Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: A possible solution to the Liar Paradox
Replies: 0

 Scott Posts: 66 Registered: 2/2/07
A possible solution to the Liar Paradox
Posted: Nov 24, 2011 10:53 AM

I think I have found the basis for a solution to the Liar Paradox that
stands as an alternative to Tarski's truth-hierarchy, and which allows
self-reference. It rests upon a disambiguation of certain terms in
mathematical logic.

First, I distinguish a *concrete* statement, which is a particular
instance of a statement such as an individual utterance, from an
*abstract* statement, which is the very idea of the statement apart
from any of its instantiations in writings or utterance. An example of
a concrete statement is a statement as it appears on a particular
sheet of paper in someone's particular handwriting, or perhaps in a
particular coding system (e.g., Goedel-numbering), as distinguished
from its abstract referent in what we might call "statement space."

This disambiguation immediately results in another ambiguity --
namely, what "this statement" refers to in the sentence, "This
statement is not true" (S). Does it refer to the *concrete* instance
of that statement, or to the abstract statement (whatever it may be)?

Let us take the most obvious example and consider "this statement" to
refer to the concrete statement, i.e., the statement as it appears on
paper as an instance of a statement. We must then modify our
conception of truth to apply to such statements. The usual paradox is
that if that statement is true, then it must be not true, and if it is
not true, then it must be true. But it cannot take on two values at
once!

To solve this, I propose that we simpy make an exception in our
concept of truth for the Liar statement, and call the statement "not
true." We then deny to the concept of truth the ability to infer the
truth of the Liar statement from its grammatical syntax. So if someone
offers the usual objection, "That's just what S is *saying*!" we
respond that even though the intensional subject of "this statement"
satisfies the intensional predicate "is not true," we cannot define it
to have more than one truth value. Again, all we need do is make an
exception in our notion of truth for the Liar statement. Additionally,
we may clarify our response by adding that "what S is saying" remains
ambiguous until we specify whether S refers to a concrete statement or
an abstract statement.

We may then proceed to write concrete sentence T, "Statement S is not
true." In this way we evade self-reference in T and make a true
statement about S. Now, statement S is still not true, but since T is,
concretely speaking, a different statement, it is allowed to be true

What do you think of this solution? Can you think of any "revenge"
paradoxes that would leave my solution in jeopardy? Has anything like
this solution been attempted before? If so, are there any good
references? I'm looking particularly for a discussion of concrete and
abstract statements, however they are so called.

I also accept better suggestions on what to name these concepts.

(crossposted to sci.logic)