Monday, August 27, 2007
Knowability and Actuality
Here's a point which dropped out of my last post. Anti-realists adopt some local or global version of the knowability principle:
KP: P only if it is possible that P is known.
They've understood this principle such that it's constrained by the actual distribution of truth-values; the actual distribution of truth-values over statements determines what is knowable. Why?
In my post, I took it to be because otherwise knowability isn't nearly demanding enough. Suppose that P is false. Still, P might be knowable in the sense that there is some world in which P is knowable - it might only have to be a world in which P is true. But the anti-realist was looking for a more substantial sense of knowability, one according to which falsehoods are not knowable. So anti-realists understand knowability in such a manner so as P is knowable if and only if P.
In the Analysis note I discussed in the previous post, Jesper Kallestrup clearly sees things quite differently. He writes:
'It would be too easy to refute anti-realism if [KP] failed in cases where P is actually true, but not known in worlds in which P is false.'
The suggestion is that having what's knowable determined by the actual distribution of truth-values blocks putative counterexamples to KP. But I'm not seeing how the point is supposed to go. Suppose that P is actually true, but not known in worlds in which P is false. I'm not seeing any danger from such cases, though they refute the following thesis:
KP!!: P only if it is necessary that P is known.
Since KP doesn't require that P is known in all possible worlds, it's not clear to me how the observation that P is not knowable in worlds in which P is false is meant to provide the material for an even prima facie counterexample to KP. So it's quite unclear to me how consideration of such cases could motivate adopting the thesis that what is knowable is fixed by the actual distribution of truth-values. Unless I'm missing something, I think the motivation has to stem from the worry about diluting the notion of knowability too much.
(Andreas blogged about related issues a while back).
KP: P only if it is possible that P is known.
They've understood this principle such that it's constrained by the actual distribution of truth-values; the actual distribution of truth-values over statements determines what is knowable. Why?
In my post, I took it to be because otherwise knowability isn't nearly demanding enough. Suppose that P is false. Still, P might be knowable in the sense that there is some world in which P is knowable - it might only have to be a world in which P is true. But the anti-realist was looking for a more substantial sense of knowability, one according to which falsehoods are not knowable. So anti-realists understand knowability in such a manner so as P is knowable if and only if P.
In the Analysis note I discussed in the previous post, Jesper Kallestrup clearly sees things quite differently. He writes:
'It would be too easy to refute anti-realism if [KP] failed in cases where P is actually true, but not known in worlds in which P is false.'
The suggestion is that having what's knowable determined by the actual distribution of truth-values blocks putative counterexamples to KP. But I'm not seeing how the point is supposed to go. Suppose that P is actually true, but not known in worlds in which P is false. I'm not seeing any danger from such cases, though they refute the following thesis:
KP!!: P only if it is necessary that P is known.
Since KP doesn't require that P is known in all possible worlds, it's not clear to me how the observation that P is not knowable in worlds in which P is false is meant to provide the material for an even prima facie counterexample to KP. So it's quite unclear to me how consideration of such cases could motivate adopting the thesis that what is knowable is fixed by the actual distribution of truth-values. Unless I'm missing something, I think the motivation has to stem from the worry about diluting the notion of knowability too much.
(Andreas blogged about related issues a while back).
Labels: Anti-Realism, Knowability
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
Monday, June 04, 2007
Oh Larsen B, you can fall on me..........
In my earlier post, I argued that commitments one takes on in accepting that superassertibility is a genuine truth-predicate for some domain of discourse (at least if one holds, as Crispin does, that any genuine truth-predicate sustains a full disquotational schema) rules out the possibility of what I'll call "iceberg ahead!" states of information; states of information such that they warrant the assertion of some sentence P, but also warrant the assertion that one of the other necessary conditions for superassertibility isn't met (so either the grounds that confer warranted assertibility upon P would not withstand arbitrarily exacting scrutiny, or there is some extension to one's state of information which no longer warrants an assertion of P).
I was reading Wright's 'Intuitionism, Realism, Relativism and Rhubarb' after I wrote the original post, and it suggested to me an example of an "iceberg ahead!" state of information. Focus on 'disputes of inclination', so, for example, a dispute over whether rhubarb is delicious. Suppose that one holds firstly that 'Rhubarb is delicious' is warrantedly assertible by subject S just in case S likes the taste of rhubard; likewise, 'Rhubarb is not delicious' is warrantedly assertible by S just in S does not like the taste of rhubarb (Wright 2006: 57). Suppose also that one holds that truth regarding expressions of inclination is constituted by superassertibility (as Wright proposes in his paper, though he does not wholeheartedly endorse the proposal).
Finally, suppose that I do not at present like the taste of rhubarb, but I have resolved to eat it every day for at least the next forty days. I also have very good (perhaps inductive or testimonial) grounds for holding that if one tastes a food-stuff forty times one will come to like it. Given the above commitments, isn't this a case in which 'Rhubarb is not delicious' is warrantedly assertible by me, but assuming the credentials of my belief that my dislike of rhubarb will not survive the next 40 days to be sufficiently good, the claim that 'Rhubarb is not delicious' is not superassertible is also warrantly assertible (since my current state of information is such that I know that there is an extension of that state of information which will not warrant my asserting 'Rhubarb is not delicious')?
It might be a little unlikely that I'd ever find myself in quite this epistemic condition. But I don't find it a really bizarre case, and all I really need is that it strikes us as coherent.
I was reading Wright's 'Intuitionism, Realism, Relativism and Rhubarb' after I wrote the original post, and it suggested to me an example of an "iceberg ahead!" state of information. Focus on 'disputes of inclination', so, for example, a dispute over whether rhubarb is delicious. Suppose that one holds firstly that 'Rhubarb is delicious' is warrantedly assertible by subject S just in case S likes the taste of rhubard; likewise, 'Rhubarb is not delicious' is warrantedly assertible by S just in S does not like the taste of rhubarb (Wright 2006: 57). Suppose also that one holds that truth regarding expressions of inclination is constituted by superassertibility (as Wright proposes in his paper, though he does not wholeheartedly endorse the proposal).
Finally, suppose that I do not at present like the taste of rhubarb, but I have resolved to eat it every day for at least the next forty days. I also have very good (perhaps inductive or testimonial) grounds for holding that if one tastes a food-stuff forty times one will come to like it. Given the above commitments, isn't this a case in which 'Rhubarb is not delicious' is warrantedly assertible by me, but assuming the credentials of my belief that my dislike of rhubarb will not survive the next 40 days to be sufficiently good, the claim that 'Rhubarb is not delicious' is not superassertible is also warrantly assertible (since my current state of information is such that I know that there is an extension of that state of information which will not warrant my asserting 'Rhubarb is not delicious')?
It might be a little unlikely that I'd ever find myself in quite this epistemic condition. But I don't find it a really bizarre case, and all I really need is that it strikes us as coherent.
Labels: Anti-Realism, Crispin Wright, Rhubarb
Saturday, June 02, 2007
How to Fitch-Church Superassertibility
I haven't spent as long as I'd like on the details here, but I don't really have the time just now, so I thought I'd post and let the more logically competent point out in what way I'm misguided (on this, or some other issue of their choosing).
A number of philosophers (Brueckner, Kvanvig, etc.) have argued against Crispin Wright's suggestion that superassertibility is a truth predicate. I'm wondering if Church-Fitch reasoning doesn't give us a very quick route to the conclusion that they're right to challenge Crispin on this point.
A sentence 'P' is superassertible (SA 'P') iff it is warrantly assertible (WA 'P') and that warrant i. will 'survive arbitrarily close and extensive investigation' and ii. is not defeasible (that is, there is no extension, i', to our current information state i such that i' fails to warrant the assertion of 'P').
To set up the problem, we need the following three principles:
1. SA 'P' iff P
2. If SA 'P&Q', then SA 'P' and SA 'Q'
3. If WA 'P' and WA 'Q', then WA 'P&Q'
These are all pretty straightforward. (1) is just the disquotation principle for SA, which Crispin thinks has to hold if SA is genuinely to count as a truth-predicate. (2) says that SA is closed under &-elimination, which follows from the closure of SA under logical implication (which again seems pretty compelling if SA is a genuine truth-predicate). Lastly, (3) says that WA is closed under &-introduction, which again seems compelling (at least when we're restricting attention to sentences with only two conjuncts; perhaps the preface and lottery paradoxes should make us doubt this principle in its full generality).
Now, the objection I have in mind starts from the observation that SA is strictly logically stronger than WA; that is, SA 'P' -> WA 'P' but not vice versa. This doesn't imply, but seems to leave open the possibility of a sentence 'X' and state of information such that:
i. WA 'X' & WA '~SA 'X''
By (3), it immediately follows that,
ii. WA 'X & ~SA 'X''
But we have,
iii. ~WA 'X & ~SA 'X''
Contradiction.
The sub-argument for iii. is just standard Church-Fitch reasoning, with (1) in place of factivity and the knowability principle. Suppose,
iv. X & ~SA 'X'
Then by (1),
v. SA 'X & ~SA 'X''
Whence by (2),
vi. SA 'X' and SA '~SA 'X''
But then, by (1) again,
vii. SA 'X' and ~SA 'X'
So we can conclude,
viii. ~(X & ~SA 'X'),
which presumably suffices for iii.
I'm not spotting any gapping holes in this, though I'm sure others will be quick to point them out. (1) isn't a legitimate supposition if we endorse the following variant of the reflection principle for WA:
4. If WA '~SA 'P'', then ~WA 'P'
But I'm not seeing any good reason to adopt (4). My supposition in i. does seem to be a legitimate possibility; it might be unusual to be in a state of information which both warrants the assertion of a particular sentence and warrants the assertion that some expansion of that information state fails to warrant that sentence, but (in empirical discourse) the idea does not seem incoherent. Yet if I haven't messed up, it is incoherent if superassertibility is a truth-predicate (given the closure of WA under &-introduction).
A number of philosophers (Brueckner, Kvanvig, etc.) have argued against Crispin Wright's suggestion that superassertibility is a truth predicate. I'm wondering if Church-Fitch reasoning doesn't give us a very quick route to the conclusion that they're right to challenge Crispin on this point.
A sentence 'P' is superassertible (SA 'P') iff it is warrantly assertible (WA 'P') and that warrant i. will 'survive arbitrarily close and extensive investigation' and ii. is not defeasible (that is, there is no extension, i', to our current information state i such that i' fails to warrant the assertion of 'P').
To set up the problem, we need the following three principles:
1. SA 'P' iff P
2. If SA 'P&Q', then SA 'P' and SA 'Q'
3. If WA 'P' and WA 'Q', then WA 'P&Q'
These are all pretty straightforward. (1) is just the disquotation principle for SA, which Crispin thinks has to hold if SA is genuinely to count as a truth-predicate. (2) says that SA is closed under &-elimination, which follows from the closure of SA under logical implication (which again seems pretty compelling if SA is a genuine truth-predicate). Lastly, (3) says that WA is closed under &-introduction, which again seems compelling (at least when we're restricting attention to sentences with only two conjuncts; perhaps the preface and lottery paradoxes should make us doubt this principle in its full generality).
Now, the objection I have in mind starts from the observation that SA is strictly logically stronger than WA; that is, SA 'P' -> WA 'P' but not vice versa. This doesn't imply, but seems to leave open the possibility of a sentence 'X' and state of information such that:
i. WA 'X' & WA '~SA 'X''
By (3), it immediately follows that,
ii. WA 'X & ~SA 'X''
But we have,
iii. ~WA 'X & ~SA 'X''
Contradiction.
The sub-argument for iii. is just standard Church-Fitch reasoning, with (1) in place of factivity and the knowability principle. Suppose,
iv. X & ~SA 'X'
Then by (1),
v. SA 'X & ~SA 'X''
Whence by (2),
vi. SA 'X' and SA '~SA 'X''
But then, by (1) again,
vii. SA 'X' and ~SA 'X'
So we can conclude,
viii. ~(X & ~SA 'X'),
which presumably suffices for iii.
I'm not spotting any gapping holes in this, though I'm sure others will be quick to point them out. (1) isn't a legitimate supposition if we endorse the following variant of the reflection principle for WA:
4. If WA '~SA 'P'', then ~WA 'P'
But I'm not seeing any good reason to adopt (4). My supposition in i. does seem to be a legitimate possibility; it might be unusual to be in a state of information which both warrants the assertion of a particular sentence and warrants the assertion that some expansion of that information state fails to warrant that sentence, but (in empirical discourse) the idea does not seem incoherent. Yet if I haven't messed up, it is incoherent if superassertibility is a truth-predicate (given the closure of WA under &-introduction).
Labels: Anti-Realism, Crispin Wright, Philosophy of Logic
Sunday, May 06, 2007
Anti-Realism: Decidability and Detectability
Back in 1983, Crispin Wright wrote:
'It is a common misunderstanding of the thrust of the anti-realist's criticisms of the role assigned to truth in classical semantics that he believes that the central notion in the theory of meaning should be an effectively decidable one.'
Amen. My entire life-span later, and this is still a point which simply has not been absorbed by opponents of anti-realism (a rare exception is Darragh Byrne, but it's hard to think of any others). I'm writing a paper at the moment, tentatively titled 'Anti-Realist DOs and DON'Ts', in which I'll substantiate Crispin's point, and show how doing so enables the anti-realist to avoid most of the objections recently leveled at him. More specifically, it'll be a defense of compatiblist anti-realism; that is, anti-realism which wants to hold on to truth-conditional semantics (the term is Byrne's). It's been claimed that such a view either inflates into realism (Alex Miller) or must deny that truth is undecidable (Williamson), but these criticisms simply make the assumption Wright warned against all those years ago. More on this in the finished paper.
Here, I want to offer a taster. I'll present two very recent arguments against anti-realism which are not major targets of my paper, but which casually overlook the possibility of a compatiblist anti-realism which holds that truth is necessarily detectable, but is not decidable. First up is Michael Loux's piece on the debate in The Oxford Handbook of Metaphysics. On page 656 we get the following passage:
'...Dummett's more particular claim that the assertability theorist should trade in verification conditions (that is, conditions that conclusively justify assertion) must face the objection that the resulting theory of meaning has precisely the difficulties Dummett takes to be the undoing of the truth-conditional theory. The claim is that to know the meaning of a statement is to know what would conclusively warrant its assertion; but statements that are in principle undecidable are such that there neither is nor can be anything that would conclusively justify their assertion or their denial. [My emphasis]. But, then, where is the strategic advantage of the assertibility-conditional theory? As we have seen, Dummett tries to forestall this objection by identifying the apparently epistemic state with a practical, discriminatory ability - the ability to recognize, if presented with it, a condition which would justify assertion. But, of course [sic], that is an ability that is in principle incapable of being exercised. [My emphasis]. And how is the attribution of that sort of ability any improvement on the truth-conditional theorist's attribution of epistemic states that can never can manifested?'
Look carefully at the crucial lines I put in italics, and ask yourself: how does it follow from a statement's undecidability - that is, the lack of a procedure which could be followed in a finite amount of time, and which if correctly implemented would be bound to conclusively verify or falsify that statement - that there could not be anything that would conclusively justify its assertion? Only, I submit, if we run together undecidability and undetectability; we assume that just because there is no such procedure, there can be no evidence which would decide the matter one way or the other (and hence our discriminatory capacities for such statements must be 'in principle incapable of being exercised') . Dummett himself is perfectly explicit that this is a mistake in his most recent book:
'Although we may have no means, even in principle, of putting ourselves into a position in which we can effectively decide whether the proposition expressed by the utterance of a given sentence is or is not true, it does not follow that we may not come to recognize that proposition as true or false; we may sometimes, and indeed often do, decide the truth or falsity of utterances of undecidable sentences, in the sense I gave to this expression.' (58)
So I submit, Loux's objection simply rests on the conflation of decidability and detectability. Now, the problem may be that Loux appears to interpret undecidable statements to be those which are 'verification- or falsification-transcendent' (635). But there's no real justification for such an interpretation, even if a few passages in Dummett may unfortunately encourage it.
A much more striking example of a failure to appreciate this point, and to miss the possibility of a compatiblist anti-realism, comes in Charles McCarthy's 'The Coherence of Anti-Realism'. In the opening line of the abstract, we are told:
'The project of anti-realism is to construct an assertibility semantics on which (1) the truth of statements obeys a recognition condition so that (2) counterexamples are forthcoming to the law of excluded middle and (3) intuitionistic formal predicate logic is provably sound and complete with respect to the associated notion of validity.'
On page 948, we are further told that in order to 'ensure that the recognition condition obtains', the anti-realist assumes that the central notion in the semantics is decidable in the sense that a decision procedure 'is available in principle'. It turns out, McCarthy argues, that this package is incoherent. All the worse for anti-realism so conceived, I say. But we're given no clues that this conception isn't mandated.
This gives you a taste of just how casually critics of anti-realism have been willing to run together the crucial notions of decidability and detectability, despite (and without so much as acknowledgment of) Crispin's explicit disavowal in print over twenty years ago. My paper, when it's done, will leave many important aspects of the debate completely untouched, but I think I'll have achieved something if I manage to start to change people's attitudes on this central issue. Seems unlikely on the face of it, but we'll see.......
'It is a common misunderstanding of the thrust of the anti-realist's criticisms of the role assigned to truth in classical semantics that he believes that the central notion in the theory of meaning should be an effectively decidable one.'
Amen. My entire life-span later, and this is still a point which simply has not been absorbed by opponents of anti-realism (a rare exception is Darragh Byrne, but it's hard to think of any others). I'm writing a paper at the moment, tentatively titled 'Anti-Realist DOs and DON'Ts', in which I'll substantiate Crispin's point, and show how doing so enables the anti-realist to avoid most of the objections recently leveled at him. More specifically, it'll be a defense of compatiblist anti-realism; that is, anti-realism which wants to hold on to truth-conditional semantics (the term is Byrne's). It's been claimed that such a view either inflates into realism (Alex Miller) or must deny that truth is undecidable (Williamson), but these criticisms simply make the assumption Wright warned against all those years ago. More on this in the finished paper.
Here, I want to offer a taster. I'll present two very recent arguments against anti-realism which are not major targets of my paper, but which casually overlook the possibility of a compatiblist anti-realism which holds that truth is necessarily detectable, but is not decidable. First up is Michael Loux's piece on the debate in The Oxford Handbook of Metaphysics. On page 656 we get the following passage:
'...Dummett's more particular claim that the assertability theorist should trade in verification conditions (that is, conditions that conclusively justify assertion) must face the objection that the resulting theory of meaning has precisely the difficulties Dummett takes to be the undoing of the truth-conditional theory. The claim is that to know the meaning of a statement is to know what would conclusively warrant its assertion; but statements that are in principle undecidable are such that there neither is nor can be anything that would conclusively justify their assertion or their denial. [My emphasis]. But, then, where is the strategic advantage of the assertibility-conditional theory? As we have seen, Dummett tries to forestall this objection by identifying the apparently epistemic state with a practical, discriminatory ability - the ability to recognize, if presented with it, a condition which would justify assertion. But, of course [sic], that is an ability that is in principle incapable of being exercised. [My emphasis]. And how is the attribution of that sort of ability any improvement on the truth-conditional theorist's attribution of epistemic states that can never can manifested?'
Look carefully at the crucial lines I put in italics, and ask yourself: how does it follow from a statement's undecidability - that is, the lack of a procedure which could be followed in a finite amount of time, and which if correctly implemented would be bound to conclusively verify or falsify that statement - that there could not be anything that would conclusively justify its assertion? Only, I submit, if we run together undecidability and undetectability; we assume that just because there is no such procedure, there can be no evidence which would decide the matter one way or the other (and hence our discriminatory capacities for such statements must be 'in principle incapable of being exercised') . Dummett himself is perfectly explicit that this is a mistake in his most recent book:
'Although we may have no means, even in principle, of putting ourselves into a position in which we can effectively decide whether the proposition expressed by the utterance of a given sentence is or is not true, it does not follow that we may not come to recognize that proposition as true or false; we may sometimes, and indeed often do, decide the truth or falsity of utterances of undecidable sentences, in the sense I gave to this expression.' (58)
So I submit, Loux's objection simply rests on the conflation of decidability and detectability. Now, the problem may be that Loux appears to interpret undecidable statements to be those which are 'verification- or falsification-transcendent' (635). But there's no real justification for such an interpretation, even if a few passages in Dummett may unfortunately encourage it.
A much more striking example of a failure to appreciate this point, and to miss the possibility of a compatiblist anti-realism, comes in Charles McCarthy's 'The Coherence of Anti-Realism'. In the opening line of the abstract, we are told:
'The project of anti-realism is to construct an assertibility semantics on which (1) the truth of statements obeys a recognition condition so that (2) counterexamples are forthcoming to the law of excluded middle and (3) intuitionistic formal predicate logic is provably sound and complete with respect to the associated notion of validity.'
On page 948, we are further told that in order to 'ensure that the recognition condition obtains', the anti-realist assumes that the central notion in the semantics is decidable in the sense that a decision procedure 'is available in principle'. It turns out, McCarthy argues, that this package is incoherent. All the worse for anti-realism so conceived, I say. But we're given no clues that this conception isn't mandated.
This gives you a taste of just how casually critics of anti-realism have been willing to run together the crucial notions of decidability and detectability, despite (and without so much as acknowledgment of) Crispin's explicit disavowal in print over twenty years ago. My paper, when it's done, will leave many important aspects of the debate completely untouched, but I think I'll have achieved something if I manage to start to change people's attitudes on this central issue. Seems unlikely on the face of it, but we'll see.......
Labels: Anti-Realism, Crispin Wright, Dummett, Tim Williamson
Tuesday, March 06, 2007
Anti-realism and Prichard's Dilemma
I'd really like to write a paper on the matters I'm going to discuss in this post. Since I have no idea where to begin writing said (non-existent) paper, or whether the ideas it would contain are even worth pursuing, I thought I'd air them here. To set the scene, I'll post here something I wrote as a comment over at Duncan's Epistemic Value:
A question I got interested in a while ago is whether there is worrying analogue of Prichard's dilemma in epistemology. Scanlon presents the original version as a dilemma facing any account of what makes an action wrong. We want to know why the fact (assuming for now that it is one) that an act is wrong gives us reason not to do it. The worry is that we'll either end up giving an uninformative answer like 'Because it's wrong!', or we'll end up citing some feature of the act which supposedly substantively explains why we have reason not to perform it which intuitively has no connection to its wrongness (that performing the act would lead to social ostracism, for example); as Scanlon notes, it is not such features of the act which we'd expect 'a moral person first and foremost to be moved by'.
In his critical study of Scanlon, David Sosa suggests that there's no worrying theoretical analogue of Prichard's dilemma - that when we ask 'why does the fact that p is false give us reason not to believe that p?', we can be satisfied with the answer 'Because p is false!' - and that this should make us suspect that trivial answers should be satisfactory in the ethical case too. I'm inclined to think Prichard's dilemma bites in both cases.
David writes of the 'Because p is false!' response:
'Is this not the kind of answer that is wanted? Ultimately (if not sooner), it is truth in which at least one sort of theoretical rationality is grounded. Any intermediate theoretical purpose that might be served by not believing the proposition could function as a reason only derivatively through its own relation to the truth. And to whatever extent that relation is not necessary, then we will feel that the alternative lacks the right sort of modal force.' ('A Big Good Thing': 374)
I want to suggest that in certain cases, we can do quite a bit better here that David is suggesting we can. Take mathematical statements for example. If one asks why some set-theoretical statement s being false gives one reason not to believe that s, can we do better than to simply point out that s is false? Surely we can; usually we will be able to prove that ~s follows from some mutually shared set of axioms. This may not always be the case, given that these axioms will fail to settle some statements about the set-theoretic universe which we may nonetheless take ourselves to have good reason not to believe (CH and its relatives, for example). In general though, we take provability to be precisely the kind of thing counting in favor of a particular mathematical statement that a rational agent considering which mathematical statements to believe would be 'first and foremost' moved by. And it's more informative than simply saying that the statement is true or false, since it appeals to the implications of some mutually accepted set of statements, often thought to have some privileged epistemic status.
So it seems clear that proof has the right kind of close connection to truth. Is it the case, as David suggests, that that fact that a statement's negation is provable is only a reason to believe it derivatively because of that close connection to truth? That's an enormous question, but I don't see any immediate grounds for a positive answer. In fact, I see lots of room to explore in defense of a negative answer.
Let's imagine there's something to all this, i.e. that we can't be satisfied with trivial answers to the theoretical analogue of Prichard's dilemma, and that a satisfactory epistemology of mathematics vindicates my suggestion that provability is just the right kind of notion to invoke in a satisfactory non-trivial response to Prichard's dilemma targeted at mathematical statements. What follows?
Well, nothing probably. But the seductive thought is the following. Firstly, that realist (bivalent, potentially verification-transcendent) truth doesn't connect up in the right way to provability to allow the above response to Prichard's dilemma to be given; the relationship of truth to proof on that picture 'lacks the right sort of modal force' to borrow David's expression from the quote above. In contrast, the anti-realist's conceptions of truth and proof gel in just way we want (of course, some anti-realists identify truth with proof. I'll bracket such potential complications here, since I'm already quite muddled enough). More generally, that an anti-realist will have some notion that connects up in the right way to truth to allow him to offer a satisfying substantive general response to the theoretical version of Prichard's dilemma; I'm imagining this to be some notion which, like Crispin Wright's notion of superassertability, is built out of the notion of warranted assertability but which mimics the notion of proof in crucial respects (see 'Can a Davidsonian Theory of Meaning.....' in Crispin's Realism, Meaning, & Truth). The contention, of course, would be that the realist has put any such notion well beyond her reach.
This is obviously ludicrously sketchy and overambitious, holding out hostages in the epistemology of mathematics, both the theoretical and the practical domains of normative theory, plus it inherits general worries about the possibility of the formulation of an anti-realistic notion that applies to non-mathematical statements but which possesses the requisite proof-like qualities, plus an argument is wanting that realist truth creates special difficulties here. But anyway, you'll all see why I don't know how to write this paper. Maybe that's for the best - I'll remain agnostic for now.
A question I got interested in a while ago is whether there is worrying analogue of Prichard's dilemma in epistemology. Scanlon presents the original version as a dilemma facing any account of what makes an action wrong. We want to know why the fact (assuming for now that it is one) that an act is wrong gives us reason not to do it. The worry is that we'll either end up giving an uninformative answer like 'Because it's wrong!', or we'll end up citing some feature of the act which supposedly substantively explains why we have reason not to perform it which intuitively has no connection to its wrongness (that performing the act would lead to social ostracism, for example); as Scanlon notes, it is not such features of the act which we'd expect 'a moral person first and foremost to be moved by'.
In his critical study of Scanlon, David Sosa suggests that there's no worrying theoretical analogue of Prichard's dilemma - that when we ask 'why does the fact that p is false give us reason not to believe that p?', we can be satisfied with the answer 'Because p is false!' - and that this should make us suspect that trivial answers should be satisfactory in the ethical case too. I'm inclined to think Prichard's dilemma bites in both cases.
David writes of the 'Because p is false!' response:
'Is this not the kind of answer that is wanted? Ultimately (if not sooner), it is truth in which at least one sort of theoretical rationality is grounded. Any intermediate theoretical purpose that might be served by not believing the proposition could function as a reason only derivatively through its own relation to the truth. And to whatever extent that relation is not necessary, then we will feel that the alternative lacks the right sort of modal force.' ('A Big Good Thing': 374)
I want to suggest that in certain cases, we can do quite a bit better here that David is suggesting we can. Take mathematical statements for example. If one asks why some set-theoretical statement s being false gives one reason not to believe that s, can we do better than to simply point out that s is false? Surely we can; usually we will be able to prove that ~s follows from some mutually shared set of axioms. This may not always be the case, given that these axioms will fail to settle some statements about the set-theoretic universe which we may nonetheless take ourselves to have good reason not to believe (CH and its relatives, for example). In general though, we take provability to be precisely the kind of thing counting in favor of a particular mathematical statement that a rational agent considering which mathematical statements to believe would be 'first and foremost' moved by. And it's more informative than simply saying that the statement is true or false, since it appeals to the implications of some mutually accepted set of statements, often thought to have some privileged epistemic status.
So it seems clear that proof has the right kind of close connection to truth. Is it the case, as David suggests, that that fact that a statement's negation is provable is only a reason to believe it derivatively because of that close connection to truth? That's an enormous question, but I don't see any immediate grounds for a positive answer. In fact, I see lots of room to explore in defense of a negative answer.
Let's imagine there's something to all this, i.e. that we can't be satisfied with trivial answers to the theoretical analogue of Prichard's dilemma, and that a satisfactory epistemology of mathematics vindicates my suggestion that provability is just the right kind of notion to invoke in a satisfactory non-trivial response to Prichard's dilemma targeted at mathematical statements. What follows?
Well, nothing probably. But the seductive thought is the following. Firstly, that realist (bivalent, potentially verification-transcendent) truth doesn't connect up in the right way to provability to allow the above response to Prichard's dilemma to be given; the relationship of truth to proof on that picture 'lacks the right sort of modal force' to borrow David's expression from the quote above. In contrast, the anti-realist's conceptions of truth and proof gel in just way we want (of course, some anti-realists identify truth with proof. I'll bracket such potential complications here, since I'm already quite muddled enough). More generally, that an anti-realist will have some notion that connects up in the right way to truth to allow him to offer a satisfying substantive general response to the theoretical version of Prichard's dilemma; I'm imagining this to be some notion which, like Crispin Wright's notion of superassertability, is built out of the notion of warranted assertability but which mimics the notion of proof in crucial respects (see 'Can a Davidsonian Theory of Meaning.....' in Crispin's Realism, Meaning, & Truth). The contention, of course, would be that the realist has put any such notion well beyond her reach.
This is obviously ludicrously sketchy and overambitious, holding out hostages in the epistemology of mathematics, both the theoretical and the practical domains of normative theory, plus it inherits general worries about the possibility of the formulation of an anti-realistic notion that applies to non-mathematical statements but which possesses the requisite proof-like qualities, plus an argument is wanting that realist truth creates special difficulties here. But anyway, you'll all see why I don't know how to write this paper. Maybe that's for the best - I'll remain agnostic for now.
Labels: Anti-Realism, Epistemology
Saturday, November 18, 2006
Acquisition Again
Exactly a month ago I posted a long post on Dummett's acquisition challenge, and I've been promising since then to have a go to get clearer on the issues I raised there. John Bengson convinced me in discussion that I'd come across in my original post too much as playing defence on the behalf of the acquisition challenger. In actual fact, my intention was just to register my confusion and to get other people's reactions to the point that was confusing me.
In what follows I'm going to take some familiarity with what I wrote in the previous post on this topic for granted.
I take it that Dummett's challenge is to explain how we could have acquired a particular concept, one which the realist credits us with possession of, but which the anti-realist finds problematic. Here's Dummett (Alex Miller calls this Dummett's 'canonical formulation' of the argument):
'[The anti-realist] maintains that the process by which we came to grasp the sense of statements of the disputed class, and the use which is subsequently made of these statements, are such that we could not derive from it [i.e. the process] any notion of what it would be for such a statement to be true independently of the sort of thing we have learned to recognise as establishing the truth of such statements.'
('The Reality of the Past', T&OE: 362. My italics)
It's important to note that Dummett is explicit here that the challenge is not to show that we understand statements in the disputed class; that's conceded by all sides. Nor is the challenge for the theorist (realist or otherwise) who holds that understanding one of the disputed statements consists in grasp of its truth-conditions to show how we could have come to have grasped those truth conditions given that our linguistic training is a training in use. The challenge as I see it is to explain in a manner consistent with the nature of our linguistic training how we could come to form a conception of what it would be like for such a sentence to be associated with realist truth-conditions, i.e. conditions that can obtain or fail to obtain undetectably. This is hardly a sharp, adequately spelled-out statement of the nature of the challenge that's being presented to the realist, but it hope it's clear enough for my present purposes.
What I found so surprising in the Wright quote from last time was that it basically concedes to the realist that it suffices to meet the challenge as I've put it above that the realist can show the following: she has some story about how we could have come by grasp of the truth-conditions of one of the disputed statements in a manner consistent with the nature of our linguistic training:
'Given that the understanding of statements in general is to be viewed as consisting in possession of a concept of their truth-conditions, acquiring a concept of an evidence-transcendent state of affairs is simply a matter of acquiring an understanding of a statement for which that state of affairs would constitute the truth-condition.' (Realism, Meaning & Truth: 16)
Once we've made this concession, it's clear how compositionality helps the realist meet the challenge.
Now, part of my confusion last time stemmed from the following. Wright thinks there are insuperable difficulties standing in the way of a semantic theory which replaces the notion of truth with that of warranted-assertibility (or something similar). So we should adopt truth-conditional semantics. Understanding a sentence consists in grasping its truth-conditions; understanding one of the disputed sentences consists in grasping its truth-conditions. The realist is to be accused of 'overdescribing' what understanding a sentence consists in; she's right to hold that it consists in grasp of truth-conditions, but unwarranted in supposing it ever consists in grasp of realist truth-conditions (since such grasp could never be manifested in linguistic behaviour). As Wright is prone to putting it (regarding epistemicism, but clearly the point is more general than that), the realist is to be accused of superstition rather than error.
My puzzlement was over this; why can't we offer the same response to the compositionality response to the acquisition challenge? We'd then concede that we understand the disputed statements, that understanding those statements consists in grasping their truth-conditions, and that one could come (via compositionality) to grasp their truth-conditions in a way consistent with recognising that our linguistic training is a training in use. But what forces us here to say that the truth-conditions we've arrived at are realist truth-conditions? (I've discovered since writing my first post on this that Miller explicitly considers such a response to the compositionality response in his 'What is the Acquisition Argument?'. See pp. 492-4)
Actually, there seems to be a very simple reason to think we are forced to regard the truth-conditions of the disputed statements as realist truth-conditions, namely that (as is conceded on all sides) these statements are undecidable (See Miller p491 and p7 of Williamson's 'Must Do Better'.) I think that there's a serious challenge there, but setting that point aside for now, we still have something of a mystery as far as I can see. Why did Wright think that it sufficed to meet the acquisition challenge - formulated as a challenge to explain how we could have acquired a conception of what it would be for a statement to be associated with realist truth-conditions - to explain how we could come to grasp (in the right sort of way) the truth-conditions of one of the disputed statements, given that he holds that nothing warrants us in regarding those truth-conditions as realist truth-conditions?
In what follows I'm going to take some familiarity with what I wrote in the previous post on this topic for granted.
I take it that Dummett's challenge is to explain how we could have acquired a particular concept, one which the realist credits us with possession of, but which the anti-realist finds problematic. Here's Dummett (Alex Miller calls this Dummett's 'canonical formulation' of the argument):
'[The anti-realist] maintains that the process by which we came to grasp the sense of statements of the disputed class, and the use which is subsequently made of these statements, are such that we could not derive from it [i.e. the process] any notion of what it would be for such a statement to be true independently of the sort of thing we have learned to recognise as establishing the truth of such statements.'
('The Reality of the Past', T&OE: 362. My italics)
It's important to note that Dummett is explicit here that the challenge is not to show that we understand statements in the disputed class; that's conceded by all sides. Nor is the challenge for the theorist (realist or otherwise) who holds that understanding one of the disputed statements consists in grasp of its truth-conditions to show how we could have come to have grasped those truth conditions given that our linguistic training is a training in use. The challenge as I see it is to explain in a manner consistent with the nature of our linguistic training how we could come to form a conception of what it would be like for such a sentence to be associated with realist truth-conditions, i.e. conditions that can obtain or fail to obtain undetectably. This is hardly a sharp, adequately spelled-out statement of the nature of the challenge that's being presented to the realist, but it hope it's clear enough for my present purposes.
What I found so surprising in the Wright quote from last time was that it basically concedes to the realist that it suffices to meet the challenge as I've put it above that the realist can show the following: she has some story about how we could have come by grasp of the truth-conditions of one of the disputed statements in a manner consistent with the nature of our linguistic training:
'Given that the understanding of statements in general is to be viewed as consisting in possession of a concept of their truth-conditions, acquiring a concept of an evidence-transcendent state of affairs is simply a matter of acquiring an understanding of a statement for which that state of affairs would constitute the truth-condition.' (Realism, Meaning & Truth: 16)
Once we've made this concession, it's clear how compositionality helps the realist meet the challenge.
Now, part of my confusion last time stemmed from the following. Wright thinks there are insuperable difficulties standing in the way of a semantic theory which replaces the notion of truth with that of warranted-assertibility (or something similar). So we should adopt truth-conditional semantics. Understanding a sentence consists in grasping its truth-conditions; understanding one of the disputed sentences consists in grasping its truth-conditions. The realist is to be accused of 'overdescribing' what understanding a sentence consists in; she's right to hold that it consists in grasp of truth-conditions, but unwarranted in supposing it ever consists in grasp of realist truth-conditions (since such grasp could never be manifested in linguistic behaviour). As Wright is prone to putting it (regarding epistemicism, but clearly the point is more general than that), the realist is to be accused of superstition rather than error.
My puzzlement was over this; why can't we offer the same response to the compositionality response to the acquisition challenge? We'd then concede that we understand the disputed statements, that understanding those statements consists in grasping their truth-conditions, and that one could come (via compositionality) to grasp their truth-conditions in a way consistent with recognising that our linguistic training is a training in use. But what forces us here to say that the truth-conditions we've arrived at are realist truth-conditions? (I've discovered since writing my first post on this that Miller explicitly considers such a response to the compositionality response in his 'What is the Acquisition Argument?'. See pp. 492-4)
Actually, there seems to be a very simple reason to think we are forced to regard the truth-conditions of the disputed statements as realist truth-conditions, namely that (as is conceded on all sides) these statements are undecidable (See Miller p491 and p7 of Williamson's 'Must Do Better'.) I think that there's a serious challenge there, but setting that point aside for now, we still have something of a mystery as far as I can see. Why did Wright think that it sufficed to meet the acquisition challenge - formulated as a challenge to explain how we could have acquired a conception of what it would be for a statement to be associated with realist truth-conditions - to explain how we could come to grasp (in the right sort of way) the truth-conditions of one of the disputed statements, given that he holds that nothing warrants us in regarding those truth-conditions as realist truth-conditions?
Labels: Anti-Realism, Dummett, Philosophy of Language
Saturday, October 21, 2006
The scope of Dummett's arguments against realism
As soon as you start to look at Dummett's arguments against realism (discussed in the previous post), the issue of their scope becomes really important. There seem to me to be two main issues here (though there are no doubt others I'm just not sensitive to yet). The first is the modus tollens argument sketched by Dummett in 'Wang's Paradox' and discussed in detail in Wright's 'Strict Finitism'. (I'm grateful to Ofra Magidor, for I assume it was she, for forcing me to rethink the significance of this argument after an exchange on TAR). The suggestion is that intuitionism in the philosophy of mathematics isn't a stable resting place; the best arguments in favour of intuitionism over realism about mathematics also favour some kind of strict finitist position over intuitionism. And if strict finitism should prove to be incoherent (as Dummett suggested), that casts suspicion on the intuitionist's original arguments.It's the second issue about scope I want to discuss here. Dummett has often been taken as offering arguments in favour of global anti-realism, i.e. realism about every domain of discourse. Now, Wright's Truth and Objectivity offered a view of realism/anti-realism debates according to which these battles were to be fought locally over some disputed class of statements: mathematical statements or statements about the past, say. It's perhaps insufficiently appreciated that Dummett saw these debates being settled locally too. Here's a quote from a 1969 paper:
'Perhaps the most interesting question about realism is precisely whether global anti-realism is coherent: for, if it is not coherent, then there must at least be some restrictions on the applicability of the anti-realist argument, and, by finding out what these are, we may hope to take a large step towards seeing how to resolve the various particular disputes. There are a number of reasons for doubting whether global anti-realism is coherent, for instance: behaviourism is one species of anti-realism, namely a rejection of realism concerning mental states and processes; phenomenalism is another species, namely the rejection of realism concerning physical objects and processes; it immediately occurs to us to wonder whether it is possible consistently to maintain an anti-realist position simultaneously in both regards.'
(Dummett T&OE: 367-8)
And from a much later, 1993 paper:
'My original intention had been to prompt what is called a ‘research programme’ in the form of a comparative study of disputes over realism. It had struck me that a variety of different traditional disputes within philosophy took the form of an opposition between a realist view of some particular subject-matter and a rejection of realism concerning that subject-matter; often one side in the dispute was conventionally labeled ‘realist’, though in other cases not. It appeared to me that no agreed method of settling these disputes was available, and that philosophers picked sides in them on the basis of predilection, not because they had discovered a means of resolving them. It further appeared to me that there was a striking parallelism in the arguments used on both sides in each of these disputes, so that, if one prescinded from the particular subject-matter, one could exhibit the abstract structure of the dispute. No two of the disputes seemed to me to be completely isomorphic, so as to have an identical abstract structure in common; but the structures were so similar that it seemed to me fruitful to propose that we should make a comparative study of them, from which, I hoped, principles would emerge for deciding in which cases the realist was in the right and in which cases his opponent.
I was understood, however, not as proposing a research programme, but as putting forward a specific philosophical thesis of great generality: not as suggesting a comparative study of a range of structurally similar problems case by case, but as advancing a single unitary thesis.'
(Dummett, The Seas of Language, 1993: 465-6)
The obvious problem is that the arguments Dummett put forward on the anti-realist's behalf turned on perfectly general questions in the theory of meaning rather than any distinctive features of a certain class of local realism/anti-realism debates; the arguments look like they'll settle every debate in the anti-realist's favour, if they go through at all. As Crispin puts the point: 'the best of the anti-realist arguments have a generality which makes it hard to see how they could succeed anywhere unless sucessful everywhere' (Realism, Meaning & Truth: 2-3). Dummett's suggestion in the 1969 passage quoted above that part of the project of resolving these debates will be to discover the limits of the anti-realist's arguments is fascinating, but the nature of the arguments makes it very hard to see how there could be such limits. This is the second issue concerning their scope.
Unfortunately, Dummett wasn't very consistent on this point as far as I can tell. After reading the 1969 passage, one could be forgiven for being somewhat surprised to find Dummett writing in a famous paper written just four years later:
'It follows that, is so far as an intuitionist position in the philosophy of mathematics (or, at least, the acceptance of an intuitionistic logic for mathematics) is supported by an argument of this first type [I think he's here referring to the manifestation argument, but I'm not quite sure], similar, though not necessarily identical, revisions must be made in the logic accepted for statements of other kinds. What is involved is a thesis in the theory of meaning of the highest possible generality. Such a thesis is vunerable in many places: if it should prove that it cannot be coherently applied to any one region of discourse, to any one class of statements, then the thesis cannot be generally true, and the general argument in favour of it must be fallacious. Construed in this way, therefore, a position in the philosophy of mathematics will be capable of being undermined by considerations which have nothing directly to do with mathematics at all.' ('The Philosophical Basis of Intuitionistic Logic' in T&OE: 227)
This is a startling passage. First of all, it's hard to square Dummett's observation here that '[w]hat is involved here is a thesis [...] of the highest possible generality' with his later surprise that people took him to be putting forward 'a specific philosophical thesis of great generality' (see the 1993 quote above).
More surprising, of course, are the related suggestions that anti-realism must be global, and that if it we discover it cannot be defended for any class of statements, that teaches us not something important about the scope of the anti-realist arguments, but that those arguments 'must be fallacious'.
The 1969 paper ('The Reality of the Past') and the 1973 paper are two of Dummett's most famous contributions to these debate. It's bizarre (almost Russell-esque) to find him expousing such different views on this clearly very important issue within such a short period of time. It's hard not to suspect that the 1973, more pessimistic attitude must be correct. That said though, the 1969 suggestion that there is something interesting and important that can be learned about the scope of Dummett's arguments remains seductive, albeit ellusive. But that's par for the course.
Labels: Anti-Realism, Dummett, Philosophy of Language
Wednesday, October 18, 2006
Acquisition and Compositionality
I'm doing an independent study this semester with Dan Bonevac on semantic anti-realism, and this is an important issue which we spend a couple of weeks puzzling over. Let me introduce the debate as briefly as I know how before getting to the issue at hand.
A semantic realist holds the following two theses. Firstly, that understanding is knowledge of truth-conditions. Secondly, that the truth-conditions of some sentences can obtain or fail to obtain unknowably. But Michael Dummett suggested that realism so characterised gives a bad picture of what understanding a declarative sentence consists in, one that fails to take account of insights that can be distilled from the philosophy of the later Wittgenstein. Here's Alex Miller's summary of Dummett's complaints against the realist's claim that understanding could consist in knowledge of potentially verification-transcendent truth-conditions:
'...if our understanding of some sentences of D [D is some disputed aread of discourse, mathematics say] is constituted by grasp of potentially recognition-transcendent truth-conditions, how could we have acquired that understanding, given that our training in the use of sentences is a training to respond to situations which we are, necessarily, capable of recognising to obtain when they obtain? And if our understanding of some sentences of D is constituted by grasp of potentially recognition-transcendent truth conditions, how could we manifest that understanding in our use of those sentences, given that the situations to which we respond are, necessarily, situations which we are capable of recognising to obtain when they obtain?' (Miller 2002: 353)
These two challenges to realism have become known, for obvious reasons, as the acquisition challenge and the manfestation challenge. It's the acquisition challenge that I'll be concerned with here.
Last week Dan and I reread Dummett's classic 1959 paper 'Truth'. It's striking just how ahead of its time this paper is. For example, we were reflecting on the fact that the Investigations has only been in print for 6 years by the time this paper is published, and that Davidson's 'Truth and Meaning' is almost a decade away. It's also striking that there's pretty much no foreshadowing of the manifestation challenge in this paper, but there is a preliminary version of the acquisition challenge (see Truth and Other Enigmas: 17-8). The challenge gets developed throughout Dummett's extensive writings on these topics (which I'm still trying to acquaint myself with), but for me Dummett's own statements of the challenge have always been a little elusive. One of the clearer statements (or at least one of the statements I think I understand best) comes in Wright's introduction to Realism, Meaning & Truth:
'How are we supposed to be able to form any understanding of what it is for a particular statement to be true if the kind of state of affairs it would take to make it true is conceived, ex hypothesi, as something beyond our experience, something which we cannot confirm and which is insulated from any distinctive impact on our consciousness. Obviously such a conception cannot be bestowed ostensively. And the challenge is simply declined if the answer is offered, 'by description'. For it is of our ability to form an understanding of precisely such a description that an account is being demanded; there could be no better description of the relevant kind of state if affairs than the very statement in question.' (13)
Wright remarks: 'This is a pleasantly simple line of argument.'
Nonetheless, as Alex Millar has been keen to point out in a couple of his papers, both Wright and Hale have basically conceded that the acquisition challenge can be met, at least when it's presented in any strong form (Wright, for example, thinks there's still a challenge to the realist in the area, but that its force is parasitic on that of the manifestation challenge. See p16).
The passages where this concession is made are puzzling. Each makes a crucial concession to the realist in a single sentence, and without any reasons offered for why the anti-realist should concede so much. Here's Wright, from the intro again, and with the sentence in question italicized:
'...the most glaring example of our ability to work with concepts which transgress the limits of what we have actually experienced is provided by the understanding each of us has of no end of sentences in our language which we have never previously heard. The anti-realist can hardly deny that we have such an ability in general. Moreover his position will lose any credibility unless he can dissociate it from the old positivist dogma that untestable statements are meaningless. So he should grant that we are able to understand statements of the kind for which he finds the realist interpretation problematic; his quarrell is not with the supposition that we do have such understanding, but only with the realist account of it. But now the realist seems to have a very simply answer. Given that the understanding of statements in general is to be viewed as consisting in possession of a concept of their truth-conditions, acquiring a concept of an evidence-transcendent state of affairs is simply a matter of acquiring an understanding of a statement for which that state of affairs would constitute the truth-condition. And such an understanding is acquired, like the understanding of any previously unheard sentence in the language, by understanding the constituent words and the significance of their mode of combination.' (15-6)
So intially the challenge was to show how we could form a concept of what it would be for there to be sentences with verfication-transcendent truth-conditions, given that we learn a language through a process of training in use, and thus through learning how to respond to, to quote Miller again, 'situations which we are capable of recognising to obtain when they obtain'. The italicized phrase concedes that all it takes to meet this challenge is to show how we can, in a manner consistent with this view of language-acquisition, come to understand one of the sentences for which the anti-realist finds the realist account of understanding bankrupt. And now the answer is easy; compositionality does all the work.
But why on earth should we think that's all it takes to meet Dummett's challenge? Wright doesn't say.
Here's the relevant passage by Hale, again with my italics:
'...a very simple response is available: we come by a grasp of realist truth-conditions by coming to understand sentences having those truth-conditions, and we come to understand such sentences in just the way in which we come to understand the vast majority of sentences in our language, by understanding their words and semantically significant syntax.' ('Realism and its Oppositions' in the Blackwell Companion to the Philosophy of Language: 279)
Again, nothing at all is offered to suggest why the anti-realist is forced to concede the point in italics. Maybe it's obvious, but it certainly isn't obvious to me.
I need to finish reading Miller's 'What is the Acquisition Argument?', though the sections on this compositionality point I've read thus far sadly didn't shed much light on my particular point of confusion. I'm also guessing there might be some discussion in the literature on truth-value links concerning the past and other minds (for example, work by McDowell and Peacocke), where I think I similar move must be made to the one made in these italicized phrases. But I feel I need to do some serious further investigation.
A semantic realist holds the following two theses. Firstly, that understanding is knowledge of truth-conditions. Secondly, that the truth-conditions of some sentences can obtain or fail to obtain unknowably. But Michael Dummett suggested that realism so characterised gives a bad picture of what understanding a declarative sentence consists in, one that fails to take account of insights that can be distilled from the philosophy of the later Wittgenstein. Here's Alex Miller's summary of Dummett's complaints against the realist's claim that understanding could consist in knowledge of potentially verification-transcendent truth-conditions:
'...if our understanding of some sentences of D [D is some disputed aread of discourse, mathematics say] is constituted by grasp of potentially recognition-transcendent truth-conditions, how could we have acquired that understanding, given that our training in the use of sentences is a training to respond to situations which we are, necessarily, capable of recognising to obtain when they obtain? And if our understanding of some sentences of D is constituted by grasp of potentially recognition-transcendent truth conditions, how could we manifest that understanding in our use of those sentences, given that the situations to which we respond are, necessarily, situations which we are capable of recognising to obtain when they obtain?' (Miller 2002: 353)
These two challenges to realism have become known, for obvious reasons, as the acquisition challenge and the manfestation challenge. It's the acquisition challenge that I'll be concerned with here.
Last week Dan and I reread Dummett's classic 1959 paper 'Truth'. It's striking just how ahead of its time this paper is. For example, we were reflecting on the fact that the Investigations has only been in print for 6 years by the time this paper is published, and that Davidson's 'Truth and Meaning' is almost a decade away. It's also striking that there's pretty much no foreshadowing of the manifestation challenge in this paper, but there is a preliminary version of the acquisition challenge (see Truth and Other Enigmas: 17-8). The challenge gets developed throughout Dummett's extensive writings on these topics (which I'm still trying to acquaint myself with), but for me Dummett's own statements of the challenge have always been a little elusive. One of the clearer statements (or at least one of the statements I think I understand best) comes in Wright's introduction to Realism, Meaning & Truth:
'How are we supposed to be able to form any understanding of what it is for a particular statement to be true if the kind of state of affairs it would take to make it true is conceived, ex hypothesi, as something beyond our experience, something which we cannot confirm and which is insulated from any distinctive impact on our consciousness. Obviously such a conception cannot be bestowed ostensively. And the challenge is simply declined if the answer is offered, 'by description'. For it is of our ability to form an understanding of precisely such a description that an account is being demanded; there could be no better description of the relevant kind of state if affairs than the very statement in question.' (13)
Wright remarks: 'This is a pleasantly simple line of argument.'
Nonetheless, as Alex Millar has been keen to point out in a couple of his papers, both Wright and Hale have basically conceded that the acquisition challenge can be met, at least when it's presented in any strong form (Wright, for example, thinks there's still a challenge to the realist in the area, but that its force is parasitic on that of the manifestation challenge. See p16).
The passages where this concession is made are puzzling. Each makes a crucial concession to the realist in a single sentence, and without any reasons offered for why the anti-realist should concede so much. Here's Wright, from the intro again, and with the sentence in question italicized:
'...the most glaring example of our ability to work with concepts which transgress the limits of what we have actually experienced is provided by the understanding each of us has of no end of sentences in our language which we have never previously heard. The anti-realist can hardly deny that we have such an ability in general. Moreover his position will lose any credibility unless he can dissociate it from the old positivist dogma that untestable statements are meaningless. So he should grant that we are able to understand statements of the kind for which he finds the realist interpretation problematic; his quarrell is not with the supposition that we do have such understanding, but only with the realist account of it. But now the realist seems to have a very simply answer. Given that the understanding of statements in general is to be viewed as consisting in possession of a concept of their truth-conditions, acquiring a concept of an evidence-transcendent state of affairs is simply a matter of acquiring an understanding of a statement for which that state of affairs would constitute the truth-condition. And such an understanding is acquired, like the understanding of any previously unheard sentence in the language, by understanding the constituent words and the significance of their mode of combination.' (15-6)
So intially the challenge was to show how we could form a concept of what it would be for there to be sentences with verfication-transcendent truth-conditions, given that we learn a language through a process of training in use, and thus through learning how to respond to, to quote Miller again, 'situations which we are capable of recognising to obtain when they obtain'. The italicized phrase concedes that all it takes to meet this challenge is to show how we can, in a manner consistent with this view of language-acquisition, come to understand one of the sentences for which the anti-realist finds the realist account of understanding bankrupt. And now the answer is easy; compositionality does all the work.
But why on earth should we think that's all it takes to meet Dummett's challenge? Wright doesn't say.
Here's the relevant passage by Hale, again with my italics:
'...a very simple response is available: we come by a grasp of realist truth-conditions by coming to understand sentences having those truth-conditions, and we come to understand such sentences in just the way in which we come to understand the vast majority of sentences in our language, by understanding their words and semantically significant syntax.' ('Realism and its Oppositions' in the Blackwell Companion to the Philosophy of Language: 279)
Again, nothing at all is offered to suggest why the anti-realist is forced to concede the point in italics. Maybe it's obvious, but it certainly isn't obvious to me.
I need to finish reading Miller's 'What is the Acquisition Argument?', though the sections on this compositionality point I've read thus far sadly didn't shed much light on my particular point of confusion. I'm also guessing there might be some discussion in the literature on truth-value links concerning the past and other minds (for example, work by McDowell and Peacocke), where I think I similar move must be made to the one made in these italicized phrases. But I feel I need to do some serious further investigation.
Labels: Anti-Realism, Dummett, Epistemology, Philosophy of Language
Friday, August 25, 2006
The Taming of the True....
I'm very pleased to have made it through my first summer of blogging without, like so many of the blogs I read, going on hiatus or anything like that - posting has been a little sporadic, but I feel I've managed to keep the momentum going ok. Given this, you may be wondering (and let's face it, who isn't?) what's happened to posting since I got back to Austin.
The truth is I'm starting to read quite seriously on the topic of Dummett's critique of the realist's picture of what understanding a declarative sentence consists in. It's always been one of the philosophical flames I get drawn moth-like towards, but there's big gaps in my knowledge of the literature that I'm trying to fill just now. In particular, I'm currently reading Crispin Wright's Realism, Meaning and Truth. It's kind of simultaneously thrilling and depressing. It's thrilling because so many philosophers today take this to be a dead issue, whereas Crispin's sense that these are live, fascinating, central issues is tangible. On the other hand, it's also evident that the kind of command of these issues required to really make anything of a contribution will be very hard-won; it's hard not to be impressed with the altitude this debate has been conducted at over the last 40-50 years, and not just a little daunting. Hence the lack of posts - I really have a lot to learn about this stuff before I have anything of my own to say that might potentially be of interest to others.
Enough biography though. Real content as soon as I'm feeling up to it.
Ps. Apologies to Neil Tennant for stealing his great title for this post - my copy of the book just arrived yesterday, and it seemed too fitting to resist.
The truth is I'm starting to read quite seriously on the topic of Dummett's critique of the realist's picture of what understanding a declarative sentence consists in. It's always been one of the philosophical flames I get drawn moth-like towards, but there's big gaps in my knowledge of the literature that I'm trying to fill just now. In particular, I'm currently reading Crispin Wright's Realism, Meaning and Truth. It's kind of simultaneously thrilling and depressing. It's thrilling because so many philosophers today take this to be a dead issue, whereas Crispin's sense that these are live, fascinating, central issues is tangible. On the other hand, it's also evident that the kind of command of these issues required to really make anything of a contribution will be very hard-won; it's hard not to be impressed with the altitude this debate has been conducted at over the last 40-50 years, and not just a little daunting. Hence the lack of posts - I really have a lot to learn about this stuff before I have anything of my own to say that might potentially be of interest to others.
Enough biography though. Real content as soon as I'm feeling up to it.
Ps. Apologies to Neil Tennant for stealing his great title for this post - my copy of the book just arrived yesterday, and it seemed too fitting to resist.
Labels: Anti-Realism
