FOM: New solution to the Liar paradox
Till Mossakowski
till at tzi.de
Fri Dec 7 10:24:37 EST 2001
A new solution to the Liar paradox has been
worked out in the PhD thesis of Andreas Beck
THE LIAR LIES AND SNOW IS WHITE
A consistent theory of truth for semantically
closed formal languages
It is available in rich text format under
http://www.tzi.de/~till/Beck/
Till Mossakowski
>From the introduction:
The content of this monograph is the formal construction of a
consistent theory of truth for a semantically closed language. The
concept "semantic closure" is understood as the closure of the
language relative to the assignment of truth values to its
expressions, i.e. we call a language semantically closed if (i) all
expressions of the language possess a truth value and (ii) each of
these truth value assignments can be expressed within the
language. This concretion of the concept of semantic closure is
necessary because according to the original very general concept used
by Tarski (1935, 1944) even the semantic closure of natural languages
is doubtful.
The central obstacle in the way of a consistent truth theory for
semantically closed languages is the evaluation of self-referential
sentences. In the past, there have been several interesting but
unsuccessful attempts to overcome this obstacle by means of truth gaps
(e.g. Kripke (1975), Martin and Woodruff (1975)), unstable truth
values (e.g. Gupta (1982), Herzberger (1982), Belnup (1982)) or
situation-dependent truth (e.g. Barwise and Etchemendy (1987)).
In the first part of this monograph, I will argue that these
approaches fail because their formal truth evaluation systems, though
different one from another, have in common that they evaluate
sentences by means of a reductionist inductive process. I argue that
no such process can formalize the way we evaluate sentences
intuitively, because intuition works holistically. In particular,
self-referential sentences (like those of Anil Gupta's Puzzle or Liar
sentences) are intuitively evaluated through holistic consideration of
the structure of the referential network in which they are embedded.
In consonance with this philosophy, the semantics developed in this
monograph defines truth holistically using graphs that have sentences
as their nodes and referential relationships among sentences as their
paths. By making the structure of a graph determine the interpretation
of the sentences constituting its nodes, it becomes possible to
consider all the sentences in a self-referential cycle simultaneously,
and hence to represent formally the holistic truth evaluation
procedure applied to such cases by our intuition.
It thus turns out that the problem of how to evaluate self-referential
sentences can be solved both philosophically and formally by using a
concept of sentence that treats two sentences as different if they
differ as regards the referential structures of their corresponding
graphs (even if the sentences have the same content). In particular,
the Liar paradox turns out to be caused by arbitrary identification of
sentences with different referential structures.
The concepts applied in this book are both philosophically and
mathematically traditional. As in situation theory, it is propositions
that are the bearers of truth; and in agreement with classical
approaches, the concept of truth is described within a
correspondence-theoretical framework. The mathematical representation
of propositions is based on the notion of non-well-founded sets
developed by Peter Aczel (1988). It should be noted, however, that no
particular mathematical background is necessary in order to understand
the theory.
The theory presented is two-valued, i.e. the distinction 'true versus
false' is identical to the distinction 'true versus not true'. The
decision in favour of a two-valued theory is based on the pragmatic
principle that additional truth values should never be introduced
without emergency; two-valuedness is not a philosophical
requirement. If strong negation in the sense of Ulrich Blau (1978)
were added to the formal language considered in this work, then the
truth theory would extend canonically to a three-valued one. For the
present discussion of semantic closure, however, it suffices to
consider a two-valued theory.
--
Till Mossakowski Phone +49-421-218-4683
Dept. of Computer Science Fax +49-421-218-3054
University of Bremen till at tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
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