The Liar lies and snow is white

A consistent theory of truth for semantically closed formal languages

PhD thesis by Andreas Beck, München
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From the preface

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.