[FOM] The liar "revenge"?

[Ran Lanzet]

Dear Arnon,

I'm glad you raised this point. I share your doubts about the alleged
revenge in the case of the ordinary liar sentence, and will be glad to see a
convincing formulation of the revenge argument. In the meantime, I have a
question regarding variants of the paradox that seem (to me) to be more
bothering -- e.g. Yablo's paradox. 

In the case of the original liar, you may be able to say that the
(allegedly) paradoxical sentence-token (call it L) is meaningless, and
perhaps even express this using a sentence-token equiform with L. You may
then be able to say something about L that will convince us that that token
fails to have any meaning, while yours does not. 

But consider Yablo's paradox, which can be put as follows: 

	S1. For every (natural) k > 1, it is not the case that sentence Sk
expresses a truth.
	S2. For every k > 2, it is not the case that sentence Sk expresses a
truth.
	S1. For every k > 3, it is not the case that sentence Sk expresses a
truth.
	...

Consider any n and assume for contradiction that Sn expresses a truth. Then
none of the sentences Sk with k > n expresses a truth. Hence, none of the
sentences Sk with k > n + 1 expresses a truth. So Sn+1 expresses a truth.
Hence, Sn does not express a truth, contradicting the hypothesis. We
conclude that none of the given sentences expresses a truth. But then, none
of the sentences Sk with, say, k > 17 expresses a truth. So S17 expresses a
truth, a contradiction.

At least on the face of it, this paradox is harder to dismiss than the
original liar. For, if you classify all the sentences Sk as meaningless,
then you seem to be committed to the following:

	S0. For every k > 0, it is not the case that sentence Sk expresses a
truth.

But why should we accept your S0 as true (and thus: as meaningful) any more
than we accept, say, S1?
Do you have a quick way of dispensing with this paradox? 

Cheers,
Ran

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Arnon Avron
Sent: Sunday, July 19, 2015 22:41
To: fom at cs.nyu.edu
Cc: nweaver at math.wustl.edu; Arnon Avron
Subject: [FOM] The liar "revenge"?


I started today to read Weaver's new book "Truth & Assertibility", and
already at the very first page I came across an argument that amazed me. It
is not that it was the first time that I encounter that argument, but it is
the first time I see it taken seriously by someone that I really respect (I
hope that the rest of the book does not depend on it!). It runs as follows:

"The first impression most people have about the liar sentence is that it is
completely meaningless. But surely a meaningless sentence cannot also be
true. So if it is meaningless then in particular it is not true, and that is
just what it says of itself... which would make it true. We have reached a
contradiction again."

Well, to me it seems that Weaver could have shorten the "argument" by saying
that if the liar sentence is meaningless, then since it is meaningful we
have reached a contradiction again...
Surely a meaningless sentence cannot say anything about anything, in
particular not about itself (or anything else). So relying on what "it says
of itself" depends on taking for granted that it is meaningful. Therefore
the above  argument is hopelessly circular!

I do wonder now if I am missing something here, and if so - what can it
possibly be.

Needless to say, for me the "liar sentences" of all types are indeed
completely  meaningless, which is why I was never bothered by them (in
contrast to the so-called logical paradoxes).

Arnon Avron

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