Obviously the chief problems raised by Fitch's Paradox surround the so-called "Knowability Principle":
(KP) p → ◊Kp If a claim is true, it is possible that it is known to be true.
However, I want to set these questions aside, and focus on a clarificatory point. We know that every known is known by somebody under certain conditions. So I think we might get a better handle on this sort of argument if we go through it again, this time indexing our epistemic operators to the epistemic agents. That is, let's keep track of who knows what. The way I'll do this is by inserting an index after each epistemic operator. For instance, instead of (KP) as I've written it above, we'll have an indexed (KP'):
(KP') p → ◊K{∃x|c}p If a claim is true, it is possible that it is known to be true by someone, i.e., by at least one person (under conditions).
This would contrast with a different, and much less plausible principle:
(KP'') p → ◊K{∀x|c}p If a claim is true, it is possible that it is known to be true by everyone (under conditions).
Obviously the two are very different. And we can do things with the |c side, too. For instance:
(KP''') p → ◊K{∃x|∃c}p If a claim is true, it is possible that it is known to be true by someone (under at least some particular set of conditions).
Which contrasts with:
(KP'''') p → ◊K{∃x|∀c}p If a claim is true, it is possible that it is known to be true by at least one person (under any set of conditions).
And, needless to say, we can mix and match. Now, I don't think any of this can be said to be unreasonable. To be known, a thing must be known by somebody; and that person knows it either under some particular set of conditions or under every set of conditions. But when we start doing this, it suddenly becomes clear that our previous argument was glossing over a lot that might turn out to be important for understanding what's really going on in the argument.
So let's begin indexing. First of all, when people accept KP, what specified version(s) of it are they really accepting? Any version which involves ∀c seems unlikely. So let's assume that we are talking about ∃c. That is, we aren't demanding that if a thing is true it is possible that it is known under any set of conditions; we just mean that if a thing is true, there is at least some set of conditions under which it is possible to know it. Suppose that p = "The sky is blue during the day". What appeals to people in (KP) is not the claim that
If the sky is blue during the day, it is possible that it is known (by someone/everyone) in every circumstance that the sky is blue during the day.
That is pretty clearly false; since it is possible for the sky to be blue and yet people only to know that it is under certain circumstances (for instance, people who have lived in caves all their lives, never seeing the sky). So let's assume that we are only talking about things that are known in some circumstance or other, i.e., ∃c.
The second question is: Who are the knowers we are assuming. Do we mean:
If the sky is blue during the day, it is possible that it is known by someone in some circumstance or other that the sky is blue during the day?
Or do we mean:
If the sky is blue during the day, it is possible that it is known by everyone in some circumstance or other that the sky is blue during the day?
The latter commits us to there being some circumstance in which everyone could know that the sky is blue; and this doesn't seem what people want (KP) to say. (Likewise, I'll assume for the rest of the post that ∃c is the only version in view.) So the proper specification would seem to be:
(KP''') p → ◊K{∃x|∃c}p If a claim is true, it is possible that it is known to be true by someone or other (under at least some particular set of conditions).
So we've got that out of the way. Now let's go on to the first stage of the argument:
(1) p ∧ ¬Kp A particular claim is true, and it is not known to be true.
(2) (p ∧ ¬Kp) → ◊K(p ∧ ¬Kp) (1) implies that it is possible to know both that the particular claim is true and that it is not known to be true.
(3) ◊K(p ∧ ¬Kp) It is possible to know that both the particular claim is true and that it is not known to be true.
First with (1). (1) is intended to tell us that there is a truth, and it is not known. Do we mean that it is not known by at least some particular person, or that it is not known by anyone?
It's hard to say, actually, since people who make this claim say things that sometimes suggest one, sometimes the other. Also, we have to keep in mind that we can't assume (at this stage) that the same 'someone or other' or 'particular person(s)' are in view. So I'll use a different variable, and for now I'll assume ∃y, i.e., at least one person (who may or may not be the same person(s) in (KP''')), but let's not forget the possible alternative. So:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other.
Now to find (2'). This is supposed to follow from (1') and (KP'''). But the variables are tricky here. So I suggest:
(2') (p ∧ ¬K{∃y|∃c}p) → ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other, it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person).
And (3') has to follow from (1') and (2'). So:
(3') ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone, to be true.
Put it all together:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other.
(2') (p ∧ ¬K{∃y|∃cp) → ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other, it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person).
(3') ◊K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone, to be true.
So far, so good. But what happens when we move on to (4)? Remember in the first post when I noted that it wasn't at all surprising that (4) led to a contradiction? Well, what about (4')?
(4') K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone, to be true.
Suppose that p = "The sky is blue," and suppose that it is true. Then (4') tells us that someone knows both that the sky is blue and that it is not known by someone (who may or may not be the same person).
And that seems less likely to yield a contradiction doesn't it? In fact, we know it doesn't, where the particular knower(s) we are describing by 'x' are not the same knower(s) we are describing by 'y'. For instance, suppose that at least one of the x-knowers is Tom, and that at least one of the y-knowers is Cindy, and that Tom is not Cindy. Then from (4') we can conclude that
It is known by Tom both that the sky is blue and that this is not known by Cindy.
Perhaps Cindy has never seen the sky and has been misinformed that the sky is mauve. So it would seem that (4') can't lead to a contradiction; there are non-contradictory instances of it.
But we know where the argument is heading, and it is toward a contradiction. Maybe it does raise a contradiction, even though (4'), as such, implies no contradiction. How so? Things are not nearly so simple as they had seemed at first; but it may well be that we don't need (4') to yield a contradiction in every instance; the paradox possibly just needs some instance of (4') to yield a contradiction. A paradox that holds only for some instances of (4') is a paradox still. Will it turn out to be so? Stay tuned for our next crazy installment. If your head is spinning a bit, good. But the roller coaster is just starting. I'll continue the indexing in a future post; and we'll also get into lots more issues, and come back to (KP) in particular, further on. And keep in mind that these are all sketchy thoughts.
