# [FOM] The liar paradox

Nik Weaver nweaver at math.wustl.edu
Tue Feb 21 12:54:09 EST 2017

```Arnon Avron wants to handle the liar paradox by distinguishing between
a sentence and the proposition or propositions that various instances of
it express.  This approach obviously invites the "revenge" sentence

(R) No instance of this sentence expresses a true proposition.

A contradiction clearly follows from assuming that any instance of (R)
expresses a true proposition.  But this means that no instance of (R)
expresses a true proposition, and it would then seem to follow that
at least some instances --- for example, the instance I just typed ---
actually do express a true proposition.

Arnon might respond by saying that no instance of (R) expresses *any*
proposition and therefore one cannot deduce (R) from the premise that
no instance of (R) expresses a true proposition.  This was his original
resolution of the standard liar paradox --- and I remind him that in his
original message this was presented as an obviously valid and complete
resolution of the paradox, with no mention of "propositions".  But that
would face the same term substitution difficulty that I explain in my
paper.

I am sure Arnon is aware of this problem and has a response.  (Maybe
involving a non-classical logic?  That seems like the obvious next
step.)  I would be surprised if his response worked, though, because
this type of solution has been thoroughly explored in the liar literature
and it hasn't panned out.  I stand on my claim

> whatever simple idea you have for an easy resolution of the liar
> paradox --- we've tried it, and it doesn't work

and I don't think the "truth is a property of propositions" idea refutes
this.  Really, it's been studied very thoroughly.  It doesn't work.

In any case, if Arnon's idea could be made to work, the proof of this
would be the development of a *formal system* for reasoning about truth
which (1) blocks the paradoxes and (2) allows ordinary unparadoxical
reasoning.  As I said in my paper, there is not now any de facto standard
formal system for reasoning about truth which does this.  (Martin Dowd
interprets me as stating "there is no good theory of truth", which I guess
is okay if "good" means "broadly accepted" and "theory" means "formal
system", but he then starts talking about Kripke and I don't remember
Kripke presenting any formal system.  Kripke's paper outlines a
construction, not a formal system, and not of truth but of "grounded"
truth.  Fefferman tried to formalize Kripke, is that where this discussion