One might, perhaps, think that the mere existence of many-valued logics shows that there exist infinitely, in fact, uncountably many truth values. However, this is not at all clear recall the more cautious use of terminology advocated by Gottwald. Whereas the algebraic values are elements of an algebraic structure and referents of formulas, the logical value true is used to define valid consequence: If every premise is true, then so is at least one of the conclusion s. The other logical value, false , is preserved in the opposite direction: If the every conclusion is false, then so is at least one of the premises.

The logical values are thus represented by a bi-partition of the set of algebraic values into a set of designated values truth and its complement falsity.

Essentially the same idea has been taken up earlier by Dummett in his influential paper, where he asks. The dichotomy between designated values and values which are not designated and its use in the definition of entailment plays a crucial role in the Suszko Reduction. In the literature on many-valued logic it is sometimes proposed to consider a set of antidesignated values which not obligatorily constitute the complement of the set of designated values see, e. The set of antidesignated values can be regarded as representing a generalized concept of falsity. This distinction leaves room for values that are neither designated nor antidesignated and even for values that are both designated and antidesignated.

Can one provide evidence for a multiplicity of logical values? A positive answer to this question emerges from considerations on truth values as structured entities which, by virtue of their internal structure, give rise to natural partial orderings on the set of values. As soon as one admits that truth values come with valuation systems , it is quite natural to assume that the elements of such a system are somehow interrelated.

This lattice may be referred to as TWO. It can be schematically presented by means of a so-called Hasse-diagram as in Figure 1. It is not quite clear how serious Frege is about this view, but it seems to suggest that truth values may well be interpreted as complex, structured entities that can be divided into parts. There exist several approaches to semantic constructions where truth values are represented as being made up from some primitive components. Then the empty proposition is interpreted as the value false , and the maximal proposition the set of all worlds in a structure as the value true.

Moreover, one can consider non-empty subsets of the maximal proposition as intermediate truth values.

### Notre Dame Journal of Formal Logic

Clearly, the intuitionistic truth values so conceived are composed from some simpler elements and as such they turn out to be complex entities. Goldblatt 81, And again, these truth values evidently have some inner structure. Moreover, the idea of truth values as compound entities nicely conforms with the modeling of truth values considered above in three-valued Kleene, Priest and four-valued Belnap logics as certain subsets of the set of classical truth values.

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 see Shramko and Wansing The values of generalized truth value functions can be called generalized truth values. By employing the idea of generalized truth value functions, one can obtain a hierarchy of valuation systems starting with a certain set-theoretic representation of the valuation system for classical logic.

The representation in question is built on a single initial value which serves then as the designated value of the resulting valuation system. Generalized truth values have a strong intuitive background, especially as a tool for the rational explication of incomplete and inconsistent information states. As Belnap points out, a computer may receive data from various maybe independent sources. Besides the standard classical cases, when a computer obtains information either that the sentence is 1 true or that it is 2 false, two other non-standard situations are possible: 3 nothing is told about the sentence or 4 the sources supply inconsistent information, information that the sentence is true and information that it is false.

In Zaitsev and Shramko it is demonstrated how ontological and epistemic aspects of truth values can be combined within a joint semantical framework. The conception of generalized truth values has its purely logical import as well. A comprehensive study of the conception of generalized logical values can be found in Shramko and Wansing The notion of a truth value is an indispensable instrument of realistic, model-theoretic approaches to semantics.

Indeed, truth values play an essential role in applications of model-theoretic semantics in areas such as, for example, knowledge representation and theorem proving based on semantic tableaux, which could not be treated in the present entry. Moreover, considerations on truth values give rise to deep ontological questions concerning their own nature, the feasibility of fact ontologies, and the role of truth values in such ontological theories.

For various directions of recent logical and philosophical investigations in the area of truth values see Truth Values I and Truth Values II Wansing rub. Truth values have been put to quite different uses in philosophy and logic, being 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.

Moreover, there are, according to Frege, only two such objects: the True das Wahre and the False das Falsche : Every assertoric sentence … is to be regarded as a proper name, and its Bedeutung , if it has one, is either the True or the False. Beaney This new and revolutionary idea has had a far reaching and manifold impact on the development of modern logic. Truth values as objects and referents of sentences 1. Truth values as logical values 2.

Ordering relations between truth-values 3. Generalized truth values Supplementary Document: Generalized truth values and multilattices 4. Accordingly, Frege asks: What else but the truth value could be found, that belongs quite generally to every sentence if the reference of its components is relevant, and remains unchanged by substitutions of the kind in question? Geach and Black 64 The idea underlying this question has been neatly reconstructed by Alonzo Church in his Introduction to Mathematical Logic 24—25 by considering the following sequence of four sentences: C1.

Sir Walter Scott is the author of Waverley. Sir Walter Scott is the man who wrote 29 Waverley Novels altogether. The number, such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is The number of counties in Utah is Burge As it has been observed repeatedly in the literature cf. As Frege explains: A statement contains no empty place, and therefore we must take its Bedeutung as an object. The idea of this method of characterizing abstract objects is also largely due to Frege, who wrote: If the symbol a is to designate an object for us, then we must have a criterion that decides in all cases whether b is the same as a , even if it is not always in our power to apply this criterion.

Beaney More precisely, one obtains a new object by abstracting it from some given kind of entities, in virtue of certain criteria of identity for this new abstract object. As he paradigmatically put it: All true propositions denote one and the same object, namely truth, and all false propositions denote one and the same object, namely falsehood. Figure 1: Lattice TWO.

Bibliography Anderson, Alan R. Anderson, Alan R. Belnap, and J. Asenjo, Florencio G. Beaney, Michael ed. Belnap, Nuel D.

Ryle ed. Dunn and G.

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