The Liar

If you have a definite method for working out the meaning of an expression, and the method is explained in the language of the expression you want to decode, then the method is self-referential.  Also, it is self-referential in a specific way:  its scope of adjudication, its normative range, includes any accounts that can be given of itself.

This makes it ambiguous, since it cannot reliably decode itself.

If I have a 'method X' for determing the meaning of any statement S, then I can say:
C:  "Method X, applied to 'S', produces meaning M"

I can ask of a method X whether it is correct, complete, and consistent.

I will set correct aside for the moment, since that question seems to imply the availablity of a 'super-method', when we have already specified X as this method.

With respect to completeness, we seem to be balked by the ambiguity problem.  When we apply X to C we get an indefinite regress, because we must apply X to 'X' (as an element of C).  We have to know what 'X' means before we can apply method X.

But if we are balked by ambiguity in this instance, then we should also be balked by the following case:

R(i):  "I know what 'R(i)' means"

This doesn't, on the face of it, look ambiguous - but any method of disambiguating it will get caught in the loop of disambiguating 'R(i)'.

(The case of R(ia) "I do not know what 'R(1a)' means" shifts the puzzle from 'R(1a)' 'I do not know', suggesting a link between traditional semantic paradoxes and so called 'Moorean' paradoxes.)
 
Similarly:

R(ii) "No definite method can disambiguate the meaning of 'R(ii)'"

R(ii) seems unambiguous.  If it is true, then its unambiguity cannot be attributed to any method of disambiguation.  If it is false, then some definite method can disambiguate it - but then any such method must get caught in the loop of disambiguating 'R(ii)'.

Coming back briefly to the correctness of any proposed disambiguation method:  The absence of such a method cannot, of course, be grounds for believing that we do not, in general, know what we mean when we speak.  Otherwise we wouldn't know what we meant when we tried to articulate such a position.

Demonstrating either the meaningfullness or the meaninglessness of the 'Liar' paradox would require a theory of meaning (to ground the demonstration).

However, the more catastrophic consequences of the 'Liar' depend exactly upon the presumption that meaning can be attributed methodically.  So we seem to have a dilemma - either it is catastrophic, or it cannot be shown to be meaningful or meaningless.

(Kripke believes he has a kind of solution to this problem?  It is arcane, and depends upon further fundamentals which - although they may also be fundamentals of mathematics - do not have a special claim in this context.  A hinge in a space of infinite dimensions can still only be mapped from 'outside' that space, though it must be located 'within' it.  Being able to find something and knowing where it is may seem to amount to the same thing, until one wants to construct a formal account of informal methods.  A map is not a list of methods.  [Is Kripke's use of Cantor a solution for Kripke or a problem for Cantor?])
 
Perhaps we can set the paradox aside as a curiosity arising from our partial and incomplete attempts to fully systematise meaning and truth.

We can even deal with it pragmatically, or contextually, if we like - particulary since there is no context in which it arises outside of quotation marks.  Even in philosophical conversation.

Here are some options:

• It is meaningless.
• It is meaningful, but its meaning cannot be calculated.
• It is meaningful, but neither true nor false. (Kripke?)
• It is meaningful, but both true and false. (Other desperate souls)
• It signposts a boundary to algorithmic meaning production which is more restrictive than the boundary of meaningfullness generally (Sensible people)

We will never calculate a Gödel number, and we will never use the liar outside of quotes, so perhaps its a matter of taste.  Although I hope not.