Monday, June 7, 2010

A Revenge Paradox For Dialetheist Solutions To Curry

(Update: Never mind. See the comments.)

Dialetheists like Graham Priest and JC Beall think that some sentences are both true and false, but cleave to classical orthodoxy to the extent of continuing to insist that every sentence is either true or false. (The latter assumption is, in fact, crucial to the derivation of contradictions from the paradoxes.) Thus, Liar sentences get classified as both true and false, and the resulting contradictions are contained by rejecting classical logic in favor of paraconsistent logic. So far, so good.

For obvious reasons, this solution doesn't help with "Curry" sentences, like S1:

S1: If S1 is true, then everything is true.

If S1 gets classified as either (just) true or both true and false, triviality ensues. As such, it had best get classified as (just) false. Of course, this by itself doesn't get around the paradox, since, for familiar reasons, triviality ensues from he mere statement of S1's disquotational truth conditions. That is, however, beside the point for the purposes of this post. The important point is that, for a dialetheist who accepts that every sentence is either true or false but wishes to avoid triviality, the only option for the truth-value of S1 is that it is (just) false.

So far, so good. What, however, about S2?

S2: If S2 is either true or false, then everything is true.

Discuss.

12 comments:

Emil said...

I don't know. How would classical logic handle it? Claiming it is meaningless?

Ben said...

Well, presumably *any* classical solution to normal Currys, or normal Liars for that matter, would work just as well (or just as badly) as a solution to this version of Curry. It doesn't seem to pose any special problem for *any* classical solution--whether meaninglessness-based, or T-Schema-instance-denying, or Tarksi-ish, or *whatever.*

Similarly, those non-classical solutions that revolve around rejecting certain instances of Excluded Middle (or even just Bivalence) would seem to work just as well (or just as badly) here as they would for normal Currys or normal Liars.

The interesting thing is that normal dialetheist solutions seem to run into a problem in this case that they don't run into in normal cases. Their normal approach to Curry sentences is to block the inference to triviality by both (a) fiddling with the conditionals so as to avoid the inference from the T-Schema instance to triviality, and (b) saying that the sentence is (just) false, rather than (just) true or both true and false. In this case, (b) delivers triviality. In fact, it looks like you get triviality on *all* of the conceptual options available to the dialetheist get you triviality. If you expand their range of options to say, for example, that *in this case* they can avoid either saying that it's true or that it isn't, then why not do the same for normal Liars?

(After all, the *main argument* for dialetheism is the argument from the Liar Paradox.)

Ben said...

To expand, a little, here's a sample of classical solutions to the Liar:

(1) You can deny that Liars are meaningful.

(2) You can deny that Liars have their apparent truth conditions--this is the line, e.g. Gil Harman takes in "Change in View", that we should just deny certain instances of the T-Schema.

(3) There's Hofweber's solution, where he accepts all the premises of the argument to a contradiction from Liar, or directly to triviality from Curry, and accepts that the inferences are valid, and even accepts that "valid inferences are truth-preserving", but argues that this is merely a generic statement, like "bears are dangerous," not a universally quantified statement. The truth of "bears are dangerous" is compatible with a few old toothless bears not being dangerous, and the truth of "valid inferences are truth-preserving" is compatible with a few inferences from true premises being non-truth-preserving, which we know because of the impossibility of their conclusions being true.

(4) There's Doug Patterson's view according to which the paradoxes show that speakers of natural language are in the thrall of a pervasive semantic illusion.

(5) There's the Barwise and Etchemendy view according to which propositions are necessarily about situations. For example, if you walk in on a card game, and you think you see your friend Jane holding the 3 of Clubs, but really you're mistaking her for her identical twin sister Janet, whose the one holding the 3 of Clubs in this game, your statement "Jane is holding the 3 of Clubs" is false, *even if,* by some massive coincidence, Jane is holding the 3 of Clubs at some other game across town, because your statement "Jane is holding the 3 of Clubs" is made true or false by the specific situation its indexed to, which is the card game you're looking at. When they come to the Liar, Barwise and Etchemendy argue that the situations to which Liars are indexed can never include *their own* truth-value (because if so, inconsistency would result). Thus, Liars are *just false*, not contradictory, since nothing in the situations they're indexed to makes them true.

....and there are many, many more.

One can doubt for all sorts of reasons that any given classical solution on that list is *plausible,* well-motivated, non-ad-hoc or whatever, but the point for the moment is that any one of them *works* as a solution to the version of Curry discussed in this point. The standard non-trivialist dialetheist line on the Liar and Curry, however, gets you triviality here.

Emil said...

Very interesting, Ben.

Still, one could still accept the argument, and accept triviality but argue that we ought to ignore inferences that utilize Curry sentences, Liar sentences and the like.
The analogy being: In mathematics one can sometimes derive a negative number as well as a positive number as the solution (when taking the square root for instance). In these cases we know to ignore the negative numbers because they are irrelevant to reality but there is nothing in the mathematics that is wrong with them. Could it not be similar with with logic? We ought to ignore certain inferences even though they are valid. Think of the example of finding out how long the sides of a square are. Given the area is 16, and we know that then area = a, where a is the length of the sides. If we do a square root on 16 we get 4 and -4. We then proceed to ignore -4 and use 4.

Such analogies are always tricky though.

John Jones said...

You said -
$ The sentence marked with a dollar sign is either false or meaningless.

I ask -
Which sentence?

My criticism is that animism ought not to be practiced in logic. A sentence does not have the power to point itself out, nor can meaning be physically placed, nor does a sentence have an autonomy that is independent of meaning.

If we assume these curio's (and how could logic prevent us?), then yes, we get the usual round of odd looking paradoxes. It is unfortunate that all logic practised today is animistic and transcendentally real.

Ben said...

I like "animism ought not to be practiced in logic" as a turn of phrase. Reminds me pleasantly of a lot of Quine's rhetorical flourishes.

That said, I'm not sure I see the gist of the solution here. You say:

"A sentence does not have the power to point itself out...."

Now, the first obvious objection is that there seem to be cases of clearly meaningful, unproblematically true or unproblematically false sentences that do just that, like:

"This sentence is six words long."

or

"This sentence is twelve words long."

Do these sentences succeed in combining use and mention? They certainly *seem* to.

The second problem is that I'm not sure what the motivation is for saying that sentences can't point themselves out. Given that sentences can clearly point *one another* out, what's the special problem with *self*-reference?

The third problem is what to say about sentences that seem to succeed in talking about themselves. Are these sentences simply meaningless? Or do they actually refer to sentences other than themselves, or....? The first answer leads you to contradiction given "this sentence is false or meaningless" and the second answer seems a bit mysterious. For example, in a sentence like:

**The sentence marked with "**" in Ben's response to John about animism in logic does not succeed in pointing itself out.

.....if "**" doesn't refer to itself, what does it refer to? One could, I suppose, describe this as a reference failure, but:

(a) That seems a bit ad hoc, given that, if in a different sentence we used exactly the same words to refer to "**", presumably you'd grant that we succeeded in referring to it,

&

(b) Depending on your view of the semantic status of reference failures, they might end up being false, and there's something very strange about saying that "this sentence is false" is false without saying that it's true. Not that you couldn't necessarily tell a story on which you got that result--see the discussion of Barwise and Etchemendy, above--but it is a bit counter-intuitive.

L33tminion said...

S2 wouldn't be meaningless or paradoxical under classical logic. Assuming S2 is false is neither self-contradictory nor problematic, it cleanly resolves to a false statement of the form "if T then F".

S1 is problematic under classical logic. If you assume it's false, then it resolves to a statement of the form "if F than F", which is true, which is a contradiction. Unlike the Liar's Paradox, there's nothing self-contradictory about assuming that S1 is true. Thus, I'd think S1 would be more of a problem for classical logic than the Liar's Paradox.

I see two ways a paraconsistent logic could work around that:

1. Say that S1 is neither true nor false, or say that it is nonsense. (Which is a boring way of dealing with paradoxes, at least.)

2. Say that "if F than F" is neither true nor false in general, at which point you can simply say that S1 is false.

If I knew more about paraconsistent logics, I'd be better able to evaluate whether 2 makes any sense. But I find that appealing, it seems counter-intuitive that sentences like "if the sky is green, the moon is made out of cheese" should be classified as true, for example. (Note that I know almost nothing about paraconsistent logic and am trying to puzzle this through with just a bit of boolean algebra.)

Ben said...

L33tminion,

Good points all. Actually, re-reading this post a few days later, I have no idea why I thought S2 was a problem for *anyone.* There's no reason the dialetheist, or anyone else, couldn't just say that the conditional was *false* and hence that the antecedent was true but that the consequent was false.

Granted, there might be more general issues one could raise here in terms of how to *justify* that diagnosis in non-ad-hoc ways, etc., but those issues don't look any more severe with regard to S2 than they do with any other Curry-ish sentence. Again, in retrospect, I don't know what I was thinking here, but I guess that's one of the best arguments for blogging, that it gives you a chance to try out your half-baked thoughts in the light of day to, if necessary, get knocked down before you can make the same mistakes anywhere more important. Gah.

John Jones said...

Ben,

I can only repeat, with different emphases, what I have already written. The fact that the logic of paradoxes of this type are built on spatio-temporal-dependent meanings makes the whole paradox investigative enterprise deeply suspect.

But there are other unusual, even bizarre moves that are made in the liar pardoxes that are made in the name of logic.

One of these, as I said, is the idea that a sentence is stand-alone, independent of meaning. Another is the fact that sentences in these paradoxes switch, unannounced, from ostensive devices or signs to syntactically evidenced meaningless sentential forms or skeletons.
Another oddity imposed on logic is the idea that the syntactic form or template of a sentence can be true or false. It can't.

There are more fundamental issues at stake here which I won't go into as it is work in progress. But I will say that logic isn't only about syntax and semantics. Logical systems can be complete and identical in both semantics and syntax, and yet remain mutually exclusive. The liar paradoxes emerge when we consider only semantics and syntax.

Jason Zarri said...

Hi John,

I know your post wasn't addressed to me, but I have a question. And before I say anything else, let me apologize if I've misunderstood the points you were trying to make.

You said:

"One of these, as I said, is the idea that a sentence is stand-alone, independent of meaning. Another is the fact that sentences in these paradoxes switch, unannounced, from ostensive devices or signs to syntactically evidenced meaningless sentential forms or skeletons.
Another oddity imposed on logic is the idea that the syntactic form or template of a sentence can be true or false. It can't."

Suppose you walk into Humanities 101 and see the following sentence written on the blackboard:

(*) At least one proposition expressed by a sentence written on the blackboard in Humanities 101 is not true.

Suppose (*) is the only sentence written on the blackboard. Granting that, if (*) expresses any proposition at all, it presumably expresses the following proposition (using ALL CAPS for propositions):

(1) AT LEAST ONE PROPOSITION EXPRESSED BY A SENTENCE WRITTEN ON THE BLACKBOARD IN HUMANITIES 101 IS NOT TRUE.

Given the semantics of English, it seems that (1) is the _only_ proposition expressed by (*).

(Notice also that since (1) is a _proposition_ it would retain its identity even if a different sentence had been written on the blackboard of Humanities 101. The same would be true if there had been more than one sentence written there, or none at all.)

Now for my question: Is the proposition which I have called (1) true, or is it not true? Given that (*) expresses (1), and that (*) is the only sentence written on the blackboard in Humanities 101, and that (1) is the _only_ proposition expressed by (*), it seems (1) must be both true and not true. First, if (1) is true, it follows that at least one proposition expressed by a sentence written on the blackboard in Humanities 101 is not true. Since (1) is the one and only proposition expressed by (*), it follows that (1) is not true. So if (1) is true, it isn't. Contradiction. Assume then that (1) is not true. In that case, it is _not_ true that at least one proposition expressed by a sentence written on the blackboard in Humanities 101 is not true. Since (*), the one and only sentence written on the blackboard in Humanities 101, expresses one and only one proposition, namely (1), it follows that (1) must be _true_. So if (1) isn't true, it is. Again we have a contradiction.

In deriving this contradiction, I have not used any "ostensive devices" or "syntactically evidenced meaningless sentential forms or skeletons". There are no ostensive devices in (*); it involves quantification, reference, predication, and that's it. It is not directly self-referential. And had another sentence been written on the blackboard in Humanities 101--such as:

(@) The Earth has more than one moon

-- no paradox would have arisen, for (@) expresses a false proposition.

Nor have I assumed that "the syntactic form or template of a sentence can be true or false". On the contrary, I have distinguished between (*) and the proposition it expresses. It is the _proposition_ expressed by (*), namely (1), that appears to be both true and false. So whether (*) is true or not true (or both) on its own, or in a derivative sense, or not at all, we still have a paradox. The requirements are just that (*) is the one and only sentence written on the blackboard in Humanities 101, and that (1) is the one and only proposition it expresses. As long as (*) means what it seems to mean--and I see no reason to doubt that--we have not diagnosed what is really going wrong in the Liar paradox.

John Jones said...

Jason Zarri said: "Suppose you walk into Humanities 101 and see the following sentence written on the blackboard:
(*) At least one proposition expressed by a sentence written on the blackboard in Humanities 101 is not true."
John Jones replies: Straightaway we find or imagine a set of marks that switch from being a sentential form to an ostensive device: and then we claim that there is a relationship between them, a claim presumably made on the grounds that they are both derived from the same spatio-temporally evidenced set of marks! How did all this happen? How did we allow a set of empty marks to behave in this independent way? Animism creeps in at this point. There are no logical rules for these transitions of form. We are playing fast and loose with a number of issues, right from the off. Does this undercut the rationale behind the standard analysis of (*)? Maybe, but to continue..

Jason Zarri said: There are no ostensive devices in (*); it involves quantification, reference, predication, and that's it. It is not directly self-referential.
John Jones replies: Are you sure it isn't an ostensive device? Perhaps I am misconstruing the nature of reference. I am claiming that as a referential device, the device (*) is spatio-temporally ostensive because (we are told to presume) it asks us to look for meaning at a spatio-temporal position (there are also other problems here to do with the self-reidentification of position..). In fact, all we have at that position is a set of empty marks, and not a sentence. But if we declare that a sentence, with its meaning, IS found at that position then we plunge into logical animism. The particular type of animism is, of course, human-derived. We dine among logical cannibals.

see rest of reply (too many letters for one post)

John Jones said...

Jason Zarri wrote: And had another sentence been written on the blackboard in Humanities 101--such as:
(@) The Earth has more than one moon
-- no paradox would have arisen, for (@) expresses a false proposition.
Nor have I assumed that "the syntactic form or template of a sentence can be true or false". On the contrary, I have distinguished between (*) and the proposition it expresses.
John Jones replies: Yes, I saw this coming and allowed for it. (*) isn't necessarily a sentence. It is a set of marks that can be used in two different ways, at least. If the empty marks are read as a sentence then its meaning is straightforward. If it is also read as a spatio-temporally indicative ostension then meaning becomes spatio-temporally placed. That is animism. But if we reject animism then we are left without meaning, without a sentence and hence have only the ostensive device. Thus, if we reject animism then (*) reduces to a set of empty marks, that can be neither true nor false. But I don't see truth or falsity as bearing upon the essential nature of the problem of these paradoxes.
You also spoke of propositions as having an identity. Is there a logical precedent for that or is it, as I suspect, a tradition among logicians?
Jason Zarri wrote: It is the _proposition_ expressed by (*), namely (1), that appears to be both true and false. So whether (*) is true or not true (or both) on its own, or in a derivative sense, or not at all, we still have a paradox. The requirements are just that (*) is the one and only sentence written on the blackboard in Humanities 101, and that (1) is the one and only proposition it expresses. As long as (*) means what it seems to mean--and I see no reason to doubt that--we have not diagnosed what is really going wrong in the Liar paradox. June 11, 2010 7:44 PM

(John Jones wrote: I hope to have suggested other possibilities that defuse the paradox. More work needs to be done and I haven't explored here the single source that I consider to be the roots of the problem of these paradoxes. I am worried about copyright. This is work in progress and I am concerned about saying too much. If anything I have said appears original then please quote name and source if used elsewhere. I know myself how one can pick up an idea and years later forget where one read it and think it is one's own)