## Wednesday, February 22, 2017

### Evening Note for Wednesday, February 22

Thought for the Evening

'Fitch's Knowability Paradox', which derives from some common assumptions about knowledge is usually summarized in the predicate calculus as:

∀p(p → ◊Kp) ⊢ ∀p(p → Kp)

This is usually summarized in much the way the SEP article by Brogaard and Salerno on the paradox does: "It tells us that if any truth can be known then every truth is in fact known." Hence the 'knowability'.

However, this way of summarizing it introduces more confusion than it should, because ◊Kp does not mean 'p is knowable' in the way we usually mean knowable. It means 'it is possible that that p is known'. This doesn't undo the paradox -- that if it is possible that a truth is known, that truth is known, which is not something one would immediately expect. But 'knowable' in ordinary discourse doesn't mean 'it is possible that it is known'; it means 'it can be known' or even 'it can come to be known', neither of which would usually be understood as saying the same thing as 'it is possible that it is in fact known'. What the paradox actually says is that, if you accept some very common things people believe about knowledge, then if there is any proposition that has the feature that when it is true, it is possible that it is known, then it must be the case that when it is true, it is indeed known. This is very different from the usual summary. As I once noted on this blog (so long ago in ancient days of yore that I was still in graduate school):

I already said that the result is usually put in terms of knowability: KP [i.e., the left hand side above] is usually read, "If p is true, it is knowable." I translated differently, as you can see. This is because I think "knowable" is a very bad translation of the double operator, ◊K. To see this, think about what we really mean when we say the following two things:

This is knowable: The sky is blue.
It is possible that this is known: The sky is blue.

The two are not equivalent, and for good reason. The English word 'knowable' in all but a very small handful of uses hides a third operator, a temporal operator -- an incipit, to be exact:

This can come to be known (can begin to be known): The sky is blue.

So when I say that some claim is knowable, I usually don't mean that it is possible that it is known; I mean that it is possible that it could come to be known. So I think there's reason to stay away from the added complications that are introduced by the word 'knowable'. Using 'knowable' makes it sound even more paradoxical; but (1) it doesn't need to be made to sound more paradoxical; and (2) it is misleading.

(An incipit operator is an operator that tells us that something begins to be; incipit [it begins] and desinit [it ends] were modal operators that medieval logicians studied quite a bit but people nowadays not so much.)

This is not to say that the use of 'knowable' in this context is illegitimate; you can see why someone might use knowable to mean ◊K -- after all, K is knowledge and ◊ is possibility, so 'knowable' could be used to mean 'it is possible that it is known'. Possibly one could even find occasional conversational cases in which it is. But it's not usually. And one can tell from summaries that even philosophers regularly slip from the strictly correct, 'it is possible that it is known', to 'it can be known', and even 'it can come to be known'; you can tell this from their colloquial descriptions of what they think their results show. But these are different modalities from ◊K.

Mixing any modalities is very, very tricky; but ◊ and K, possibility and knowledge, are especially so, and it is often difficult to keep straight about how we are to understand them, and about the rules of inference we are using. In real life, when we are talking about knowledge, we are usually in fact talking about coming to know or having come to know; but the standard epistemic logic just talks about being known. There are lots of ways the difference can throw off interpretations.

This was all brought to mind thinking about Rutten's modal-epistemic argument for God's existence (PDF), which Red brought up in comments here a couple weeks back:

1. For all p, if p is unknowable, then p is necessarily false (first premise; the principle),
2. The proposition ‘God does not exist’ is necessarily unknowable (second premise),
3. Therefore, ‘God does not exist’ is necessarily false (from both premises)
4. Therefore, necessarily, God exists (conclusion; from (3)).

If someone is going to evaluate this, they have to be clear about what 'unknowable' means. Does it mean ~◊K, i.e,. 'it is not possible that it is known'? Does it mean 'it is impossible to come to know it'? Does it mean 'it is not such that it can be known'? And so forth. And it gets a bit worse, because (2) talks about 'necessarily unknowable'. So if we were to interpret 'knowable' as it would usually be interpreted in common conversation, we might have a whole string of at least 4 modal operators in one proposition: (1) it is necessary that (2) one cannot (3) begin (4) to know p. That's not necessarily how it would have to be understood; but you can see how it would be important to keep straight on exactly how you do understand it.

All of this is just interpretive; it doesn't directly give us a line on how to evaluate either Fitch's Paradox or Rutten's modal-epistemic argument. But it's good to avoid making such evaluation harder than it needs to be.

* Joseph Millum, The Foundation of the Child's Right to an Open Future. I discussed this topic here. Millum's is a good, and very thorough discussion of some problems with the notion.

* Tristan Haze, The Resurgence of Metaphysics as a Notational Convenience. I'm very interested, of course, in accounts of how philosophical scenes get transformed, how ideas transmogrify, and the like. This hypothesis for the rise of analytic metaphysics makes considerable amount of sense, and is probably true.

* Ellen Carmichael, Lafayette's America, on America's French Founding Father.