🚀 go-pugleaf

I've just come across what looks like an extremely fascinating book.
_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and
John Etchemendy, Oxford University Press, 1987.  It only costs $20;
its ISBN is 0-19-505072-X.  Buy this book.

I will summarize as best I can as to what is in it; it's definitely
on my summer reading list.  The level is that of standard first-year
logic.  (My first impressions include, inter alia, that it gives pre-
cisely what I was asking for in earlier postings about infinite re-
gressions and self-justifying in my disagreements with Paul Torek: viz,
while *some* infinite regressions are harmful, not *all* of them are.)

Saul Kripke, in "Outline of a theory of truth" _The Journal of Phil-
osophy_ 72 (1975), pp 690-716, re-opened the question that had suppos-
edly been answered by Tarski: "what is truth?"  He proposed a theory
that permitted circular reference.  It has not been generally accept-
ed, but the realization that simple schemes in the spirit of Zermelo
for banishing of the Liar paradox are fundamentally unacceptable be-
came widespread.  Variants have been proferred; what B&E present is
a model-theoretic approach to analyzing these theories that permit
circular reference, based on a version of set theory that permits
non-well-founded sets.  The axioms they use, in an intuitive form due
to Aczel, lead to unique sets with any given pattern of abstract cir-
cular reference, eg, there is only one x such that x={x}, only one y
such that y={0,y}, etc.

B&E develop, first a Russellian inspired interpretation of truth,
where propositions are definitive statements about the "real world"
out there, and then second an Austinian inspired interpretation,
where propositions always have context-sensitive features.

An example they use to illustrate this difference concerns "Claire
has the 3 of clubs".  Start by assuming that a fixed moment of time
and a fixed set of unambiguously identified people are being referred
to--this kind of understanding is for ease of speaking, and not a
significant feature of any interpretation.  In their Russellian view,
this statement has an absolute truth value, determined by identifying
Claire and the 3 of clubs.  In their Austinian view, the statement's
truth value can depend on its speaker's understanding of the situa-
tion: if the speaker mistook Emily for Claire, or the deuce for the
three-spot, then the statement is false, even if it turns out that
the real Claire had the real 3 of clubs all along.

In both cases B&E work out a non-well-founded model theory that per-
mits liar-type paradoxes in both situations to be resolved--moreover,
it permits an understanding of *why* the resolution works as it did.

In the Russellian case, the liar paradox becomes false, while the
truth-teller puzzle ("This sentence is true.") becomes model-depen-
dent.  Note that this is not the same as context-sensitive: once
one has identified the "real world" correctly in their Russellian
analysis, one gets a rigid true/false determination.  They find,
however, that the Russellian resolution is unnatural, and can even
define semantically the notion of a proposition being "paradoxical".

In the Austinian case, they give explicit constructions of Liars,
ie, utterers of "This sentence is false", some of whom are telling
the truth, and some of whom are telling a falsehood!  And the same
can be done for other utterers of paradoxical/puzzling sentences.
In particular, they show the relevance of the difference between
negation and denial, ie, between asserting a falsehood and denying
a truth.

They contrast the two views by saying that their Russellian solution
is analogous to the introduction of the set/class distinction into
naive set theory and the proof that the Russell class of sets that
don't contain themselves is the universal class, while their Austin-
ian solution compares with relativizing the Russell class to within
a fixed (not necessarily well-founded) set.  The first just shows
one has left ZFC, while the latter is the standard diagonalizing
technique of construction entirely within ZFC.  In the Austinian
case, they derive Reflection Principles which allow for arbitrary
great generality in scope of reference.

I hope to understand this book by the time Raymond Smullyan's version
comes out!

   "Corollary 23: A sentence "phi" is intrinsically paradox-
    ical in the Russellian semantics just in case "phi" is
    intrinsically deniable, while "not phi" is necessarily
    false.

   "The difference between denial and negation, once pointed
    out, is easy enough to acknowledge, but even easier to
    forget. This is especially true in logic, where the em-
    phasis is place on truth as a property of sentences, and
    denials are relegated to the pragmatic wastebasket. And
    in general, ignoring this distinction does little serious
    harm, any more than ignoring relativistic effects causes
    problems in trips to the supermarket. But at speeds ap-
    proaching that of light, ignoring relativistic effects
    gives paradoxical results. Similarly, in the realm of
    semantics. Corollary 23 points out that, when approach-
    ing sentences like the Liar, we risk paradox if we ig-
    nore the difference between negation and denial."

ucbvax!garnet!weemba	Matthew P Wiener/Brahms Gang/Berkeley CA 94720
   "Logicians, it is said, abhor ambiguity but love paradox."