[FOM] The liar "revenge"?

W.Taylor at math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Tue Jul 21 11:59:54 EDT 2015

May I suggest it might clarify things a little, or at least stop some
talking at crossed purposes, if people are careful to distinguish
the use of "sentences" and "strings" in this context, with the agreement
being that some string will be called a sentence only if the caller has
reason to believe it truly is one, i.e. is meaningful.

Otherwise, I tend to agree with Arnon's approach.

Quoting Arnon Avron <aa at tau.ac.il>:

> 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.

I agree that this is the key killer point, but should be using the word
"string" in place of "sentence" throughout.

>  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.

I would suggest, however, that the difference is between these two is
not all that significant.  Russell's paradox, in particular, appeared
at a time where sets were still regarded as extensions of intensional
descriptions, these being given in words in natural language, for the most
part.  That being so, they landed us in paradox country for the same
reason as the liar - namely, assuming that a description that seemed
to be meaningful was, in fact, not so on closer inspection.

So, it seems to me that the liar and Russell are very close.
Cantor & Burali-Forti are more purely extensional-set-theoretical cases,
and are easily disposed of in (e.g.) Zermelo's system, therefore.

Bill Taylor.

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