A few months ago I wrote a short argument against Stephen Yablo’s definition of a priority as truth that could be recognised in virtue of understanding. Less wordilly, S is a priori iff understanding S is sufficient for recognising S as true. The objection turned on Pierce’s Law:

(1) ((p -> q) -> p) -> p

I said that Yablo couldn’t have all the following things. First, this is a logical truth. Second, all logical truths are a priori. Third, his account of what it is to be a priori. Fourth, that the meaning of the -> is its introduction/elimination rules in a single-conclusion natural-deduction system. He can’t have them all because the rules for -> alone don’t let you prove (1), you also need the rules for ~.

You might think I’m being fussy here in saying that he needs rules for some other connective. But actually that amount of fussiness was completely in the spirit of Yablo’s definition. The definition was designed to get the result that the a priori is not closed under logical consequence, for just the reason that A might be a priori, and If A then B might be a priori in part in virtue of the meanings of the terms in A, while B is not a priori. It might be, that is, that once you understand the terms in A you will realise that B is true, but just understanding the terms in B might not be enough to show you that B is true. The non-closure of the a priori would be VERY BAD NEWS for we small band of two-dimensionalists, so I thought I was making a small blow for freedom with this little argument.

Partial disclosure: Just how much of a two-dimensionalist I am is not entirely clear. I’m at least a fellow-traveller, but saying more would require me feeling less one-thirty-in-the-morningish.

Second partial disclosure: It wasn’t very clear in the original post just what the argument was. It was partially an ad hominem, since I suspected Yablo believed the other three horns of the quadrilemma. And it was partially a plausibility argument against his account of the a priori, because I suspected most readers would think his account of the a priori was the weakest of the four claims. But that was sort of in bad faith, because it isn’t what I believe. Precisely what I believed may become clear below.

Anyway, I just realised going over the argument again tonight was that it seems to cause just as many problems for views I hold as it does for Yablo’s position. (Think of this as the philosophical logical version of blowback.) Roughly, I want to hold all of the following theses.

(2) All logical truths are analytic.
(3) All analytic truths are true in virtue of the meanings of their terms.
(4) If (2) and (3) are true, then if all instances of a particular schema are logical truths, they must all be true in virtue of the meanings of the logical connectives.
(5) The meaning of the -> is given by the introduction/elimination rules for it in a single-conclusion natural-deduction system (i.e. the deduction theorem and modus ponens).

Accepting (2) to (5) means giving up that (1) is a logical truth. That’s OK I think. I actually think (2) through (5) constitute a pretty plausible argument for the claim that (1) is not a logical truth. The problems start arising soon after that.

(6) The meaning of ~ is given by its introduction/elimination rules in a classical single-conclusion natural-deduction system (i.e. reductio ad absurdum and double negation elimination).
(7) If A is true in virtue of the meanings of its logical connectives, then it is a logical truth.
(8) If If A then B is a logical truth, and A is a logical truth, then B is a logical truth.

And here things start to go badly wrong for me. Because (6) and (7) commit me to (9) and (10) being logical truths, so (8) commits me to (1) also being a logical truth.

(9) ~~[((p -> q) -> p) -> p]
(10) [~~[((p -> q) -> p) -> p]] -> [((p -> q) -> p) -> p]

And now I’m stuck in a contradiction. I have to give up one of (2) through (8). (Ever been in a heptalemma before? It’s kinda neat in a screwy way.) There’s little to recommend giving up (4) or (7), which seem fairly unobtrusive steps in the argument. And (3) looks definitional.

I could sort of understand giving up (2). I think it’s the position that most naturally fits with what Yablo says about the a priori. And I’m somewhat tempted by it actually. Say the intuitionists are sort of right about analyticity, so (1) is not analytic. But they are wrong about logical truth. Logical truth is not truth in virtue of logical constants, its being a consequence of something(s) true in virtue of logical constants. So (1) is a logical truth because (9) and (10) are, not because it is true in virtue of its logical constants.

Dummett, of course, recommends giving up (6). Deep down, I suspect that’s the most sane option available. But even if I’m right, in philosophy that is not always the most telling of considerations.

Stephen Read in some recent work has recommended giving up (5). The meaning of -> is given by its in/elim rules in multiple-conclusion natural-deduction systems. In those systems (1) can be deduced from the meaning of -> alone. I’m going to be in print sort of endorsing this position soon, which may not be wise. (It’s a throwaway comment in the Problem of the Many paper. And strictly I say only that this is the best response to certain arguments of Dummett’s, of which (2) to (5) is a loose paraphrase. But it’s probably still too strong.)

Anyway, the bold late night conjecture here is that (8) could be the culprit. Perhaps logical truth is not closed under modus ponens. That would be exciting!

There’s a few take-home lessons from this. I’ll just mention two. First, the argument for intuitionism from (2) to (5) is a little less compelling than it first appears, because accepting it requires giving up (6), (7) or (8). This is probably obvious to everyone who thinks about the matter for more than two seconds, but it was only working through Yablo’s definition of a priority that made me realise it. Secondly, once you start giving up closure for various things, life gets pretty interesting…

What triggered all this off was that Steve Yablo was at Brown today to give a talk on two-dimensionalism. It covered a lot of ground and there’s no way I could do justice to the points made in this format, or at this hour. But when Beyond Rigidification appears on a website near you, I highly recommend checking it out.

A few months ago I wrote a short argument against Stephen Yablo’s definition of a priority as truth that could be recognised in virtue of understanding. Less wordilly, S is a priori iff understanding S is sufficient for recognising S as true. The objection turned on Pierce’s Law:

(1) ((p -> q) -> p) -> p

I said that Yablo couldn’t have all the following things. First, this is a logical truth. Second, all logical truths are a priori. Third, his account of what it is to be a priori. Fourth, that the meaning of the -> is its introduction/elimination rules in a single-conclusion natural-deduction system. He can’t have them all because the rules for -> alone don’t let you prove (1), you also need the rules for ~.

You might think I’m being fussy here in saying that he needs rules for some other connective. But actually that amount of fussiness was completely in the spirit of Yablo’s definition. The definition was designed to get the result that the a priori is not closed under logical consequence, for just the reason that A might be a priori, and If A then B might be a priori in part in virtue of the meanings of the terms in A, while B is not a priori. It might be, that is, that once you understand the terms in A you will realise that B is true, but just understanding the terms in B might not be enough to show you that B is true. The non-closure of the a priori would be VERY BAD NEWS for we small band of two-dimensionalists, so I thought I was making a small blow for freedom with this little argument.

Partial disclosure: Just how much of a two-dimensionalist I am is not entirely clear. I’m at least a fellow-traveller, but saying more would require me feeling less one-thirty-in-the-morningish.

Second partial disclosure: It wasn’t very clear in the original post just what the argument was. It was partially an ad hominem, since I suspected Yablo believed the other three horns of the quadrilemma. And it was partially a plausibility argument against his account of the a priori, because I suspected most readers would think his account of the a priori was the weakest of the four claims. But that was sort of in bad faith, because it isn’t what I believe. Precisely what I believed may become clear below.

Anyway, I just realised going over the argument again tonight was that it seems to cause just as many problems for views I hold as it does for Yablo’s position. (Think of this as the philosophical logical version of blowback.) Roughly, I want to hold all of the following theses.

(2) All logical truths are analytic.
(3) All analytic truths are true in virtue of the meanings of their terms.
(4) If (2) and (3) are true, then if all instances of a particular schema are logical truths, they must all be true in virtue of the meanings of the logical connectives.
(5) The meaning of the -> is given by the introduction/elimination rules for it in a single-conclusion natural-deduction system (i.e. the deduction theorem and modus ponens).

Accepting (2) to (5) means giving up that (1) is a logical truth. That’s OK I think. I actually think (2) through (5) constitute a pretty plausible argument for the claim that (1) is not a logical truth. The problems start arising soon after that.

(6) The meaning of ~ is given by its introduction/elimination rules in a classical single-conclusion natural-deduction system (i.e. reductio ad absurdum and double negation elimination).
(7) If A is true in virtue of the meanings of its logical connectives, then it is a logical truth.
(8) If If A then B is a logical truth, and A is a logical truth, then B is a logical truth.

And here things start to go badly wrong for me. Because (6) and (7) commit me to (9) and (10) being logical truths, so (8) commits me to (1) also being a logical truth.

(9) ~~[((p -> q) -> p) -> p]
(10) [~~[((p -> q) -> p) -> p]] -> [((p -> q) -> p) -> p]

And now I’m stuck in a contradiction. I have to give up one of (2) through (8). (Ever been in a heptalemma before? It’s kinda neat in a screwy way.) There’s little to recommend giving up (4) or (7), which seem fairly unobtrusive steps in the argument. And (3) looks definitional.

I could sort of understand giving up (2). I think it’s the position that most naturally fits with what Yablo says about the a priori. And I’m somewhat tempted by it actually. Say the intuitionists are sort of right about analyticity, so (1) is not analytic. But they are wrong about logical truth. Logical truth is not truth in virtue of logical constants, its being a consequence of something(s) true in virtue of logical constants. So (1) is a logical truth because (9) and (10) are, not because it is true in virtue of its logical constants.

Dummett, of course, recommends giving up (6). Deep down, I suspect that’s the most sane option available. But even if I’m right, in philosophy that is not always the most telling of considerations.

Stephen Read in some recent work has recommended giving up (5). The meaning of -> is given by its in/elim rules in multiple-conclusion natural-deduction systems. In those systems (1) can be deduced from the meaning of -> alone. I’m going to be in print sort of endorsing this position soon, which may not be wise. (It’s a throwaway comment in the Problem of the Many paper. And strictly I say only that this is the best response to certain arguments of Dummett’s, of which (2) to (5) is a loose paraphrase. But it’s probably still too strong.)

Anyway, the bold late night conjecture here is that (8) could be the culprit. Perhaps logical truth is not closed under modus ponens. That would be exciting!

There’s a few take-home lessons from this. I’ll just mention two. First, the argument for intuitionism from (2) to (5) is a little less compelling than it first appears, because accepting it requires giving up (6), (7) or (8). This is probably obvious to everyone who thinks about the matter for more than two seconds, but it was only working through Yablo’s definition of a priority that made me realise it. Secondly, once you start giving up closure for various things, life gets pretty interesting…

What triggered all this off was that Steve Yablo was at Brown today to give a talk on two-dimensionalism. It covered a lot of ground and there’s no way I could do justice to the points made in this format, or at this hour. But when Beyond Rigidification appears on a website near you, I highly recommend checking it out.

The following strikes me as a pretty
persuasive argument against a thorough-going process reliabilism. Since I’m no
expert on the field, I don’t know how similar it is to existing arguments
against process reliabilism, which is to say that if this turns out to be a
boring repetition of familiar points, well at least it wasn’t intentional
plagiarism.

Process reliabilism says that the
justification of a belief is proportional to the reliability of the process
that generated the belief. This raises the generality problem, as stressed in
Conee and Feldman’s 1998 paper – what is the process by which the belief
is generated? Or, to put the point more obscurely, what are the individuation
conditions for process types being used in this formulation. At one level the
generality problem is the problem of making the basic claim of process
reliabilism contentful – if we are prepared to count gruesome enough types,
then every belief is the product of some very reliable processes, and some very
unreliable processes. But let’s assume that problem has been handled.

At another level, the generality problem
raises a tension that I think can’t be resolved for a full-blown process
reliabilist. On the one hand, we want processes to be instantiated more than
one time, or else we’ll be led to the crazy view that a belief is justified iff
it is true. So we don’t want the instantiation to be too fine-grained.
On the other hand, the definition of justification entails rather immediately
(so immediately that it might surprise you to learn how long it took me to
realise this) every belief generated by the same process is equally justified. To
the extent that justificatory status can be very sensitive to the particular
ways a belief is formed, that implies we want processes to be individuated
quite finely. I think, and I think I have an example that supports this, that
these two constraints can’t be satisfied at once. Onto the example…

DIAGNOSIS

Morgan is
displaying symptoms S. Dr Watson knows that symptoms S normally
imply that the patient has a liver disease. But he also knows that in some
cases, happily enough in all and only cases where the patient has genetic condition
C, a patient with symptoms S doesn’t have a liver disease, but in
fact has a kidney disease. Dr. Watson also knows that genetic condition C
is rare, only 1% of males and 7% of females are C. And he knows that
there’s no easy way to test for whether a patient has condition C, for
usually it has no readily observable effects. And he knows he has no other
relevant information about whether Morgan is has condition C. So Watson
concludes that Morgan has a liver disease.

How justified is Dr. Watson’s belief?

I think you don’t know enough to say yet,
because you don’t know whether Morgan is male or female. If Morgan is male,
then Watson’s belief is very well justified. If Morgan is female, then Watson’s
belief isn’t particularly well justified, for he should be taking more
seriously the possibility that Morgan has condition C. Even in that case, it isn’t a disastrous
belief, but not as well justified as in the case where Morgan is male. Since
the two possible beliefs are not equally well justified, we need to say that
they are the results of different processes.

That alone might not be a problem. Perhaps
we can find a different way of categorising beliefs such that the belief that a
male patient displaying S has a liver disease falls into a different
category than the belief that a female patient displaying S has a liver
disease, though I’m not entirely convinced that existing (pure) reliabilist
theories have the resources to do this.

The problem is that the example generalises.
If x and y are both relatively small numbers, and Watson knows
that x% of males have condition C and y% of females do,
then his conclusion that Morgan has a liver disease is more justified if Morgan
is male rather than female for any such x and y, even if they are very
close, say x = 4.5 and y = 5, or even, I’d guess, if x =
4.5 and y = 4.51.

That means that we’re going to have to posit
infinitely many different categories of belief-forming processes, just to
account for all the different possible processes via which Watson could form
the belief that Morgan has a liver disease. The problem is that when categories
belief-forming processes get so fine-grained, we will start to get some
lucky guesses counting as justified beliefs, because they are the only beliefs
ever formed by that process, and some unlucky reasoned judgments counting as unjustified
beliefs, again because of the small sample size. This I take it should be
intolerable.

One response to related problems raised in
the 1980s was to modalise the notion of reliability. Maybe I’ll come back to
that in later posts, but I think it should be pretty clear that won’t help. The
problem is that there’s too many darn worlds to possibly count successes and
failures of a process, and no other approach to summarising the data from
nearby possible worlds seems to be much use.

This is not a problem for theories of
justification that incorporate some aspects of process reliabilism, but also
build in some more traditional internalist evaluations of modes of reasoning.
Ernie Sosa’s virtue reliabilism is like this, and my theory, which is
reliabilist about observational beliefs and (sorta kinda) foundationalist about
non-observational beliefs isn’t either. But a theory that is all process
reliabilism all the time really looks like it has problems with DIAGNOSIS.

The following strikes me as a pretty
persuasive argument against a thorough-going process reliabilism. Since I’m no
expert on the field, I don’t know how similar it is to existing arguments
against process reliabilism, which is to say that if this turns out to be a
boring repetition of familiar points, well at least it wasn’t intentional
plagiarism.

Process reliabilism says that the
justification of a belief is proportional to the reliability of the process
that generated the belief. This raises the generality problem, as stressed in
Conee and Feldman’s 1998 paper – what is the process by which the belief
is generated? Or, to put the point more obscurely, what are the individuation
conditions for process types being used in this formulation. At one level the
generality problem is the problem of making the basic claim of process
reliabilism contentful – if we are prepared to count gruesome enough types,
then every belief is the product of some very reliable processes, and some very
unreliable processes. But let’s assume that problem has been handled.

At another level, the generality problem
raises a tension that I think can’t be resolved for a full-blown process
reliabilist. On the one hand, we want processes to be instantiated more than
one time, or else we’ll be led to the crazy view that a belief is justified iff
it is true. So we don’t want the instantiation to be too fine-grained.
On the other hand, the definition of justification entails rather immediately
(so immediately that it might surprise you to learn how long it took me to
realise this) every belief generated by the same process is equally justified. To
the extent that justificatory status can be very sensitive to the particular
ways a belief is formed, that implies we want processes to be individuated
quite finely. I think, and I think I have an example that supports this, that
these two constraints can’t be satisfied at once. Onto the example…

DIAGNOSIS

Morgan is
displaying symptoms S. Dr Watson knows that symptoms S normally
imply that the patient has a liver disease. But he also knows that in some
cases, happily enough in all and only cases where the patient has genetic condition
C, a patient with symptoms S doesn’t have a liver disease, but in
fact has a kidney disease. Dr. Watson also knows that genetic condition C
is rare, only 1% of males and 7% of females are C. And he knows that
there’s no easy way to test for whether a patient has condition C, for
usually it has no readily observable effects. And he knows he has no other
relevant information about whether Morgan is has condition C. So Watson
concludes that Morgan has a liver disease.

How justified is Dr. Watson’s belief?

I think you don’t know enough to say yet,
because you don’t know whether Morgan is male or female. If Morgan is male,
then Watson’s belief is very well justified. If Morgan is female, then Watson’s
belief isn’t particularly well justified, for he should be taking more
seriously the possibility that Morgan has condition C. Even in that case, it isn’t a disastrous
belief, but not as well justified as in the case where Morgan is male. Since
the two possible beliefs are not equally well justified, we need to say that
they are the results of different processes.

That alone might not be a problem. Perhaps
we can find a different way of categorising beliefs such that the belief that a
male patient displaying S has a liver disease falls into a different
category than the belief that a female patient displaying S has a liver
disease, though I’m not entirely convinced that existing (pure) reliabilist
theories have the resources to do this.

The problem is that the example generalises.
If x and y are both relatively small numbers, and Watson knows
that x% of males have condition C and y% of females do,
then his conclusion that Morgan has a liver disease is more justified if Morgan
is male rather than female for any such x and y, even if they are very
close, say x = 4.5 and y = 5, or even, I’d guess, if x =
4.5 and y = 4.51.

That means that we’re going to have to posit
infinitely many different categories of belief-forming processes, just to
account for all the different possible processes via which Watson could form
the belief that Morgan has a liver disease. The problem is that when categories
belief-forming processes get so fine-grained, we will start to get some
lucky guesses counting as justified beliefs, because they are the only beliefs
ever formed by that process, and some unlucky reasoned judgments counting as unjustified
beliefs, again because of the small sample size. This I take it should be
intolerable.

One response to related problems raised in
the 1980s was to modalise the notion of reliability. Maybe I’ll come back to
that in later posts, but I think it should be pretty clear that won’t help. The
problem is that there’s too many darn worlds to possibly count successes and
failures of a process, and no other approach to summarising the data from
nearby possible worlds seems to be much use.

This is not a problem for theories of
justification that incorporate some aspects of process reliabilism, but also
build in some more traditional internalist evaluations of modes of reasoning.
Ernie Sosa’s virtue reliabilism is like this, and my theory, which is
reliabilist about observational beliefs and (sorta kinda) foundationalist about
non-observational beliefs isn’t either. But a theory that is all process
reliabilism all the time really looks like it has problems with DIAGNOSIS.