Yaroslav Shramko

Department of Philosophy
Kryvyi Rih National University
Ukraine

Heinrich Wansing

Department of Philosophy II
Logic and Epistemology
University of Bochum
Germany

This tutorial  provides a detailed introduction  into the conception of truth values, an important notion of modern logical semantics and philosophy of logic, explicitly introduced by Gottlob Frege. Frege conceived this notion as a natural component  of his language analysis where sentences, being saturated expres- sions, are interpreted as a special kind of names referring to a special kind of objects: the True (das Wahre) and the False (das Falsche). These are essentially the truth  values of classical logic, which obey the principle of bivalence saying that there may exist only two distinct logical values.
Truth  values have been put to quite different uses in philosophy and logic and have been characterized,  for example, as:

• primitive  abstract objects denoted by sentences  in natural  and formal languages,

• abstract entities hypostatized as the equivalence  classes of sentences,

• what is aimed at in judgements,

• values indicating the degree of truth  of sentences,

• entities that can be used to explain the vagueness of concepts,

• values that are preserved in valid inferences,

• values that convey information concerning a given proposition.

Depending on their particular use, truth values can be treated as unanalyzed, as defined, as unstructured, or as structured entities.  Moreover, the classical conception of truth values can be developed further and generalized in various ways.   One way is to give up the principle of bivalence, and to proceed to many-valued logics dealing with more than two truth  values. Another way is to generalize the very notion of a truth  value by reconstructing them as complex units with an elaborate nature of their own.

In fact, the idea of truth values as compound entities nicely conforms with the modelling of truth values in some many-valued  systems, such as three-valued (Kleene, Priest) and four-valued (Belnap) logics, as certain subsets of the set of classical truth  values. The latter approach is essentially due to Michael Dunn, who proposed to generalize the notion of a classical truth-value function in order to represent the so-called “underdetermined” and “overdetermined” valuations. Namely, Dunn considers a valuation to be a function not from sentences  to elements of the set {the True, the False } but from sentences to subsets of this set. By developing this idea, one arrives at the concept of a generalized truth value function, which is a function from sentences into the subsets of some basic set of truth values. The values of generalized truth value functions can be called generalized truth values.

The tutorial consists of three sessions, in the course of which we unfold step by step the idea of generalized truth  values and demonstrate its fruitfulness for an analysis of many logical and philosophical problems.

Session 1.
The  notion of a truth value and the ways of its generalization
In the first lecture we explain how Gottlob Frege’s notion of a truth  value has become part of the standard philosophical and logical terminology. This notion is an indispensable instrument of realistic, model-theoretic approaches to logical semantics. Moreover, there  exist well-motivated theories of generalized truth values that lead far beyond Frege’s classical the True and the False. We discuss the possibility of generalizing the notion of a truth  value by conceiving them as complex units which possess a ramified inner structure.  We explicate some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value understood as a subset of some basic set of initial truth values of a lower degree. It turns out that this generalization is well- motivated and leads to the notion of a truth value multilattice.  In particular, one can proceed from the bilattice  F OU R2  with  both an information  and a truth-and-falsity  ordering to another algebraic structure, namely the trilattice SI X T EEN3   with an information ordering together with a truth  ordering and a (distinct)  falsity ordering.

Session 2.
Logics of generalized truth values
In this lecture we present various approaches to the construction of logical sys- tems related to truth  value multilattices.   More concretely, we investigate the logics generated by the algebraic operations under the truth  order and under the falsity order in bilattices and trilattices,  as well as various interrelations between them. It is also rather natural to formulate the logical systems in the language obtained by combining the vocabulary of the logic of the truth  order and the falsity order. We consider the corresponding  first-degree  consequence systems, Hilbert-style axiomatizations and Gentzen-style sequent calculi for the multilattice-logics.

Session 3.
Generalized  truth-values: logical and philosophical applications
Besides its purely logical impact, the idea of truth  values has induced a radical rethinking of some central issues in ontology, epistemology and the philosophy of logic, including:  the categorial status of truth  and falsehood, the theory of abstract objects, the subject-matter of logic and its ontological foundations, and the concept of a logical system. In the third lecture we demonstrate the wealth of philosophical problems, which can be analyzed by means of the apparatus of truth  values.  Among these problems are the liar paradox and the notion of hyper-contradiction, the famous slingshot-argument, Suszko thesis, harmonious many-valued logics, and some others.

References

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