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
# posted by Aidan @ 11:57 AM 
Comments:
<< Home
Could you explain how viii is sufficient for iii? I'm having trouble seeing it. Given the conditions on SA, we can go from SA'~p' to ~SA'p' but the latter doesn't imply ~WA'p', which looks like what is needed for iii. Instead it gives a disjunction of of the conditions on SA.
This is minor but viii follows from just the <- direction of (1), so (3) isn't even needed.
I have one broader question. Why does Wright think (1) holds? It seems like something's being the case doesn't mean that one is warranted in the strong way required for the biconditional. The -> direction seems good, but the converse seems odd.
This is minor but viii follows from just the <- direction of (1), so (3) isn't even needed.
I have one broader question. Why does Wright think (1) holds? It seems like something's being the case doesn't mean that one is warranted in the strong way required for the biconditional. The -> direction seems good, but the converse seems odd.
If you have closure under &-Intro for two conjuncts, you have it for arbitrarily many conjuncts: from WA P and WA Q you have WA P&Q. Then from that and WA R you have WA (P&Q)&R. Etc.
Richard,
Hmm, yes, thanks. I guess I had in mind the fuss Hawthorne makes over the difference between single-premise epistemic closure and multi-premise closure in 'Knowledge and Lotteries', but I guess that won't help me here.
Shawn,
I'm a little confused. How does viii. follow from the right to left direction of 1? I'm just not seeing that at all. (Wouldn't I need ~(SA 'X & ~SA 'X'') for that inference? And I don't appear to have that.)
The inference I made from viii. to iii. was intended to be of the following form:
If |- ~P, then ~WA 'P'.
I wasn't anticipating a detour through something of the form SA '~P'.
On why Crispin thinks something like (1) holds, I'll need to reread chapter 2 of 'Truth and Objectivity' before I can answer that. If I'm remembering correctly, Brueckner's little paper raises a similar issue to the one you raise. It's in '98 Nous if you want to take a look.
Hmm, yes, thanks. I guess I had in mind the fuss Hawthorne makes over the difference between single-premise epistemic closure and multi-premise closure in 'Knowledge and Lotteries', but I guess that won't help me here.
Shawn,
I'm a little confused. How does viii. follow from the right to left direction of 1? I'm just not seeing that at all. (Wouldn't I need ~(SA 'X & ~SA 'X'') for that inference? And I don't appear to have that.)
The inference I made from viii. to iii. was intended to be of the following form:
If |- ~P, then ~WA 'P'.
I wasn't anticipating a detour through something of the form SA '~P'.
On why Crispin thinks something like (1) holds, I'll need to reread chapter 2 of 'Truth and Objectivity' before I can answer that. If I'm remembering correctly, Brueckner's little paper raises a similar issue to the one you raise. It's in '98 Nous if you want to take a look.
Ah, my problem was that I wasn't sure how one introduces WA without appeal to SA. Proving the negation of a proposition would prevent one from warrentedly asserting that proposition wouldn't it. My mistake. I thought there was an appeal to ~SA'X' (from SA'~X'). You're right, sufficient.
Unless I'm misunderstanding what it says, the right to left direction of 1 is X -> SA'X'. Assume iv, get X, get SA'X, but ~SA'X', so ~iv, i.e. viii, cause iv is exactly contradictory of 1. Classically, vii is obtainable from 1 by switching P->Q to ~PvQ, and then pulling the ~ out front to get ~(P & ~Q).
Unless I'm misunderstanding what it says, the right to left direction of 1 is X -> SA'X'. Assume iv, get X, get SA'X, but ~SA'X', so ~iv, i.e. viii, cause iv is exactly contradictory of 1. Classically, vii is obtainable from 1 by switching P->Q to ~PvQ, and then pulling the ~ out front to get ~(P & ~Q).
Ok, I see. Yes, that's much neater. (Though sadly it's actually 2 we can dispense with - I mislabeled, and had to fix it. Sorry.)
But I want to steer clear of CL. What stands in the way of assuming iv., then &-elim to get
X, and
~SA 'X',
and then apply (1) to X and &-intro our way to glory (i.e. vii.)?
But I want to steer clear of CL. What stands in the way of assuming iv., then &-elim to get
X, and
~SA 'X',
and then apply (1) to X and &-intro our way to glory (i.e. vii.)?
Actually, doubly thanks for the amendment Shawn - my step from iv. to v is invalid in this context. Urg.
Post a Comment
<< Home
