Saturday, January 27, 2007
Symmetry and Paradox
In Vagueness and Contradiction, Roy Sorensen discusses the No-No paradox; abstracting away from details we'll review shortly, this is the puzzle that stands to the truth-teller sentence as the Yes-No paradox stands to the liar. So the (strengthened) Liar sentence is:
S1: S1 is not true,
while the following open-pair generates the Yes-No paradox:
S2: S3 is true
S3: S2 is not true.
What groups these together as a family is that there is no consistent assignment of truth-values to these sentences. In contrast there are consistent assignments of truth-values to the truth-teller sentence:
S4: S4 is true,
and to the pair of sentences that give rise to the No-No paradox:
S5: S6 is not true
S6: S5 is not true.
S4 can consistently receive T or F, we're just at a loss to assign one rather than the other. We can't assign S5 and S6 TT or FF, but TF or FT are fine. Problem is, it seems like symmetry considerations seem to compel us to assign matching truth-values to the two sentences. (We can increase the symmetry between the two sentences by presenting the paradox the way Russell does; we have a sheet of white card, with just "The sentence on the other side of this card is not true" printed on each side. The idea is that there's no grounds on which to conjecture that there has been a context-shift or anything that would justify assigning different true-values to the members of the pair.) So although consistency allows TF or FT, symmetry considerations seem to rule those options out.
More recently, in 'A Definite No-No' (In Beall's Liars and Heaps, 2003) Sorensen presents the following pair of sentences:
S7: S8 is not definitely true
S8: S7 is not definitely true.
(Sorensen's presentation attempts to maximize their 'equal-billing' a little more than mine, again to increase the force of symmetry considerations, but that won't matter here; anyone who feels I've loaded the dice against Sorensen can simply look at his presentation on p225 of his paper.)
The assignment FF (or more carefully, assigning not-true to both) looks like it's immediately ruled out by the logic of definiteness; as Sorensen points out, the untruth of one of the pair implies the definite truth of the other. What about the other options? Sorensen writes:
'. . .the symmetry of the two sentences is destabilizing. We have the following feelings: As twins, [S7] and [S8] should have matching truth-values (or should be equally bereft of truth-values). Since we have already ruled out the possibility that both are untruths, we conclude that [S7] and [S8] are both true.
The truth of the pair is no problem in itself. But there is a ironical difficulty in us possessing such a tidy argument [S7] and [S8] are true. The discovery of a proof for a proposition is generally regarded as a sufficient condition for that proposition being definitely true. Since [S7] and [S8] each deny that the other is definitely true, they would each be false if proven true. Strangely, any cogent argument to the effect that both sentences are true must backfire.' (226)
I just don't find this all that plausible. The crucial move in all of this is the equation of 'cogent argument' with 'proof'. The move is particularly clear in the second and third sentences of the quote; Sorensen passes without a ripple from our possession of a 'tidy argument' to the possession of a proof. Now, this might just be written off as a terminological gripe if Sorensen's argument that the only assignment we can make to S7 and S8 is TT really did constitute a proof. But surely that issue turns on the status of the symmetry principle. After all, it's the symmetry principle that rules out the true-untrue and untrue-true assignments.
So what is the status of the symmetry principle? All Sorensen says is that we have a 'feeling' it should hold. I agree; the principle is pretty intuitively compelling, and likewise I find the argument it underwrites that TT is the only available assignment is pretty compelling. But does such an argument have the credentials to constitute a proof, so that it's uncontroversial that possession of this argument suffices to bestow definiteness on its conclusion? I don't see that Sorensen has motivated that at all. So Sorensen needs to provide an argument that the symmetry principle has some more privileged status, or that there's some intimate but hitherto unrecognized link between intuitively cogent philosophical argument and definiteness. Otherwise all he's done is given us really good reason to think that both S7 and S8 are true.
S1: S1 is not true,
while the following open-pair generates the Yes-No paradox:
S2: S3 is true
S3: S2 is not true.
What groups these together as a family is that there is no consistent assignment of truth-values to these sentences. In contrast there are consistent assignments of truth-values to the truth-teller sentence:
S4: S4 is true,
and to the pair of sentences that give rise to the No-No paradox:
S5: S6 is not true
S6: S5 is not true.
S4 can consistently receive T or F, we're just at a loss to assign one rather than the other. We can't assign S5 and S6 TT or FF, but TF or FT are fine. Problem is, it seems like symmetry considerations seem to compel us to assign matching truth-values to the two sentences. (We can increase the symmetry between the two sentences by presenting the paradox the way Russell does; we have a sheet of white card, with just "The sentence on the other side of this card is not true" printed on each side. The idea is that there's no grounds on which to conjecture that there has been a context-shift or anything that would justify assigning different true-values to the members of the pair.) So although consistency allows TF or FT, symmetry considerations seem to rule those options out.
More recently, in 'A Definite No-No' (In Beall's Liars and Heaps, 2003) Sorensen presents the following pair of sentences:
S7: S8 is not definitely true
S8: S7 is not definitely true.
(Sorensen's presentation attempts to maximize their 'equal-billing' a little more than mine, again to increase the force of symmetry considerations, but that won't matter here; anyone who feels I've loaded the dice against Sorensen can simply look at his presentation on p225 of his paper.)
The assignment FF (or more carefully, assigning not-true to both) looks like it's immediately ruled out by the logic of definiteness; as Sorensen points out, the untruth of one of the pair implies the definite truth of the other. What about the other options? Sorensen writes:
'. . .the symmetry of the two sentences is destabilizing. We have the following feelings: As twins, [S7] and [S8] should have matching truth-values (or should be equally bereft of truth-values). Since we have already ruled out the possibility that both are untruths, we conclude that [S7] and [S8] are both true.
The truth of the pair is no problem in itself. But there is a ironical difficulty in us possessing such a tidy argument [S7] and [S8] are true. The discovery of a proof for a proposition is generally regarded as a sufficient condition for that proposition being definitely true. Since [S7] and [S8] each deny that the other is definitely true, they would each be false if proven true. Strangely, any cogent argument to the effect that both sentences are true must backfire.' (226)
I just don't find this all that plausible. The crucial move in all of this is the equation of 'cogent argument' with 'proof'. The move is particularly clear in the second and third sentences of the quote; Sorensen passes without a ripple from our possession of a 'tidy argument' to the possession of a proof. Now, this might just be written off as a terminological gripe if Sorensen's argument that the only assignment we can make to S7 and S8 is TT really did constitute a proof. But surely that issue turns on the status of the symmetry principle. After all, it's the symmetry principle that rules out the true-untrue and untrue-true assignments.
So what is the status of the symmetry principle? All Sorensen says is that we have a 'feeling' it should hold. I agree; the principle is pretty intuitively compelling, and likewise I find the argument it underwrites that TT is the only available assignment is pretty compelling. But does such an argument have the credentials to constitute a proof, so that it's uncontroversial that possession of this argument suffices to bestow definiteness on its conclusion? I don't see that Sorensen has motivated that at all. So Sorensen needs to provide an argument that the symmetry principle has some more privileged status, or that there's some intimate but hitherto unrecognized link between intuitively cogent philosophical argument and definiteness. Otherwise all he's done is given us really good reason to think that both S7 and S8 are true.
Labels: Philosophy of Logic
# posted by Aidan @ 4:38 PM 
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You accept that if we take on board the symmetry principle, then the only assignment we can make to S7 and S8 is TT.
So, it seems that there is an argument in which the symmetry principle figures as a premise, the conclusion of which is that assignment to S7 and S8.
To be sure, for such an argument to constitute a proof, in a formal sense of that term, we need to make sure that the argument conforms to some set of antecedently accepted rules of inference - for which we have better be able to demonstrate at least soundness.
However, it should also be clear that we rarely possess such security in philosophical disputes. The issue is rather a different one, as you also seem to point out.
Suppose we grant that the argument, in which symmetry is a premise and the assignment in question is the conclusion, is valid - although we may not be in a position to represent that validity formally. Then the question you raise is whether we should be happy with accepting the symmetry-premise just on the basis of its intuitively compelling status. Granted, this is a fragile way of proceeding. But the question is whether we can do better.
It seems to me that this kind of reassurance, meagre though it may be, is, in the vast majority of cases, all that we can aspire to in philosophy.
Of course, that doesn't mean that we cannot have a fruitful discussion about the viability of intutions in philosophical debates etc. But I don't think we can appeal to considerations about the general fragility of intuitions when dealing with particular arguments. If intuitions are fragile, then that is a problem for the majority of philosophical arguments.
So, it seems that there is an argument in which the symmetry principle figures as a premise, the conclusion of which is that assignment to S7 and S8.
To be sure, for such an argument to constitute a proof, in a formal sense of that term, we need to make sure that the argument conforms to some set of antecedently accepted rules of inference - for which we have better be able to demonstrate at least soundness.
However, it should also be clear that we rarely possess such security in philosophical disputes. The issue is rather a different one, as you also seem to point out.
Suppose we grant that the argument, in which symmetry is a premise and the assignment in question is the conclusion, is valid - although we may not be in a position to represent that validity formally. Then the question you raise is whether we should be happy with accepting the symmetry-premise just on the basis of its intuitively compelling status. Granted, this is a fragile way of proceeding. But the question is whether we can do better.
It seems to me that this kind of reassurance, meagre though it may be, is, in the vast majority of cases, all that we can aspire to in philosophy.
Of course, that doesn't mean that we cannot have a fruitful discussion about the viability of intutions in philosophical debates etc. But I don't think we can appeal to considerations about the general fragility of intuitions when dealing with particular arguments. If intuitions are fragile, then that is a problem for the majority of philosophical arguments.
"Then the question you raise is whether we should be happy with accepting the symmetry-premise just on the basis of its intuitively compelling status."
Actually, this wasn't my question at all. I'm quite happy with accepting the symmetry principle on this basis, even if it's perceived as fragile, and even if in lots of cases in philosophical contexts that might be the best we can do, I wouldn't find that overly troubling.
The problem is not that if Sorensen's argument doesn't constitute a proof 'in a formal sense of that term', as you put it, it doesn't give us sufficient grounds to believe its conclusion. In fact, I'm happy to say (and do say in the post) that Sorensen's argument should move us to adopt the conclusion. The point is rather that if his argument doesn't have the status of a proof 'in a formal sense of that term', there's no reason to think our possession of it suffices to bestow *definiteness* on its conclusion. If you read the last paragraph of my post again, you'll see that this was explicitly my concern. The worry is not that if such an argument can't bestow definiteness on its conclusion, this marks some weakness in philosophical argumentation; it's that Sorensen's paradox is not a paradox - the TT option becomes a stable and well-motivated assignment.
The Definite No-No is meant to put pressure on some widely accepted connection between possession of a proof and the definiteness of its conclusion. I'm claiming Sorensen's puzzle does nothing of the sort; he's just offered us goods grounds to believe that both S7 and S8 are true, but not definitely so.
Actually, this wasn't my question at all. I'm quite happy with accepting the symmetry principle on this basis, even if it's perceived as fragile, and even if in lots of cases in philosophical contexts that might be the best we can do, I wouldn't find that overly troubling.
The problem is not that if Sorensen's argument doesn't constitute a proof 'in a formal sense of that term', as you put it, it doesn't give us sufficient grounds to believe its conclusion. In fact, I'm happy to say (and do say in the post) that Sorensen's argument should move us to adopt the conclusion. The point is rather that if his argument doesn't have the status of a proof 'in a formal sense of that term', there's no reason to think our possession of it suffices to bestow *definiteness* on its conclusion. If you read the last paragraph of my post again, you'll see that this was explicitly my concern. The worry is not that if such an argument can't bestow definiteness on its conclusion, this marks some weakness in philosophical argumentation; it's that Sorensen's paradox is not a paradox - the TT option becomes a stable and well-motivated assignment.
The Definite No-No is meant to put pressure on some widely accepted connection between possession of a proof and the definiteness of its conclusion. I'm claiming Sorensen's puzzle does nothing of the sort; he's just offered us goods grounds to believe that both S7 and S8 are true, but not definitely so.
This is interesting. So, if I've understood it correctly, what is at stake is whether or not the reasons given for the TT distribution constitute a proof in the relevant discourse. And the discourse here seems to be one with a logic containing some sort of D-operator. One thing is sure, if all the premises of the arguments were 'definitized', then the conclusion could be definitized as well. However, as it's probably not the case that the symmetry principle is definite, this doesn't follow. In other words, I see no reason why we should conclude that the conclusion in this case is definite.
Yeah, that's right Ole - that's exactly the kind of thing I had in mind when I suggested that Sorensen needs to convince us that the symmetry principle has some more privileged status.
Ok, sorry about that. I see that I missed this point about definiteness. But now that you've pulled the wool from my eyes, I think I agree.
So, does everyone agree that the D-operator distributes across logical consequence? I seem to remember Ole posting something about this.
So, does everyone agree that the D-operator distributes across logical consequence? I seem to remember Ole posting something about this.
Can't we just make the paradox stronger by:
9: There is no good argument that (10) is true.
10: There is no good argument that (9) is true.
Surely either both or neither has a good argument. But if neither does, then they're both true, which seems to constitute a good argument.
9: There is no good argument that (10) is true.
10: There is no good argument that (9) is true.
Surely either both or neither has a good argument. But if neither does, then they're both true, which seems to constitute a good argument.
Kenny,
Sure, I've actually been playing around a little with some of these alternative pairs of sentences. But I take it Sorensen thought there was something particularly philosophically interesting about the Definite No-No; that it put pressure on the connections between proof and definiteness in a way that might bear on the vagueness debate, for example. So what would be more to Sorensen's purposes would be a strengthened version which exploits the proof/definiteness connection, and I haven't seen how to do that, though as always maybe I'm just missing the obvious.
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Sure, I've actually been playing around a little with some of these alternative pairs of sentences. But I take it Sorensen thought there was something particularly philosophically interesting about the Definite No-No; that it put pressure on the connections between proof and definiteness in a way that might bear on the vagueness debate, for example. So what would be more to Sorensen's purposes would be a strengthened version which exploits the proof/definiteness connection, and I haven't seen how to do that, though as always maybe I'm just missing the obvious.
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