Tuesday, October 11, 2011

AFAIK and Modal Logic

In comments on another blog, an interesting question about the modal logic of phrases like 'as far as I know' (AFAIK), with someone making an argument with the following structure:

If all that is A is B, then C. Therefore, if all that we know to be A is B, then, as far as we know, C.

This is certainly not formally valid. If, for instance

A = true
B = what we know is true
C = we (collectively) know all truths

then we get an ontological argument that we are (collectively) omniscient, based entirely on the true premises that (1)if everything that is true is what we know to be true, we (collectively) know all truths; and that (2) what we know to be true is what we know is true. Since we don't know all truths, even collectively, this pretty much guarantees that the argument is invalid.

But it did start me thinking on the side question of how the AFAIK modality works. So let's go through some standard kinds of principles in modal logic and see if we get anything that sounds about right, and also see if any interesting side issues arise that shed light on the whole thing.

(1) Distribution: [AFAIK](p ->q) -> ([AFAIK]p->[AFAIK]q)
If p's being true implies (as far as I know) q's being true, then p's being true (as far as I know) implies q's being true (as far as I know).

This seems a reasonable enough principle, but there is a bit of room to think it doubtful, because if something is true as far as I know, this does not mean that I know that it is true. In fact, p's being true as far as I know doesn't seem to rule out q's being false as far as I know; this would be a puzzling situation to be in, but it's not clear that it's contradictory. Certainly such puzzles do seem to arise: where, as far as you know, X is true and, as far as you know, Y is true, but X and Y can't both actually be true. This is one way we can characterize paradoxes, for instance. The problem arises because clearly if I say "p is true as far as I know" I am saying, "Given everything I know, p seems to be true". It's probabilistic and defeasible: p doesn't definitely follow from everything I know (because then I would just say that I know p is true), it just seems to. And we know for a fact that things can seem to follow from what you know that are just plain inconsistent -- these aporia, anomalies, and paradoxes lead us to look a little deeper, and try to expand our knowledge so that we don't just have to go on what seems to be true.

So maybe we should say that Distribution fails for AFAIK, and that we need a different principle, e.g.,

Spread: [AFAIK](p ->q) -> [AFAIK]([AFAIK]p->[AFAIK]q)

That seems to get around the problem pretty reasonably. On the other hand, usually when Distribution-like principles fail, you can actually tweak things a bit (e.g., introduce a condition, or slightly modify the implication, or what have you) so that you restore Distribution, or something very, very close to it. And there are many advantages of doing so, because systems that make use of Distribution-like principles tend to be pretty straightforward, all things considered. So maybe we shouldn't give it up. I'm not sure off the top of my head, though, what would get around the fuzziness of AFAIK.

Further, it's interesting to note that AFAIK is not, like knowledge or necessity, a stable operator: what is true as far as I know changes over time, and varies from circumstance to circumstance. Therefore it doesn't seem that a situation in which [AFAIK](p->q) really does imply that in a situation where [AFAIK]p , it would also be true that [AFAIK]q. For instance, I tell you today that as far as I know, if John is in a place called Paris, John is in France. This can certainly be true. But then I get a letter from John postmarked Paris, Texas. What is true as far as I know has changed, and it's true that [AFAIK](John is in a place called Paris) but not true that [AFAIK](John is in France). This is, of course, easily handled; [AFAIK] would have to be treated as time-relative. It really doesn't mean 'as far as I know' (simply) but rather 'as far as I know at a given time'. And actually, this makes a lot of sense, whether one wants Distribution or not.

(2) M-Boxish: [AFAIK]p ->p
'If p is true as far as I know, p is true.'

This clearly doesn't work unless I am omniscient. What this shows is that AFAIK doesn't work like standard necessity. It doesn't seem like it would be any Box-like modality at all.

(3) M-Diamondish: p->[AFAIK]p
'If p is true, p is true as far as I know'

Again, this doesn't work unless I am omniscient. What this shows is that AFAIK doesn't work like standard possibility. But I think one could argue that this is a less obviously absurd position; which might be taken as a reason to think that it's some kind of Diamond modality -- possibility like, but not possibility in our usual sense. So let's deal with this a bit more. Since AFAIK obviously has something to do with knowledge, we should ask ourselves what relation it has with knowledge, which is very Box-like.

(4) Epistemic D-Diamondish: [KIp]->[AFAIK]p
'If I know that p is true, p is true as far as I know'.

Now we have something that seems right. The reverse, of course, (which I would, on the same principles of naming I've been using, call Epistemic CD-Diamondish) doesn't work unless I'm omniscient, for much the same reason M-Diamondish doesn't.

Of course, this is exactly what we would expect: AFAIK is an epistemic modality, dealing with knowledge; but it is definitely weaker than K (which is knowledge).

What it suggests to mind immediately is that AFAIK may very well work like 'Permissible' in deontic logic.* And if we can salvage distribution it in fact just is the same as the logical systems typically used for deontic logic -- either D or D4, to be precise, depending on whether we accept that claim, "If I know something, I know that I know it," -- if we don't accept it, we're in D, and if we do, we're in D4. And even if we have to modify Distribution, we're still going to be left with the conclusion that AFAIK is Permissible-like.

This actually makes a lot of sense, if you think about. If I say 'P is true as far as I know', this is actually not too far from saying, 'Given what I know, it's OK (permissible) for me to conclude that P is true'. One might want to argue that the former claim is a little bit stronger than the latter claim, and I think there's something to be said for that, too. If that's so, we might have to look for what, precisely, distinguishes AFAIK from a 'It-Is-Permissible-for-Me-to-Conclude' modality. I would suggest, if we go that route, that we look at the fact that AFAIK seems to suggest that there's at least a reasonable argument from what we know to the conclusion (it's just not an absolutely certain argument). AFAIK would then be something like there-is-good-reason-for-me-to-conclude, which is slightly stronger than it-is-OK-for-me-to-conclude. We certainly do make such distinctions on occasion. On the other hand, there are plenty of contexts where this would be splitting hairs. I'm not sure the best way to go here.

So what do you think?

* Incidentally, looking at this SEP article again, which is by Paul McNamara, I have to say that it is extraordinarily good. If I've not recommended it before, I recommend it.

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