[FOM] The liar "revenge"?

Parsons, Charles parsons2 at fas.harvard.edu
Mon Jul 20 21:03:57 EDT 2015

On Jul 20, 2015, at 5:05 PM, Arnon Avron <aa at tau.ac.il> wrote:

> Making an argument much longer, with several steps, does not
> make it any more compelling, if the crucial point
> remains as weak as before. In the case of Cole's reply,
> I skip all the five first steps and go directly to the
> sixth one:
>> 6. About to feel content with this "solution" to the paradox, an
>> unfortunate observation is made. A meaningless sentence is not true, in the
>> sense that it is not the case that a meaningless sentence makes an
>> assertion that is true. Thus, it seems the liar sentence is true after all,
>> since it asserts that a particular meaningless string of characters is not
>> true.
> So again I read that a meaningless sentence asserts something, which is
> a contradiction in terms. A meaningless sentence does not assert
> anything, and does not say anything. Period. This is the meaning 
> of "meaningless".
> Well, if people prefer that instead of saying that the liar sentence
> is  meaningless I'll say that it does not assert anything,
> (or that it does not say anything) then fine - as long as the
> answer would not be again something of the type:
> "if it does not assert anything then it is true, because this
> is precisely what it asserts"... With such a logic
> I simply cannot cope.
> Let me add here the following comment. It seems to me that there are
> a lot of people who simply *want* to keep the liar paradox alive,
> and to see it as an unbreakable paradox.  I see
> little point in arguing with them if all they can do is to repeat
> arguments that like the "proofs" of the existence of god,
> convince only those who want to be "convinced" (and in fact 
> are convinced of what the argument "proves" well before hearing it...). 
> But those people should better be aware that any conclusion
> they reach from the "paradox" will be completely irrelevant to
> mathematicians who do not see a real problem with the liar - that is,
> practically all mathematicians. Indeed, the liar is known 
> for two thousands years or so, and (as far as I know) 
> mathematicians never really care about it. The story was completely
> different when they faced Russel's paradox (or the other
> "logical paradoxes") - and for good reasons. 
> Arnon
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Those who have contributed to this thread evidently do not know that there is a vast and sophisticated literature on the Liar and similar paradoxes. I have put my own two cents in on the issues, but that was forty years ago ("The liar paradox," Journal of Philosophical Logic 3 (1974), 381-412). I would still recommend the collection Recent Essays on Truth and the Liar Paradox, ed. Robert L. Martin (Oxford UP, 1984). It includes my paper (to be sure with an abbreviated version of a Postscript that takes account of some later literature) and, more important, the classic paper of Saul Kripke, "Outline of a theory of truth" (Originally Journal of Philosophy 72 (1975)). I would also recommend the writings of Michael Glanzberg (references on his web site, Northwestern philosophy) and the book Saving Truth from Paradox by Hartry Field (Oxford, about 2008).

About the idea that "the liar sentence is meaningless," whether a sentence gives rise to paradox can depend on quite contingent facts. That's even true of one of the ancient versions, the Epimenides. Epimenides is said to have said, "All Cretans are liars." This is typically read to mean, "Every other statement made by Cretan s false." But paradox only arises if every _other_ statement made by a Cretan is false. I don't see the case for declaring Epimenides' sentence meaningless, particularly since  if some Cretan sometime had told the truth, it is simply false. This was pointed out by A. N. Prior in 1958, but I would guess that it was known to the ancients. Similar contingency obtains for examples given by me and by Kripke, and if I remember right for examples in Tarski's "The semantic conception of truth." (I don't think he made anything of the point.) I think the point is now a commonplace.

It's another matter whether mathematicians should be interested in these problems. They have given rise to some nice mathematical logic, but I would not want to argue that (at least after the clarifications in the twenty years after Russell's paradox and then Tarski's work on truth and definability) that the Liar and related problems are a problem for the foundations of mathematics.

Charles Parsons

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