further thoughts to follow, I am sure.
