### Monday, August 27, 2007

## Knowability and Dialetheism

In the most recent issue of Analysis, Peter Milne has argued that an omniscient being would have to be a dialetheist. The argument proceeds as follows. The assumption that there is an omniscient being entails a contradiction. But rather than merely treating this as a reductio of our assumption, we have the option of holding that the contradiction is true. Since it's true, our omniscient being must know it, so if we want to hold on to an omniscient being, we'll have to recognize that it must know something of the form P & ~P. The argument that the assumption that there is an omniscient being leads to a contradiction runs as follows. Consider the following sentence:

S: No omniscient being knows that which the sentence S expresses.

S is clearly self-referential, but it doesn't trip itself up; it's simply true if there aren't any omniscient beings. Now, suppose that ~S, i.e.,

1. Some omniscient being knows that which S expresses.

Then, given what S expresses,

2. Some omniscient being knows that no omniscient being knows that which the sentence S expresses.

But then by the factivity of knowledge,

3. No omniscient being knows that which the sentence S expresses.

But that's just S. So S is true.

Now suppose that X is an omniscient being. Then X knows all, including what we've just shown, namely (3). So,

4. X knows that no omniscient being knows that which the sentence S expresses,

which, given what S expresses, yields,

5. X knows that which S expresses.

But X is omniscient. So,

6. Some omniscient being knows that which S expresses.

But (6) is just ~S. So, on the assumption that an omniscient being exists, S & ~S.

Milne leaves the reader to draw their own conclusions from his proof. Jesper Kallestrup has picked up the baton, and adapted Milne's proof to show that the knowability principle (KP) entails a contradiction. Again, rather than treating this as a reductio, we might accept that the contradiction is true. A final application of (KP) yields that this contradiction is knowable. Anti-realists are committed to dialetheism.

It can hardly have escaped the attention of readers of this blog that I'm more sympathetic than most to anti-realism, and (unlike Jon Cogburn), I'm not in the market for the conclusion that it entails committment to dialetheism. There's a lot that might be said in order to dodge this conclusion. Jesper notes that so-called restriction strategies that have been offered by Dummett and Tennant in response to the knowability paradox also work here, but he points out they are associated with a whole bunch of problems. Crispin Wright's endorsement of KP as a local thesis also blocks Jesper's reasoning, which requires an assumption that KP holds globally.

Here I just want to suggest that there's a further restriction to KP the anti-realist can make, one that's independently motivated and yet which blocks the proof. Let's begin by examining Jesper's argument that KP entails a contradiction.

Consider the following variant of S:

S*: Nobody possibly knows that which the sentence S expresses.

Again, this is self-referential but does not simply implode. But now suppose that we adopt the following:

KP: P expresses a truth only if it is possible that somebody knows that P expresses a truth.

Actuality Assumption (AA): It is possible that somebody knows that P expresses a truth only if there is some actual person who, perhaps which finite extensions to her capacities, possibly knows that P expresses a truth.

Factivity Assumption (FA): It is possible that somebody knows that P expresses a truth only if P expresses a truth.

AA and FA are usually taken as being part of a package with KP (though, always the exception, Jon Cogburn seems to reject FA here, p238).

Now, suppose the following:

1. That which S* expresses is true.

Then by KP we have,

2. It is possible that somebody knows that which S* expresses.

By AA, we get,

3. Somebody possibly knows that which S* expresses.

But (3) is simply ~S*.

Now suppose,

4. That which S* expresses is false.

Then, by the contrapositive of FA, we arrive at,

5. It is not possible that somebody knows that which S* expresses.

Hence by AA,

6. Nobody possibly knows that which S* expresses.

But this is just S*! So, given KP, AA, and FA (the standard anti-realist package), we get ((S* -> ~S*) & (~S* -> S*)), which intuitionistically yields (~S* & ~~S*), and classically (~S* & S*). Hence the standard package entails a contradiction.

I think the main issue here is how to understand the constraint on KP that FA imposes. We don't want KP to be trivially true or false; surely every party in this debate wants to be affirming a doctrine with some substance. But there's some trickiness here: adopting FA is meant to block trivializing KP by allowing it to be satisfied in cases where P is false in @ and yet it is possible to know that P since there is a world w such that w is not @, and P expresses a truth and thus something knowable in w. The motivating idea behind FA is that KP is supposed to be constrained by the distribution of truth-values at the actual world, since otherwise even falsehoods are knowable - not the interpretation of knowability the anti-realist wants.

Unfortunately, adopting FA threatens to make KP come out trivially false. KP is supposed to be a necessary truth by the anti-realist's lights. So it's supposed to be impossible that P express a truth and yet it be impossible for some suitably pimped-out subject to come to know that P expresses a truth. But now suppose that it happens not to be known that Q expresses a truth, even though it in fact does. Given the requirement that we've attempted to capture in FA, that KP be sensitive to the actual distribution of truth-values, the question is whether it's possible that someone might come to know that Q expresses a truth, given the actual distribution of truth-values. But this obviously isn't possible, since the actual distribution of truth-values determines that nobody knows that Q expresses a truth. So, contrary to KP taken as a necessary truth, it is possible that Q express a truth and yet it be impossible that a suitably pimped-out subject to come to know that Q expresses a truth. That's a possibility just because of the simple fact that sometimes truths aren't in fact known by any subject in the actual world. (This line of thought is Sven Rosenkranz's, from p354 of his 2003).

So the anti-realist has a balancing act to do. (S)he wants to hold on to the thought that P expresses a truth if and only if it is possible that somebody knows that P expresses a truth. The right-to-left direction of this is AA, and it constrains the notion of knowability so that what is knowable is determined by the actual distribution of truth-values. But if we allow that what is knowable is determined by the actual distribution of truth-values over sentences expressing our current state of ignorance (or knowledge), we're in trouble, for then KP is clearly false. That was the upshot of Sven's argument. A principled way of avoiding this difficulty is needed, and Sven offers a proposal in his paper.

This has all been pretty torturous, but I've still to show how this all might bear on Kallestrup's proof. We need to look again at the move from line (4) to line (5):

4. That which S* expresses is false.

Then, by the contrapositive of FA, we arrive at,

5. It is not possible that somebody knows that which S* expresses.

But S* is clearly a sentence expressing something about our current state of ignorance, and we resolved, in the light of Sven's point, to understand KP and FA in such a way that the truth-values of such sentences did not go towards determining what is knowable (for example, Sven's suggestion may be put (I hope!) as follows: KP and FA only hold when the truth-value of P isn't alterable by any (perhaps idealized) subject). My suggestion, then, is that once we take Sven's point seriously, and change our understanding of KP and FA accordingly, the move from (4) to (5) will be invalidated. So the upshot, if I'm right, is that an already necessary restriction on the standard anti-realist package serves to invalid Kallestrup's proof of a contradiction.

Homework: What's the bearing of this, if any, on the knowability paradox?

References:

Kallestrup, J. 2007. If omniscient beings are dialetheists, then so are anti-realists. Analysis 67: 252-4.

Milne, P. 2007. Omniscient beings are dialetheists. Analysis 67: 250-1.

Rosenkranz, S. 2003. Realism and Understanding. Erkenntnis 58: 353-78.

S: No omniscient being knows that which the sentence S expresses.

S is clearly self-referential, but it doesn't trip itself up; it's simply true if there aren't any omniscient beings. Now, suppose that ~S, i.e.,

1. Some omniscient being knows that which S expresses.

Then, given what S expresses,

2. Some omniscient being knows that no omniscient being knows that which the sentence S expresses.

But then by the factivity of knowledge,

3. No omniscient being knows that which the sentence S expresses.

But that's just S. So S is true.

Now suppose that X is an omniscient being. Then X knows all, including what we've just shown, namely (3). So,

4. X knows that no omniscient being knows that which the sentence S expresses,

which, given what S expresses, yields,

5. X knows that which S expresses.

But X is omniscient. So,

6. Some omniscient being knows that which S expresses.

But (6) is just ~S. So, on the assumption that an omniscient being exists, S & ~S.

Milne leaves the reader to draw their own conclusions from his proof. Jesper Kallestrup has picked up the baton, and adapted Milne's proof to show that the knowability principle (KP) entails a contradiction. Again, rather than treating this as a reductio, we might accept that the contradiction is true. A final application of (KP) yields that this contradiction is knowable. Anti-realists are committed to dialetheism.

It can hardly have escaped the attention of readers of this blog that I'm more sympathetic than most to anti-realism, and (unlike Jon Cogburn), I'm not in the market for the conclusion that it entails committment to dialetheism. There's a lot that might be said in order to dodge this conclusion. Jesper notes that so-called restriction strategies that have been offered by Dummett and Tennant in response to the knowability paradox also work here, but he points out they are associated with a whole bunch of problems. Crispin Wright's endorsement of KP as a local thesis also blocks Jesper's reasoning, which requires an assumption that KP holds globally.

Here I just want to suggest that there's a further restriction to KP the anti-realist can make, one that's independently motivated and yet which blocks the proof. Let's begin by examining Jesper's argument that KP entails a contradiction.

Consider the following variant of S:

S*: Nobody possibly knows that which the sentence S expresses.

Again, this is self-referential but does not simply implode. But now suppose that we adopt the following:

KP: P expresses a truth only if it is possible that somebody knows that P expresses a truth.

Actuality Assumption (AA): It is possible that somebody knows that P expresses a truth only if there is some actual person who, perhaps which finite extensions to her capacities, possibly knows that P expresses a truth.

Factivity Assumption (FA): It is possible that somebody knows that P expresses a truth only if P expresses a truth.

AA and FA are usually taken as being part of a package with KP (though, always the exception, Jon Cogburn seems to reject FA here, p238).

Now, suppose the following:

1. That which S* expresses is true.

Then by KP we have,

2. It is possible that somebody knows that which S* expresses.

By AA, we get,

3. Somebody possibly knows that which S* expresses.

But (3) is simply ~S*.

Now suppose,

4. That which S* expresses is false.

Then, by the contrapositive of FA, we arrive at,

5. It is not possible that somebody knows that which S* expresses.

Hence by AA,

6. Nobody possibly knows that which S* expresses.

But this is just S*! So, given KP, AA, and FA (the standard anti-realist package), we get ((S* -> ~S*) & (~S* -> S*)), which intuitionistically yields (~S* & ~~S*), and classically (~S* & S*). Hence the standard package entails a contradiction.

I think the main issue here is how to understand the constraint on KP that FA imposes. We don't want KP to be trivially true or false; surely every party in this debate wants to be affirming a doctrine with some substance. But there's some trickiness here: adopting FA is meant to block trivializing KP by allowing it to be satisfied in cases where P is false in @ and yet it is possible to know that P since there is a world w such that w is not @, and P expresses a truth and thus something knowable in w. The motivating idea behind FA is that KP is supposed to be constrained by the distribution of truth-values at the actual world, since otherwise even falsehoods are knowable - not the interpretation of knowability the anti-realist wants.

Unfortunately, adopting FA threatens to make KP come out trivially false. KP is supposed to be a necessary truth by the anti-realist's lights. So it's supposed to be impossible that P express a truth and yet it be impossible for some suitably pimped-out subject to come to know that P expresses a truth. But now suppose that it happens not to be known that Q expresses a truth, even though it in fact does. Given the requirement that we've attempted to capture in FA, that KP be sensitive to the actual distribution of truth-values, the question is whether it's possible that someone might come to know that Q expresses a truth, given the actual distribution of truth-values. But this obviously isn't possible, since the actual distribution of truth-values determines that nobody knows that Q expresses a truth. So, contrary to KP taken as a necessary truth, it is possible that Q express a truth and yet it be impossible that a suitably pimped-out subject to come to know that Q expresses a truth. That's a possibility just because of the simple fact that sometimes truths aren't in fact known by any subject in the actual world. (This line of thought is Sven Rosenkranz's, from p354 of his 2003).

So the anti-realist has a balancing act to do. (S)he wants to hold on to the thought that P expresses a truth if and only if it is possible that somebody knows that P expresses a truth. The right-to-left direction of this is AA, and it constrains the notion of knowability so that what is knowable is determined by the actual distribution of truth-values. But if we allow that what is knowable is determined by the actual distribution of truth-values over sentences expressing our current state of ignorance (or knowledge), we're in trouble, for then KP is clearly false. That was the upshot of Sven's argument. A principled way of avoiding this difficulty is needed, and Sven offers a proposal in his paper.

This has all been pretty torturous, but I've still to show how this all might bear on Kallestrup's proof. We need to look again at the move from line (4) to line (5):

4. That which S* expresses is false.

Then, by the contrapositive of FA, we arrive at,

5. It is not possible that somebody knows that which S* expresses.

But S* is clearly a sentence expressing something about our current state of ignorance, and we resolved, in the light of Sven's point, to understand KP and FA in such a way that the truth-values of such sentences did not go towards determining what is knowable (for example, Sven's suggestion may be put (I hope!) as follows: KP and FA only hold when the truth-value of P isn't alterable by any (perhaps idealized) subject). My suggestion, then, is that once we take Sven's point seriously, and change our understanding of KP and FA accordingly, the move from (4) to (5) will be invalidated. So the upshot, if I'm right, is that an already necessary restriction on the standard anti-realist package serves to invalid Kallestrup's proof of a contradiction.

Homework: What's the bearing of this, if any, on the knowability paradox?

References:

Kallestrup, J. 2007. If omniscient beings are dialetheists, then so are anti-realists. Analysis 67: 252-4.

Milne, P. 2007. Omniscient beings are dialetheists. Analysis 67: 250-1.

Rosenkranz, S. 2003. Realism and Understanding. Erkenntnis 58: 353-78.

Labels: Anti-Realism, Dialetheism, Dummett, Knowability, Philosophy of Logic, Religion