From newshub.ccs.yorku.ca!torn!cs.utexas.edu!sun-barr!ames!agate!stanford.edu!CSD-NewsHost.Stanford.EDU!Sunburn.Stanford.EDU!pratt Tue Nov 24 10:52:08 EST 1992
Article 7648 of comp.ai.philosophy:
Xref: newshub.ccs.yorku.ca comp.ai.philosophy:7648 sci.logic:2316
Newsgroups: comp.ai.philosophy,sci.logic
Path: newshub.ccs.yorku.ca!torn!cs.utexas.edu!sun-barr!ames!agate!stanford.edu!CSD-NewsHost.Stanford.EDU!Sunburn.Stanford.EDU!pratt
>From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Self-Reference and Paradox (was Re: Human intelligence...)
Message-ID: <1992Nov15.060331.3162@CSD-NewsHost.Stanford.EDU>
Sender: news@CSD-NewsHost.Stanford.EDU
Organization: Computer Science Department,  Stanford University.
References: <1992Nov14.151559.13227@oracorp.com>
Date: Sun, 15 Nov 1992 06:03:31 GMT
Lines: 54

In article <1992Nov14.151559.13227@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
>The liar paradox need not be paradoxical at all if we
>replace the unrestricted notion of "is false" by a more restricted
>notion.

This proposal appears implicitly in Scott's model of the untyped lambda
calculus (which admits self-referentiality), and explicitly in work by
Kripke and subsequently Gaifman on truth-valued interpretations of
self-referential sentences.

The simplest instance of the proposal arises when one asks why the
universe does not blink out when the output of an inverter is connected
back to its input.  The answer, at least for those inverters that come
in little plastic sixpacks with 14 pins, is that their inputs and
outputs quickly reach a compromise in the neighborhood of 2.5 volts
before the big bang has had much of a chance to get under way.  Any
explanation of this behavior must clearly go beyond just the two truth
values 0 volts and 5 volts appearing in the explanation of the normal
behavior of inverters.

By adding bottom to otherwise completely flat domains, Scott achieves a
similar effect for self-referential lambda terms.

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".  I vaguely recall some
discussion of notions of undefinedness, e.g. interpretation of the
self-reference as being to an earlier truth value, i.e. I/O delay to
the rescue, leading to an oscillatory interpretation of "undefined" as
an alternation of True and False.  The main point however is that a
simple 3-valued truth system, with X being a less committal truth value
than either False or True, is enough for a model of the liar paradox.

At TARK-1 (Joe Halpern's first conference on Theory and Reasoning about
Knowledge, at Asilomar around 1987) Haim Gaifman proposed a nice
refinement of Kripke's semantics.  According to Kripke once X has crept
in anywhere it is "upwards contagious".  Having assigned X to "This
sentence is false", Kripke then goes on to assign X to "`This sentence
is false' is not true," the evident truth of the sentence
notwithstanding.

Gaifman gave a rule for computing truth of a self-referential sentence
in which the strongly connected components of the sentence were
assigned truth values in topological order.  In the above example the
fragment "This sentence is false" is a strongly connected component
assessed to be X.  Only after making that assignment is the rest of
this example evaluated, at which point it is noticed that X is indeed
different from True, making the whole sentence True rather than X.

These things seem simple in hindsight, but the wheels of knowledge seem
to grind at least as slowly for the obvious as for the deep.
-- 
Vaughan Pratt	  		A fallacy is worth a thousand steps.


