🚀 go-pugleaf

From daryl@oracorp.com (Daryl McCullough)
>In comp.ai.philosophy, we were discussing the meaningfulness of
>self-referential sentences, and I noticed that many people seem to
>think that such sentences are inherently invalid, or meaningless, or
>paradoxical.
>In my opinion, the problem with such sentences are not with their
>self-referential character, but with their use of an unrestricted
>notion of truth (or falsity).

Certainly the paradox is not due solely to self-reference.  There is
nothing paradoxical about the self referring sentence; "This sentence
has five words".  Your appeal to a restricted notion of truth is
essentially Tarski's solution.

From pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
>All I know of Kripke's work is a talk I heard by a third party about a
>semantic proposal by Kripke for a logic of Per Martin-Loef.  As I
>understood it Kripke had suggested interpreting "This sentence is
>false" as having the truth value "undefined"

It's been a long time since I read Kripke's paper on truth, but as
I remember, his basic way of assigning a truth value is as follows:

	1.  Assign truth values to simple sentences (e. g. "Snow is white"
	    is assigned true and "Snow is black" is assigned false).
	2.  Assign truth values that assert the truth or falsity of
	    sentences assigned truth values in the previous step (e. g.,
	    "`Snow is white' is true" is assigned true, "`Snow is white' is
	    false" is assigned false, etc.).
	3.  Continue.

This process produces a least fixed point.  A sentence is true if it
is assigned true in that fixed point, false if it is assigned false
in that fixed point, and ungrounded otherwise.  Note that all paradoxical
sentences are ungrounded.  However, some ungrounded sentences may not
be paradoxical, e. g., "This sentence is true".  A sentence is paradoxical
only if it cannot be assigned a truth value at any fixed point.

The difference between this solution and Tarski's is that whereas
Tarski has a separate, but completely defined, notion of truth at each
linguistic level of a language, Kripke has a single, but not completely
defined, notion of truth for the entire language.
 
John McLean