From newshub.ccs.yorku.ca!torn!cs.utexas.edu!zaphod.mps.ohio-state.edu!rpi!psinntp!psinntp!scylla!daryl Tue Nov 24 10:52:07 EST 1992
Article 7646 of comp.ai.philosophy:
Xref: newshub.ccs.yorku.ca comp.ai.philosophy:7646 sci.logic:2315
Newsgroups: comp.ai.philosophy,sci.logic
Path: newshub.ccs.yorku.ca!torn!cs.utexas.edu!zaphod.mps.ohio-state.edu!rpi!psinntp!psinntp!scylla!daryl
>From: daryl@oracorp.com (Daryl McCullough)
Subject: Self-Reference and Paradox (was Re: Human intelligence...)
Message-ID: <1992Nov14.151559.13227@oracorp.com>
Organization: ORA Corporation
Date: Sat, 14 Nov 1992 15:15:59 GMT
Lines: 101

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. The paradigm example, of course, is the liar paradox

     This sentence is false.

Another paradoxical phrase is (I think it's called Richard's paradox):

     The smallest number that cannot be defined in fewer than twenty
     words.

(It seems to define a number, and only takes twelve words to do it,
contradicting the definition of the number.)

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).

The liar paradox need not be paradoxical at all if we
replace the unrestricted notion of "is false" by a more restricted
notion.

First, let me define an operation Sub(x,y) on pairs of strings x and y
as follows: Sub(x,y) is the result of enclosing x in quotations, and
substituting the result for all blanks occurring in string y. For
example, Sub(`Hello',`_ is a greeting in English') yields the true
sentence: ``Hello' is a greeting in English'.

Second, let T be any describable collection of strings which can be
interpreted as true statements. Then, in terms of T, we define the
collection S as follows:

      S = the set of all pairs of strings <x,y> such that Sub(x,y)
          is in T

Now, in terms of S, we can define the diagonalization set D as follows:

      D = the set of all strings x such that <x,x> is *not* in S.

Since we assumed that T was a describable collection, it should be
clear that D is also describable. Then let's look at a particular
string (call it G_0): 
     G_0 = `_ is an element of D'.

(The fact that D is describable means that the name D can be replaced
by the description of D, so that we need not depend on a particular
name for the collection.)

Now let's consider the question of whether string G_0 is an element of
D. That is, is the sentence `G_0 is an element of D' true? Well, by
definition of D, G_0 is an element of D if and only if <G_0, G_0> is
not an element of S, which is true if and only if Sub(G_0,G_0) is
*not* a string in T. What is the string Sub(G_0,G_0)? Well, by
definition of Sub, it is the string G = ``_ is an element of D' is an
element of D' (once again, if you like, all references to D could be
replaced by descriptions of D). Therefore, we get:

     G_0 is an element of D if and only if G is not an element of T.

Expanding G_0 and G gives us:

     `_ is an element of D' is an element of D
     if and only if
     ``_ is an element of D' is an element of D' is not an element of T.

Notice that this implies that

     G expresses a true statement if and only if G is not an element of T

That means that if T is a describable collection of true sentences,
then G expresses a true statement that is not in T. In other words, no
describable collection of statements can contain all true statements
(and only true statements).

In order for statements about truth and falsity to be meaningful, it
must be possible to replace the unrestricted notion of truth by a more
restricted notion, realizing that there are truths that are not
captured by the restricted notion. So, for instance, the sentence

    This sentence is false.

refers to an unrestricted notion of falsity, and is therefore
meaningless. We can replace "false" by a restricted notion of falsity
in two different ways; either we can interpret "_ is false" as either
"_ is not true, according to restricted notion of truth T", or "The
negation of _ is true, according to restricted notion of truth T".

    This sentence is not true, according to T.

is a true statement, but is not captured by the restricted notion T.

    The negation of this sentence is true, according to T.

is a false statement, but its negation is not captured by the restricted
notion T.

Daryl McCullough
ORA Corp.
Ithaca, NY



