Truth Values

First published Tue Mar 30, 2010; substantive revision Sat Mar 1, 2025

Truth values have been put to quite different uses in philosophy andlogic, being characterized, for example, as:

primitive abstract objects denoted by sentences in natural andformal languages,
abstract entities hypostatized as the equivalence classes ofsentences,
what is aimed at in judgements,
values indicating the degree of truth of sentences,
entities that can be used to explain the vagueness ofconcepts,
values that are preserved in valid inferences,
values that convey information concerning a givenproposition.

Depending on their particular use, truth values have been treated asunanalyzed, as defined, as unstructured, or as structuredentities.

The notion of a truth value has been explicitly introduced into logicand philosophy by Gottlob Frege—for the first time in (Frege1891), and most notably in his seminal paper (Frege 1892). Although itwas Frege who made the notion of a truth value to one of the centralconcepts of semantics, the idea of special semantical values, however,was anticipated by Boole and Peirce already, see the survey article ona “history of truth values” by Béziau (2012).According to Kneale and Kneale (1962: 413), Boole’s systemcontains all that is needed for its interpretation “in terms oftruth values of propositions”, and as Church (1956: 17) remarks,the “explicit use of two truth-values appears for the first timein a paper by C.S. Peirce in theAmerican Journal ofMathematics, vol. 7 (1885), pp. 180–202”. Fregeconceived this notion as a natural component of his language analysiswhere sentences, being saturated expressions, are interpreted as aspecial kind of names, which refer to (denote, designate, signify) aspecial kind of objects: truth values. Moreover, there are, accordingto Frege, only two such objects:the True (dasWahre) andthe False (das Falsche):

Every assertoric sentence … is to be regarded as a proper name,and itsBedeutung, if it has one, is either the True or theFalse. (Frege 1892, trans. Beaney 1997: 158)

This new and revolutionary idea has had a far reaching and manifoldimpact on the development of modern logic. It provides the means touniformly complete the formal apparatus of a functional analysis oflanguage by generalizing the concept of a function and introducing aspecial kind of functions, namely propositional functions, or truthvalue functions, whose range of values consists of the set of truthvalues. Among the most typical representatives of propositionalfunctions one finds predicate expressions and logical connectives. Asa result, one obtains a powerful tool for a conclusive implementationof the extensionality principle (also called the principle ofcompositionality), according to which the meaning of a complexexpression is uniquely determined by the meanings of its components.On this basis one can also discriminate between extensional andintensional contexts and advance further to the conception ofintensional logics. Moreover, the idea of truth values has induced aradical rethinking of some central issues in the philosophy of logic,including: the categorial status of truth, the theory of abstractobjects, the subject-matter of logic and its ontological foundations,the concept of a logical system, the nature of logical notions,etc.

In the following, several important philosophical problems directlyconnected to the notion of a truth value are considered and varioususes of this notion are explained.

1. Truth values as objects and referents of sentences

1.1 Functional analysis of language and truth values

The approach to language analysis developed by Frege rests essentiallyon the idea of a strict discrimination between two main kinds ofexpressions: proper names (singular terms) and functional expressions.Proper names designate (signify, denote, or refer to) singularobjects, and functional expressions designate (signify, denote, orrefer to) functions. [Note: In the literature, the expressions‘designation’, ‘signification’,‘denotation’, and ‘reference’ are usuallytaken to be synonymous. This practice is used throughout the presententry.] The name ‘Ukraine’, for example, refers to acertain country, and the expression ‘the capital of’denotes a one-place function from countries to cities, in particular,a function that maps Ukraine to Kyiv. Whereas names are“saturated” (complete) expressions, functional expressionsare “unsaturated” (incomplete) and may be saturated byapplying them to names, producing in this way new names. Similarly,the objects to which singular terms refer are saturated and thefunctions denoted by functional expression are unsaturated. Names towhich a functional expression can be applied are called thearguments of this functional expression, and entities towhich a function can be applied are called thearguments ofthis function. The object which serves as the reference for the namegenerated by an application of a functional expression to itsarguments is called thevalue of the function for thesearguments. Particularly, the above mentioned functional expression‘the capital of’ remains incomplete until applied to somename. An application of the function denoted by ‘the capitalof’ to Ukraine (as an argument) returns Kyiv as the objectdenoted by the compound expression ‘the capital ofUkraine’ which, according to Frege, is a proper name of Kyiv.Note that Frege distinguishes between an \(n\)-place function \(f\) asan unsaturated entity that can be completed by and applied toarguments \(a_1\),…, \(a_n\) and itscourse of values,which can be seen as the set-theoretic representation of thisfunction: the set

\[\{\langle a_1, \ldots, a_n, a\rangle \mid a = f(a_1,\ldots , a_n)\}.\]

Pursuing this kind of analysis, one is very quickly confronted withtwo intricate problems.First, how should one treatdeclarativesentences? Should one perhaps separate them intoa specific linguistic category distinct from the ones of names andfunction symbols? Andsecond, how—from a functionalpoint of view—should one deal withpredicateexpressions such as ‘is a city’, ‘istall’, ‘runs’, ‘is bigger than’,‘loves’, etc., which are used to denote classes ofobjects, properties of objects, or relations between them and whichcan be combined with (applied to) singular terms to obtain sentences?If one considers predicates to be a kind of functional expressions,what sort of names are generated by applying predicates to theirarguments, and what can serve as referents of these names,respectively values of these functions?

A uniform solution of both problems is obtained by introducing thenotion of atruth value. Namely, by applying the criterion of“saturatedness” Frege provides a negative answer to thefirst of the above problems. Since sentences are a kind of completeentities, they should be treated as a sort of proper names, but namesdestined to denote some specific objects, namely the truth values:the True andthe False. In this way one also obtainsa solution of the second problem. Predicates are to be interpreted assome kind of functional expressions, which being applied to these orthose names generate sentences referring to one of the two truthvalues. For example, if the predicate ‘is a city’ isapplied to the name ‘Kyiv’, one gets the sentence‘Kyiv is a city’, which designatesthe True(i.e., ‘Kyiv is a city’is true). On the otherhand, by using the name ‘Mount Everest’, one obtains thesentence ‘Mount Everest is a city’ which clearlydesignatesthe False, since ‘Mount Everest is acity’is false.

Functions whose values are truth values are calledpropositionalfunctions. Frege also referred to them as concepts(Begriffe). A typical kind of such functions (besides theones denoted by predicates) are the functions denoted by propositionalconnectives. Negation, for example, can be interpreted as a unaryfunction convertingthe True intothe False andvice versa, and conjunction is a binary function that returnsthe True as a value when both its argument positions arefilled in bythe True, etc. Propositional functions mapping\(n\)-tuples of truth values into truth values are also calledtruth-value functions.

Frege thus in a first step extended the familiar notion of a numericalfunction to functions on singular objects in general and, moreover,introduced a new kind of singular objects that can serve as argumentsand values of functions on singular objects, the truth values. In afurther step, he considered propositional functions taking functionsas their arguments. The quantifier phrase ‘every city’,for example, can be applied to the predicate ‘is acapital’ to produce a sentence. The argument of thesecond-order function denoted by ‘every city’ isthefirst-order propositional function on singular objectsdenoted by ‘is a capital’. The functional value denoted bythe sentence ‘Every city is a capital’ is a truth value,the False.

Truth values thus prove to be an extremely effective instrument for alogical and semantical analysis of language.^[1] Moreover, Frege provides truth values (as proper referents ofsentences) not merely with a pragmatical motivation but also with astrong theoretical justification. The idea of such justification, thatcan be found in Frege 1892, employs the principle ofsubstitutivity of co-referential terms, according to whichthe reference of a complex singular term must remain unchanged whenany of its sub-terms is replaced by an expression having the samereference. This is actually just an instance of the compositionalityprinciple mentioned above. If sentences are treated as a kind ofsingular terms which must have designations, then assuming theprinciple of substitutivity one “almost inevitably” (asKurt Gödel (1944: 129) explains) is forced to recognize truthvalues as the most suitable entities for such designations.Accordingly, Frege asks:

What else but the truth value could be found, that belongs quitegenerally to every sentence if the reference of its components isrelevant, and remains unchanged by substitutions of the kind inquestion? (Geach and Black 1952: 64)

The idea underlying this question has been neatly reconstructed byAlonzo Church in hisIntroduction to Mathematical Logic(1956: 24–25) by considering the following sequence of foursentences:

C1.: Sir Walter Scott is the author ofWaverley.
C2.: Sir Walter Scott is the man who wrote 29Waverley Novelsaltogether.
C3.: The number, such that Sir Walter Scott is the man who wrote thatmanyWaverley Novels altogether is 29.
C4.: The number of counties in Utah is 29.

C1–C4 present a number of conversion steps each producingco-referential sentences. It is claimed that C1 and C2 must have thesame designation by substitutivity, for the terms ‘the author ofWaverley’ and ‘the man who wrote 29Waverley Novels altogether’ designate one and the sameobject, namely Walter Scott. And so must C3 and C4, because thenumber, such that Sir Walter Scott is the man who wrote that manyWaverley Novels altogether is the same as the number ofcounties in Utah, namely 29. Next, Church argues, it is plausible tosuppose that C2, even if not completely synonymous with C3, is atleast so close to C3 “so as to ensure its having the samedenotation”. If this is indeed the case, then C1 and C4 musthave the same denotation (designation) as well. But it seems that theonly (semantically relevant) thing these sentences have in common isthat both are true. Thus, taken that there must be something what thesentences designate, one concludes that it is just their truth value.As Church remarks, a parallel example involving false sentences can beconstructed in the same way (by considering, e.g., ‘Sir WalterScott is not the author ofWaverley’).

This line of reasoning is now widely known as the “slingshotargument”, a term coined by Jon Barwise and John Perry (1981:395), who stressed thus an extraordinary simplicity of the argumentand the minimality of presuppositions involved. Stated generally, thepattern of the argument goes as follows (cf. Perry 1996). One startswith a certain sentence, and then moves, step by step, to a completelydifferent sentence. Every two sentences in any step designatepresumably one and the same thing. Hence, the starting and theconcluding sentences of the argument must have the same designation aswell. But the only semantically significant thing they have in commonseems to be their truth value. Thus, what any sentence designates isjust its truth value.

A formal version of this argument, employing the term-forming,variable-binding class abstraction (or property abstraction) operatorλ\(x\) (“the class of all \(x\) such that” or“the property of being such an \(x\) that”), was firstformulated by Church (1943) in his review of Carnap’sIntroduction to Semantics. Quine (1953), too, presents avariant of the slingshot using class abstraction, see also (Shramkoand Wansing 2009a). Other remarkable variations of the argument arethose by Kurt Gödel (1944) and Donald Davidson (1967, 1969),which make use of the formal apparatus of a theory of definitedescriptions dealing with the description-forming, variable-bindingiota-operator (ι\(x\), “the \(x\) such that”). It isworth noticing that the formal versions of the slingshot show how tomove—using steps that ultimately preserve reference—fromany true (false) sentence toany other suchsentence. In view of this result, it is hard to avoid the conclusionthat what the sentences refer to are just truth values.

The slingshot argument has been analyzed in detail by many authors(see especially the comprehensive study by Stephen Neale (Neale 2001)and references therein) and has caused much controversy notably on thepart of fact-theorists, i.e., adherents of facts, situations,propositions, states of affairs, and other fact-like entitiesconceived as alternative candidates for denotations of declarativesentences. Also see thesupplement on the slingshot argument.

1.2 Truth as a property versus truth as an object

Truth values evidently have something to do with a general concept oftruth. Therefore it may seem rather tempting to try to incorporateconsiderations on truth values into the broader context of traditionaltruth-theories, such as correspondence, coherence, anti-realistic, orpragmatist conceptions of truth. Yet, it is unlikely that suchattempts can give rise to any considerable success. Indeed, theimmense fruitfulness of Frege’s introduction of truth valuesinto logic to a large extent is just due to its philosophicalneutrality with respect to theories of truth. It does not commit oneto any specific metaphysical doctrine of truth. In one significantrespect, however, the idea of truth values contravenes traditionalapproaches to truth by bringing to the forefront the problem of itscategorial classification.

In most of the established conceptions, truth is usually treated as aproperty. It is customary to talk about a “truthpredicate” and its attribution to sentences, propositions,beliefs or the like. Such an understanding corresponds also to aroutine linguistic practice, when one operates with the adjective‘true’ and asserts, e.g., ‘That 5 is a prime numberis true’. As Scott Soames observes:

‘True’ is an adjective that, in its central use, functionsas a grammatical predicate, taking nouns and noun phrases as subjects… This grammatical fact encourages us to think that when weassertively utter such a sentence we typically refer to something andsay of it that it is true. The thing referred to is called, byphilosophers, thebearer of truth (falsity). To say of such athing that it is true (false) is to predicate truth (falsity) of it.In this view, ‘true’ is a logical as well as a grammaticalpredicate used to describe or characterize entities as having acertain property––truth. (Soames 1999: 13)

By contrast with this apparently quite natural attitude, thesuggestion to interpret truth as an object may seem very confusing, tosay the least. Nevertheless this suggestion is also equipped with aprofound and strong motivation demonstrating that it is far from beingjust an oddity and has to be taken seriously (cf. Burge 1986).

First, it should be noted that the view of truth as a property is notas natural as it appears on the face of it. Frege brought into play anargument to the effect that characterizing a sentence astrueadds nothing new to its content, for ‘It is true that 5 is aprime number’ says exactly the same as just ‘5 is a primenumber’. That is, the adjective ‘true’ is in a senseredundant and thus is not a real predicate expressing a realproperty such as the predicates ‘white’ or‘prime’ which, on the contrary, cannot simply beeliminated from a sentence without an essential loss for its content.In this case a superficial grammatical analogy is misleading. Thisidea gave an impetus to the deflationary conception of truth(advocated by Ramsey, Ayer, Quine, Horwich, and others, see the entryonthe deflationary theory of truth).

However, even admitting the redundancy of truth as a property, Fregeemphasizes its importance and indispensable role in some otherrespect. Namely, truth, accompanying every act of judgment as itsultimate goal, secures an objectivevalue of cognition byarranging for every assertive sentence a transition from the level ofsense (the thought expressed by a sentence) to the level of denotation(its truth value). This circumstance specifies the significance oftaking truth as a particular object. As Tyler Burge explains:

Normally, the point of using sentences, what “matters tous”, is to claim truth for a thought. The object, in the senseof the point orobjective, of sentence use was truth. It isilluminating therefore to see truth as an object. (Burge 1986:120)

As it has been observed repeatedly in the literature (cf., e.g., Burge1986, Ruffino 2003), the stress Frege laid on the notion of a truthvalue was, to a great extent, pragmatically motivated. Besides anintended gain for his system of “Basic Laws” (Frege1893/1903) reflected in enhanced technical clarity, simplicity, andunity, Frege also sought to substantiate in this way his view on logicas a theoretical discipline with truth as its main goal and primarysubject-matter. Incidentally, Gottfried Gabriel (1986) demonstratedthat in the latter respect Frege’s ideas can be naturally linkedup with a value-theoretical tradition in German philosophy of thesecond half of the 19^th century; see also (Gabriel 2013) onthe relation between Frege’s value-theoretically inspiredconception of truth values and his theory of judgement. Morespecifically, Wilhelm Windelband, the founder and the principalrepresentative of the Southwest school of Neo-Kantianism, was actuallythe first who employed the term “truth value”(“Wahrheitswert”) in his essay “What isPhilosophy?” published in 1882 (see Windelband 1915: 32), i.e.,nine years before Frege 1891, even if he was very far from treating atruth value as a value of a function.

Windelband defined philosophy as a “critical science aboutuniversal values”. He considered philosophical statements to benot mere judgements but ratherassessments, dealing with somefundamental values,the value of truth being one of the mostimportant among them. This latter value is to be studied by logic as aspecial philosophical discipline. Thus, from a value-theoreticalstandpoint, the main task of philosophy, taken generally, is toestablish the principles of logical, ethical and aestheticalassessments, and Windelband accordingly highlighted the triad of basicvalues: “true”, “good” and“beautiful”. Later this triad was taken up by Frege in1918 when he defined the subject-matter of logic (see below). Gabrielpoints out (1984: 374) that this connection between logic and a valuetheory can be traced back to Hermann Lotze, whose seminars inGöttingen were attended by both Windelband and Frege.

The decisive move made by Frege was to bring together a philosophicaland a mathematical understanding of values on the basis of ageneralization of the notion of a function on numbers. While Frege mayhave been inspired by Windelband’s use of the word‘value’ (and even more concretely – ‘truthvalue’), it is clear that he uses the word in its mathematicalsense. If predicates are construed as a kind of functional expressionswhich, being applied to singular terms as arguments, producesentences, then the values of the corresponding functions must bereferences of sentences. Taking into account that the range of anyfunction typically consists of objects, it is natural to conclude thatreferences of sentences must be objects as well. And if one now justtakes it that sentences refer to truth values (the True andthe False), then it turns out that truth values are indeedobjects, and it seems quite reasonable to generally explicate truthand falsity as objects and not as properties. As Frege explains:

A statement contains no empty place, and therefore we must take itsBedeutung as an object. But thisBedeutung is atruth-value. Thus the two truth-values are objects. (Frege 1891,trans. Beaney 1997: 140)

Frege’s theory of sentences as names of truth values has beencriticized, for example, by Michael Dummett who stated ratherdramatically:

This was the most disastrous of the effects of the misbegottendoctrine that sentences are a species of complex singular terms, whichdominated Frege’s later period: to rob him of the insight thatsentences play a unique role, and that the role of almost every otherlinguistic expression … consists in its part in formingsentences. (Dummett 1981: 196)

But even Dummett (1991: 242) concedes that “to deny thattruth-values are objects … seems a weak response”.

The presentation of truth as an object rather than a property seemsalso to minimize, if not avoid, certain philosophical difficultiesassociated with the problem of truth-bearers, which is typical forsupporters of the interpretation of truth as a property (cf. the aboveobservation of Soames, and also the subsection on truth-bearers in the entry ontruth). Indeed, if we construe truth as a certain property, then the questionof a bearer (or bearers) of this property naturally arises, and, inparticular, about what kind of things can act as such bearers. It isclear that the nature of these things must have a significant impacton the nature of its properties, and in particular on the property oftruth. Now, there can be different candidates for the role of truthbearers: “Truth can be predicated of many kinds of things:propositions, sentences, statements, judgements, assertions,declarations, claims, utterances, beliefs, conjectures, hypotheses,theories, stories, etc” (Bar-Hillel 1973: 303). Moreover, theproperties of, say, utterances as physical objects are inherentlydifferent from the properties of propositions as abstract objects.Should we then conclude that there are many properties of beingtrue?

One way to avoid such a conclusion is to adopt a kind oftruth-bearer monism, where only a certain type of entities,whether sentences, propositions, or something else, can play thisrole. This view, however, seems to be at odds with the usuallinguistic practice when we can say that our beliefs are true as muchas our statements. Therefore, a weaker view is often adopted,highlighting some kind of entities asprimary bearers oftruth and defining other bearers, which are considered to bederivative orsecondary, by means of these, (see, e.g.,Bar-Hillel 1973, Hanks 2013). This still leaves the question ofwhether the entities defined as secondary truth bearers really havethe property of being true, and if so, whether this property is thesame for secondary truth bearers as for the primary ones.

Andrea Strollo (2022) develops a version oftruth-bearerpluralism. He stresses that “to understand the nature oftruth attention should be paid to the kind of entities that are apt tobear it”, and in this respect, truth is similar to “manyother properties”. In particular, it is argued that“different kinds of truth bearers tend to support differentproperties of truth”, and that “there are reasons to admita plurality of truth bearers”. Thus, Strollo recognizes not onlythe plurality of truth bearers, but also theplurality of truthproperties themselves, and argues for the naturalness of suchplurality. Nevertheless, the very naturalness of such multiplicationof the notion of truth can be questioned and in any case requiresspecial justification.

The situation here is similar to what we have with many otherproperties when we try to abstract some general meaning from a termthat denotes a particular property in its relation to differentbearers. To explain this situation it can be useful to involveAristotle's account of ‘pros hen being’ that onecan find in the fourth book ofMetaphysics, where Aristotlesays: “There are many senses in which a thing may be said to‘be’, but all that ‘is’ is related to onecentral point [πρόσ ἕν]” (Aristotle,Metaphysics IV (\(\Gamma\)), 1003a 33). Literally, the Greek phrase“pros hen” means “in relation toone”. Famously, G.E.L. Owen (1960) “names theproshen relation ‘focal meaning’, and claims that itmeans that all the ‘senses [of “being”] have onefocus, one common element’, or ‘a central sense’, sothat ‘all its senses can be explained in terms of substance andof the sense of “being” that is appropriate tosubstance’.” (Yu 2001: 206). By way of example, Aristotleconsiders the property ofbeing healthy, and observes that“everything which is healthy is related to health, one thing inthe sense that it preserves health, another in the sense that itproduces it, another in the sense that it is a symptom of health,another because it is capable of it” (Aristotle, Metaphysics IV(\(\Gamma\)), 1003a 35). It can be noted that “these varioussenses have something in common: a reference to one central thing,health” (Cohen & Reeve 2025). Remarkably, the latter is akind of abstraction (abstract object) which, through its‘focal meaning’, determines the particular meanings of allassociated properties (more precisely, the meanings of thecorresponding expressions).

Now, in like manner, we can take an abstract object,Truth,and consider everything that can be true (propositions, sentences,beliefs, etc.) as connected to it by various relations (denoting,expressing, holding, etc.). We have then the relationships of thefocalsubstance to various other things which act as bearersof the corresponding property. In this way, it is hoped to reduce allthe variety of different particular meanings associated with theproperty ofbeing true to a single ‘focalmeaning’ determined by the relevant substance, a certain truthvalue.

Generally, as Sefrin-Weis (2009: 262) explains, “focal meaningestablishes a kind of “reductive translation” (Owen 1960:169 and 180) of statements about anything in the general field ofbeing to statements about substance”. In Owen's own words:

The claim of IV that ‘being’ is an expression with focalmeaning is a claim that statements about non-substances can be reducedto—translated into—statements about substances andnon-substances are no more than the logical shadows of substances.(Owen 1960: 180)

Of course, it is always possible to retain the truth predicate in themeta-language and continue to use the more familiar (predicative)wording, by stipulating that, for example, a sentence (proposition,belief, etc.) is considered to be true if and only if it denotes (orin some other relevant way is related to) the truth value “theTrue”. Such a linguistic convention, understood as a mereabbreviation, would be philosophically harmless, and should not implyany problematic ontological commitments (other than recognizing thecorresponding abstract object).

It is worth mentioning that Zalta (2005) provides an analysis of theTarski-schema that allows one to interpret truth neither as aproperty, nor as an object, but as a mode of predication. Namely, letus extend a language for the second-order predicate calculus with anabstraction operator \(\lambda\) and interpret the \(\lambda\)-expressionsrelationally rather than functionally. Then the principle of\(\lambda\)-Conversion:

\[[\lambda x_1\ldots x_n\: \phi ]y_1\ldots y_n \equiv \phi [y_1,\ldots ,y_n/x_1,\ldots ,x_n] \]

would assert: “objects \(x_1,\ldots , x_n\) exemplify being a\(y_1,\ldots , y_n\) such that \(\phi\) if and only if \(x_1,\ldots , x_n\) aresuch that \(\phi\)”. Now, \(\lambda\)-Conversion has also 0-placeinstances:

\[[\lambda\: \phi] \equiv \phi, \]

which, being interpreted as the propositional Tarski T-Schema, can beread as “that-\(\phi\) is true iff \(\phi\)”. Zalta arguesthat the expression \([\lambda\: \phi]\) is both a 0-place relation term and aformula, and thus, truth is represented here as the 0-place case ofexemplification, i.e., a certain mode of predication. It seems thatsuch a possibility of analyzing the concept of truth is not elaboratedin the literature, but it may be a promising direction of itsexamination.

1.3 The ontology of truth values

If truth values are accepted and taken seriously as a special kind ofobjects, the obvious question as to the nature of these entitiesarises. The above characterization of truth values as objects is fartoo general and requires further specification. One way of suchspecification is to qualify truth values asabstract objects.Note that Frege himself never used the word ‘abstract’when describing truth values. Instead, he has a conception of socalled “logical objects”, truth values being primary andthe most fundamental of them (Frege 1976: 121). Among the otherlogical objects Frege pays particular attention to are sets andnumbers, emphasizing thus their logical nature (in accordance with hislogicist view).

Church (1956: 25), when considering truth values, explicitlyattributes to them the property of being abstract. Since then it iscustomary to label truth values as abstract objects, thus allocatingthem into the same category of entities as mathematical objects(numbers, classes, geometrical figures) and propositions. One may posehere an interesting question about the correlation between Fregeanlogical objects and abstract objects in the modern sense (see theentry onabstract objects). Obviously, the universe of abstract objects is much broader than theuniverse of logical objects as Frege conceives them. The latter areconstrued as constituting an ontological foundation for logic, andhence for mathematics (pursuant to Frege’s logicist program).Generally, the class ofabstracta includes a wide diversityof platonic universals (such as redness, youngness, justice ortriangularity) and not only those of them which are logicallynecessary. Nevertheless, it may safely be said that logical objectscan be considered as paradigmatic cases of abstract entities, orabstract objects in their purest form.

It should be noted that finding an adequate definition of abstractobjects is a matter of considerable controversy. According to a commonview, abstract entities lack spatio-temporal properties and relations,as opposed to concrete objects which exist in space and time (Lowe1995: 515). In this respect truth values obviouslyareabstract as they clearly have nothing to do with physical spacetime.In a similar fashion truth values fulfill another requirement oftenimposed upon abstract objects, namely the one of a causal inefficacy(see, e.g., Grossmann 1992: 7). Here again, truth values are very muchlike numbers and geometrical figures: they have no causal power andmake nothing happen.

Finally, it is of interest to consider how truth values can beintroduced by applying so-calledabstraction principles,which are used for supplying abstract objects withcriteria ofidentity. The idea of this method of characterizing abstractobjects is also largely due to Frege, who wrote:

If the symbola is to designate an object for us, then wemust have a criterion that decides in all cases whetherb isthe same asa, even if it is not always in our power to applythis criterion. (Frege 1884, trans. Beaney 1997: 109)

More precisely, one obtains a new object by abstracting it from somegiven kind of entities, in virtue of certain criteria of identity forthis new (abstract) object. This abstraction is performed in terms ofan equivalence relation defined on the given entities (see Wrigley2006: 161). The celebrated slogan by Quine (1969: 23) “No entitywithout identity” is intended to express essentially the sameunderstanding of an (abstract) object as an “item falling undera sortal concept which supplies a well-defined criterion of identityfor its instances” (Lowe 1997: 619).

For truth values such a criterion has been suggested in Anderson andZalta (2004: 2), stating that for any two sentences \(p\) and \(q\),the truth value of \(p\) is identical with the truth value of \(q\) ifand only if \(p\) is (non-logically) equivalent with \(q\) (cf. alsoDummett 1959: 141). This idea can be formally explicated following thestyle of presentation in Lowe (1997: 620):

\[ \forall p\forall q[(\textit{Sentence}(p) \mathbin{\&} \textit{Sentence}(q)) \Rightarrow(tv(p)=tv(q) \Leftrightarrow(p\leftrightarrow q))], \]

where &, \(\Rightarrow, \Leftrightarrow, \forall\) standcorrespondingly for ‘and’, ‘if… then’,‘if and only if’ and ‘for all’ in themetalanguage, and \(\leftrightarrow\) stands for someobject language equivalence connective (biconditional).

Incidentally, Carnap (1947: 26), when introducing truth-values asextensions of sentences, is guided by essentially the same idea.Namely, he points out a strong analogy between extensions ofpredicators and truth values of sentences. Carnap considers a wideclass of designating expressions (“designators”) amongwhich there are predicate expressions (“predicators”),functional expressions (“functors”), and some others.Applying the well-known technique of interpreting sentences aspredicators of degree 0, he generalizes the fact that two predicatorsof degree \(n\) (say, \(P\) and \(Q)\) have the same extension if andonly if \(\forall x_1\forall x_2 \ldots \forall x_n(Px_1 x_2\ldots x_n\leftrightarrow Qx_1 x_2\ldots x_n)\) holds. Then, analogously, twosentences (say, \(p\) and \(q)\), being interpreted as predicators ofdegree 0, must have the same extension if and only if\(p\leftrightarrow q\) holds, that is if and only if they areequivalent. And then, Carnap remarks, it seems quite natural to taketruth values as extensions for sentences.

Note that this criterion employs afunctional dependencybetween an introduced abstract object (in this case a truth value) andsome other objects (sentences). More specifically, what is consideredis the truth value of a sentence (or proposition, or the like). Thecriterion of identity for truth values is formulated then through thelogical relation of equivalence holding between these otherobjects—sentences, propositions, or the like (with an explicitquantification over them).

It should also be remarked that the properties of the object languagebiconditional depend on the logical system in which the biconditionalis employed. Biconditionals of different logics may have differentlogical properties, and it surely matters what kind of the equivalenceconnective is used for defining truth values. This means that theconcept of a truth value introduced by means of the identity criterionthat involves a biconditional between sentences is alsologic-relative. Thus, if ‘\(\leftrightarrow\)’ stands formaterial equivalence, one obtains classical truth values, but if theintuitionistic biconditional is employed, one gets truth values ofintuitionistic logic, etc. Taking into account the role truth valuesplay in logic, such an outcome seems to be not at all unnatural.

Anderson and Zalta (2004: 13), making use of an object theory fromZalta (1983), propose the following definition of ‘the truthvalue of proposition \(p\)’ (‘\(tv(p)\)’ [notationadjusted]):

\[ tv(p) =_{df} \iota x(A!x \wedge \forall F(xF\leftrightarrow \exists q(q\leftrightarrow p \wedge F= [\lambda y\:q]))), \]

where \(A\)! stands for a primitive theoretical predicate ‘beingabstract’, \(xF\) is to be read as “\(x\) encodes\(F\)” and [λy q] is a propositionalproperty (“being such a \(y\) that \(q\)”). (Note that inthis system the variable \(q\) is used in both term and formulaposition; in many systems going back to Church and Frege, there areexpressions that are both terms and formulas.) That is, according tothis definition, “the extension of \(p\) is the abstract objectthat encodes all and only the properties of the form[λy q] which are constructed out ofpropositions \(q\) materially equivalent to \(p\)” (Anderson andZalta 2004: 14).

The notion of a truth value in general is then defined as an objectwhich is the truth value of some proposition:

\[TV(x) =_{df} \exists p(x = tv(p)).\]

Using this apparatus, it is possible to explicitly define the Fregeantruth valuesthe True \((\top)\) andthe False\((\bot)\):

\[ \begin{align}\top &=_{df} \iota x(A!x \wedge \forall F(xF \leftrightarrow \exists p(p \wedge F= [\lambda y\: p])));\\ \bot &=_{df} \iota x(A!x \wedge \forall F(xF \leftrightarrow \exists p(\neg p \wedge F= [\lambda y\: p]))).\\ \end{align} \]

Anderson and Zalta prove then that \(\top\) and \(\bot\) are indeedtruth values and, moreover, that there are exactly two such objects.The latter result is expected, if one bears in mind that what thedefinitions above actually introduce are theclassical truthvalues (as the underlying logic is classical). Indeed,\(p\leftrightarrow q\) is classically equivalent to \((p\wedgeq)\vee(\neg p\wedge \neg q)\), and \(\neg(p\leftrightarrow q)\) isclassically equivalent to \((p\wedge \neg q)\vee(\neg p\wedge q)\).That is, the connective of material equivalence divides sentences intotwo distinct collections. Due to the law of excluded middle thesecollections are exhaustive, and by virtue of the law ofnon-contradiction they are exclusive. Thus, we get exactly twoequivalence classes of sentences, each being a hypostatizedrepresentative of one of two classical truth values.

2. Truth values as logical values

2.1 Logic as the science of logical values

In a late paper Frege (1918) claims that the word ‘true’determines the subject-matter of logic in exactly the same way as theword ‘beautiful’ does for aesthetics and the word‘good’ for ethics. Thus, according to such a view, theproper task of logic consists, ultimately, in investigating “thelaws of being true” (Sluga 2002: 86). By doing so, logic isinterested in truth as such, understood objectively, and not in whatis merely taken to be true. Now, if one admits that truth is aspecific abstract object (the corresponding truth value), then logicin the first place has to explore the features of this object and itsinterrelations to other entities of various other kinds.

A prominent adherent of this conception was Jan Łukasiewicz. Ashe paradigmatically put it:

All true propositions denote one and the same object, namely truth,and all false propositions denote one and the same object, namelyfalsehood. I consider truth and falsehood to besingularobjects in the same sense as the number 2 or 4 is. …Ontologically, truth has its analogue in being, and falsehood, innon-being. The objects denoted by propositions are calledlogicalvalues. Truth is the positive, and falsehood is the negativelogical value. … Logic is the science of objects of a specialkind, namely a science oflogical values. (Łukasiewicz1970: 90)

This definition may seem rather unconventional, for logic is usuallytreated as the science of correct reasoning and valid inference. Thelatter understanding, however, calls for further justification. Thisbecomes evident, as soon as one asks,on what grounds oneshould qualify this or that pattern of reasoning as correct orincorrect.

In answering this question, one has to take into account that anyvalid inference should be based on logical rules which, according to acommonly accepted view, should at least guarantee that in a validinference the conclusion(s) is (are) true if all the premises aretrue. Translating this demand into the Fregean terminology, it wouldmean that in the course of a correct inference the possession of thetruth valuethe True should bepreserved from thepremises to the conclusion(s). Thus, granting the realistic treatmentof truth values adopted by Frege, the understanding of logic as thescience of truth values in fact provides logical rules with anontological justification placing the roots of logic in a certain kindof ideal entities (see Shramko 2014).

These entities constitute a certain uniform domain, which can beviewed as a subdomain of Frege’s so-called “thirdrealm” (the realm of the objective content of thoughts, andgenerally abstract objects of various kinds (see Frege 1918, cf.Popper 1972 and also Burge 1992: 634). Among the subdomains of thisthird realm one finds, e.g., the collection of mathematical objects(numbers, classes, etc.). The set of truth values may be regarded asforming another such subdomain, namely the one oflogicalvalues, and logic as a branch of science rests essentially onthislogical domain and on exploring its features andregularities.

2.2 Many-valued logics, truth degrees and valuation systems

According to Frege, there are exactly two truth values,theTrue andthe False. This opinion appears to be ratherrestrictive, and one may ask whether it is really indispensable forthe concept of a truth value. One should observe that in elaboratingthis conception, Frege assumed specific requirements of his system oftheBegriffsschrift, especially the principle of bivalencetaken as a metatheoretical principle, viz. that there exist only twodistinct logical values. On the object-language level this principlefinds its expression in the famous classical laws of excluded middleand non-contradiction. The further development of modern logic,however, has clearly demonstrated that classical logic is only oneparticular theory (although maybe a very distinctive one) among thevast variety of logical systems. In fact, the Fregean ontologicalinterpretation of truth values depicts logical principles as a kind ofontological postulations, and as such they may well be modified oreven abandoned. For example, by giving up the principle of bivalence,one is naturally led to the idea of postulatingmany truthvalues.

It was Łukasiewicz, who as early as 1918 proposed to takeseriously other logical values different from truth and falsehood (seeŁukasiewicz 1918, 1920). Independently of Łukasiewicz, EmilPost in his dissertation from 1920, published as (Post 1921),introduced \(m\)-valued truth tables, where \(m\) is any positiveinteger. Whereas Post’s interest inmany-valued logic(where “many” means “more than two”) wasalmost exclusively mathematical, Łukasiewicz’s motivationwas philosophical (see the entry onmany-valued logic). He contemplated the semantical value of sentences about thecontingent future, as discussed in Aristotle’sDeinterpretatione. Łukasiewicz introduced a third truth valueand interpreted it as “possible”. By generalizing thisidea and also adopting the above understanding of the subject-matterof logic, one naturally arrives at the representation of particularlogical systems as a certain kind ofvaluation systems (see,e.g., Dummett 1981, 2000; Ryan and Sadler 1992).

Consider a propositional language \(\mathcal{L}\) built upon a set ofatomic sentences \(\mathcal{P}\) and a set of propositionalconnectives \(\mathcal{C}\) (the set of sentences of \(\mathcal{L}\)being the smallest set containing \(\mathcal{P}\) and being closedunder the connectives from \(\mathcal{C})\). Then avaluationsystem \(\mathbf{V}\) for the language \(\mathcal{L}\) is atriple \(\langle \mathcal{V}, \mathcal{D}, \mathcal{F}\rangle\), where\(\mathcal{V}\) is a non-empty set with at least two elements,\(\mathcal{D}\) is a subset of \(\mathcal{V}\), and \(\mathcal{F} =\{f_{c _1},\ldots, f_{c _m}\}\) is a set of functions such that \(f_{c_i}\) is an \(n\)-place function on \(\mathcal{V}\) if \(c_i\) is an\(n\)-place connective. Intuitively, \(\mathcal{V}\) is the set oftruth values, \(\mathcal{D}\) is the set ofdesignated truthvalues, and \(\mathcal{F}\) is the set of truth-value functionsinterpreting the elements of \(\mathcal{C}\). If the set of truthvalues of a valuation system \(\mathbf{V}\) has \(n\) elements,\(\mathbf{V}\) is said to be \(n\)-valued. Any valuation system can beequipped with an assignment function which maps the set of atomicsentences into \(\mathcal{V}\). Each assignment \(a\) relative to avaluation system \(\mathbf{V}\) can be extended to all sentences of\(\mathcal{L}\) by means of a valuation function \(v_a\) defined inaccordance with the following conditions:

\[ \begin{align}\forall p &\in \mathcal{P} , &v_a (p) &= a(p) ; \tag{1}\\ \forall c_i &\in \mathcal{C} , & v_a ( c_i ( A_1 ,\ldots , A_n )) &= f_{c_i} ( v_a ( A_1 ),\ldots , v_a ( A_n )) \tag{2} \\ \end{align} \]

It is interesting to observe that the elements of \(\mathcal{V}\) aresometimes referred to asquasi truth values. SiegfriedGottwald (1989: 2) explains that one reason for using the term‘quasi truth value’ is that there is no convincing anduniform interpretation of the truth values that in many-valued logicare assumed in addition to the classical truth valuestheTrue andthe False, an understanding that, according toGottwald, associates the additional values with the naiveunderstanding of being true, respectively the naive understanding ofdegrees of being true (cf. also the remark by Font (2009:383) that “[o]ne of the main problems in many-valued logic, atleast in its initial stages, was the interpretation of the‘intermediate’ or ‘non-classical’values”, et seq.). In later publications, Gottwald has changedhis terminology and states that

[t]o avoid any confusion with the case of classical logic one prefersin many-valued logic to speak oftruth degrees and to use theword “truth value” only for classical logic. (Gottwald2001: 4)

Nevertheless in what follows the term ‘truth values’ willbe used even in the context of many-valued logics, without anycommitment to a philosophical conception of truth as a graded notionor a specific understanding of semantical values in addition to theclassical truth values.

Since the cardinality of \(\mathcal{V}\) may be greater than 2, thenotion of a valuation system provides a natural foundational frameworkfor the very idea of a many-valued logic. The set \(\mathcal{D}\) ofdesignated values is of central importance for the notion of avaluation system. This set can be seen as a generalization of theclassical truth valuethe True in the sense that itdetermines many central logical notions and thereby generalizes someof the important roles played by Frege’sthe True (cf.the introductory remarks about uses of truth values). For example, theset of tautologies (logical laws) is directly specified by the givenset of designated truth values: a sentence \(A\) is atautology in a valuation system \(\mathbf{V}\) iff for everyassignment \(a\) relative to \(\mathbf{V}\), \(v_a(A) \in\mathcal{D}\). Another fundamental logical notion—that of anentailment relation—can also be defined by referring to the set\(\mathcal{D}\). For a given valuation system \(\mathbf{V}\) acorresponding entailment relation \((\vDash_V)\) is usually defined bypostulating the preservation of designated values from the premises tothe conclusion:

\[ \tag{3} \Delta\vDash_V A \textrm{ iff }\forall a[(\forall B \in \Delta: v_a (B) \in \mathcal{D}) \Rightarrow v _a (A) \in \mathcal{D}]. \]

A pair \(\mathcal{M} = \langle \mathbf{V}, v_a\rangle\), where\(\mathbf{V}\) is an \((n\)-valued) valuation system and \(v_a\) avaluation in \(\mathbf{V}\), may be called an \((n\)-valued)model based on \(\mathbf{V}\). Every model \(\mathcal{M} =\langle \mathbf{V}, v_a\rangle\) comes with a corresponding entailmentrelation \(\vDash_{\mathcal{M}}\) by defining\(\Delta\vDash_{\mathcal{M} }A\textrm{ iff }(\forall B \in \Delta: v_a(B) \in \mathcal{D}) \Rightarrow v_a(A) \in \mathcal{D}\).

Suppose \(\mathfrak{L}\) is a syntactically defined logical system\(\mathfrak{L}\) with a consequence relation \(\vdash_{ \mathfrak{L}}\), specified as a relation between the power-set of \(\mathcal{L}\)and \(\mathcal{L}\). Then a valuational system \(\mathbf{V}\) is saidto bestrictly characteristic for \(\mathfrak{L}\) just incase \(\Delta\vDash_V A \textrm{ iff } \Delta\vdash_{ \mathfrak{L}}A\) (see Dummett 1981: 431). Conversely, one says that\(\mathfrak{L}\)is characterized by \(\mathbf{V}\). Thus, ifa valuation system is said todetermine a logic, thevaluation systemby itself is, properly speaking,not a logic, but only serves as a semantic basis for somelogical system. Valuation systems are often referred to as(logical)matrices. Note that in (Urquhart 1986),the set \(\mathcal{D}\) of designated elements of a matrix is requiredto be non-empty, and in Dunn & Hardegree 2001, \(\mathcal{D}\) isrequired to be a non-empty proper subset of \(\mathbf{V}\). With aview on semantically defining a many-valued logic, these restrictionsare very natural and have been taken up in (Shramko & Wansing2011) and elsewhere. For the characterization of consequence relations(see the supplementary documentSuszko’s Thesis), however, the restrictions do not apply.

In this way Fregean, i.e., classical, logic can be presented asdetermined by a particular valuation system based on exactly twoelements: \(\mathbf{V}_{cl} = \langle \{T, F\}, \{T\}, \{ f_{\wedge},f_{\vee}, f_{\rightarrow}, f_{\sim}\}\rangle\), where \(f_{\wedge},f_{\vee}, f_{\rightarrow},f_{\sim}\) are given by the classical truthtables for conjunction, disjunction, material implication, andnegation.

As an example for a valuation system based on more that two elements,consider two well-known valuation systems which determineKleene’s (strong) “logic of indeterminacy” \(K_3\)and Priest’s “logic of paradox” \(P_3\). In apropositional language without implication, \(K_3\) is specified bytheKleene matrix \(\mathbf{K}_3 = \langle \{T, I, F\},\{T\}, \{ f_c: c \in \{\sim , \wedge , \vee \}\} \rangle\), where thefunctions \(f_c\) are defined as follows:

\[ \begin{array}{c|c}f_\sim & \\ \hline T & F \\ I & I \\ F & T \\ \end{array}\quad \begin{array}{c|c|c|c}f_\wedge & T & I & F \\ \hline T & T & I & F \\ I & I & I & F \\ F & F & F & F \\ \end{array}\quad \begin{array}{c|c|c|c}f_\vee & T & I & F \\ \hline T & T & T & T \\ I & T & I & I \\ F & T & I & F \\ \end{array} \]

ThePriest matrix \(\mathbf{P}_3\) differs from\(\mathbf{K}_3\) only in that \(\mathcal{D} = \{T, I\}\). Entailmentin \(\mathbf{K}_3\) as well as in \(\mathbf{P}_3\) is defined by meansof (3).

There are natural intuitive interpretations of \(I\) in\(\mathbf{K}_3\) and in \(\mathbf{P}_3\) as theunderdetermined and theoverdetermined valuerespectively—a truth-value gap and a truth-value glut. Formallythese interpretations can be modeled by presenting the values ascertain subsets of the set of classical truth values \(\{T, F\}\).Then \(T\) turns into \(\mathbf{T} = \{T\}\) (understood as“true only”), \(F\) into \(\mathbf{F} = \{F\}\)(“false only”), \(I\) is interpreted in \(K_3\) as\(\mathbf{N} = \{\} = \varnothing\) (“neither true norfalse”), and in \(P_3\) as \(\mathbf{B} = \{T, F\}\)(“both true and false”). (Note that also Asenjo(1966) considers the same truth-tables with an interpretation of thethird value as “antinomic”.) The designatedness of a truthvalue can be understood in both cases as containment of the classical\(T\) as a member.

If one combines all these new values into a joint framework, oneobtains the four-valued logic \(B_4\) introduced by Dunn (1976) andBelnap (1977a, b). A Gentzen-style formulation can be found in (Font1997: 7). This logic is determined by theBelnap matrix\(\mathbf{B}_4 = \langle \{\mathbf{N}, \mathbf{T}, \mathbf{F},\mathbf{B}\}, \{\mathbf{T}, \mathbf{B}\}, \{ f_c: c \in \{\sim ,\wedge , \vee \}\}\rangle\), where the functions \(f_c\) are definedas follows:

\[ \begin{array}{c|c}f_\sim & \\ \hline T & F \\ B & B \\ N & N \\ F & T \\ \end{array}\quad \begin{array}{c|c|c|c|c}f_\wedge & T & B & N & F \\ \hline T & T & B & N & F \\ B & B & B & F & F \\ N & N & F & N & F \\ F & F & F & F & F\\ \end{array}\quad \begin{array}{c|c|c|c|c}f_\vee & T & B & N & F \\ \hline T & T & T & T & T\\ B & T & B & T & B \\ N & T & T & N & N \\ F & T & B & N & F \\ \end{array} \]

Definition (3) applied to the Belnap matrix determines the entailment relation of\(\mathbf{B}_4\). This entailment relation is formalized as thewell-known logic of “first-degree entailment”(\(E_{fde}\)) introduced in Anderson & Belnap (1975) (see alsoOmori and Wansing 2017).

The syntactic notion of a single-conclusion consequence relation hasbeen extensively studied by representatives of the Polish school oflogic, most notably by Alfred Tarski, who in fact initiated this lineof research (see Tarski 1930a,b; cf. also Wójcicki 1988). Inview of certain key features of a standard consequence relation it isquite remarkable—as well as important—that any entailmentrelation \(\vDash_V\) defined as above has the following structuralproperties (see Ryan and Sadler 1992: 34):

\[\begin{align}\tag{4} \Delta\cup \{A\}&\vDash_V A &\textrm{(Reflexivity)} \\ \tag{5} \textrm{If } \Delta\vDash_V A \textrm{ then } \Delta\cup \Gamma &\vDash_V A &\textrm{(Monotonicity)}\\ \tag{6} \textrm{If } \Delta\vDash_V A \textrm{ for every } A \in \Gamma &\\ \textrm{ and } \Gamma \cup \Delta \vDash_V B, \textrm{ then } \Delta &\vDash_VB & \textrm{(Cut)} \end{align}\]

Moreover, for every \(A \in \mathcal{L}\), every \(\Delta \subseteq\mathcal{L}\), and every uniform substitution function \(\sigma\) on\(\mathcal{L}\) the followingSubstitution property holds(\(\sigma(\Delta)\) stands for \(\{ \sigma(B) \mid B \in \Delta\})\):

\[ \tag{7} \Delta\vDash_V A \textrm{ implies } \sigma(\Delta)\vDash_V \sigma(A). \]

(The function of uniform substitution \(\sigma\) is defined as follows.Let \(B\) be a formula in \(\mathcal{L}\), let \(p_1,\ldots, p_n\) beall the propositional variables occurring in \(B\), and let \(\sigma(p_1)= A_1,\ldots , \sigma(p_n) = A_n\) for some formulas \(A_1 ,\ldots ,A_n\).Then \(\sigma(B)\) is the formula that results from B by substitutingsimultaneously \(A_1\),…, \(A_n\) for all occurrences of\(p_1,\ldots, p_n\), respectively.)

If \(\vDash_V\) in the conditions(4)–(6) is replaced by \(\vdash_{ \mathfrak{L} }\), then one obtains what isoften called aTarskian consequence relation. If additionallya consequence relation has the substitution property (7), then it is calledstructural. Thus, any entailment relationdefined for a given valuation system \(\mathbf{V}\) presents animportant example of a consequence relation, in that \(\mathbf{V}\) isstrictly characteristic for some logical system \(\mathfrak{L}\) witha structural Tarskian consequence relation.

Generally speaking, the framework of valuation systems not onlyperfectly suits the conception of logic as the science of truthvalues, but also turns out to be an effective technical tool forresolving various sophisticated and important problems in modernlogic, such as soundness, completeness, independence of axioms,etc.

2.3 Truth values, truth degrees, and vague concepts

The term ‘truth degrees’, used by Gottwald and many otherauthors, suggests that truth comes by degrees, and these degrees maybe seen as truth values in an extended sense. The idea of truth as agraded notion has been applied to model vague predicates and to obtaina solution to the Sorites Paradox, the Paradox of the Heap (see theentry on theSorites Paradox). However, the success of applying many-valued logic to the problem ofvagueness is highly controversial. Timothy Williamson (1994: 97), forexample, holds that the phenomenon of higher-order vagueness“makes most work on many-valued logic irrelevant to the problemof vagueness”.

In any case, the vagueness of concepts has been much debated inphilosophy (see the entry onvagueness) and it was one of the major motivations for the development offuzzy logic (see the entry onfuzzy logic). In the 1960s, Lotfi Zadeh (1965) introduced the notion of afuzzyset. A characteristic function of a set \(X\) is a mapping whichis defined on a superset \(Y\) of \(X\) and which indicates membershipof an element in \(X\). The range of the characteristic function of aclassical set \(X\) is the two-element set \(\{0,1\}\) (which may beseen as the set of classical truth values). The function assigns thevalue 1 to elements of \(X\) and the value 0 to all elements of \(Y\)not in \(X\). A fuzzy set has a membership function ranging over thereal interval [0,1]. A vague predicate such as ‘is much earlierthan March 20^th, 1963’, ‘is beautiful’,or ‘is a heap’ may then be regarded as denoting a fuzzyset. The membership function \(g\) of the fuzzy set denoted by‘is much earlier than March 20^th, 1963’ thusassigns values (seen as truth degrees) from the interval [0, 1] tomoments in time, for example \(g\)(1p.m., August 1^st, 2006)\(= 0\), \(g\)(3a.m., March 19^th, 1963) \(= 0\),\(g\)(9:16a.m., April 9^th, 1960) \(= 0.005\), \(g\)(2p.m.,August 13^th, 1943) \(= 0.05\), \(g\)(7:02a.m., December2^nd, 1278) \(= 1\).

The application of continuum-valued logics to the Sorites Paradox hasbeen suggested by Joseph Goguen (1969). The Sorites Paradox in itsso-called conditional form is obtained by repeatedly applyingmodus ponens in arguments such as:

A collection of 100,000 grains of sand is a heap.
If a collection of 100,000 grains of sand is a heap, then acollection 99,999 grains of sand is a heap.
If a collection of 99,999 grains of sand is a heap, then acollection 99,998 grains of sand is a heap.
…
If a collection of 2 grains of sand is a heap, then a collectionof 1 grain of sand is a heap.
Therefore: A collection of 1 grain of sand is a heap.

Whereas it seems that all premises are acceptable, because the firstpremise is true and one grain does not make a difference to acollection of grains being a heap or not, the conclusion is, ofcourse, unacceptable. If the predicate ‘is a heap’ denotesa fuzzy set and the conditional is interpreted as implication inŁukasiewicz’s continuum-valued logic, then the SoritesParadox can be avoided. The truth-function \(f_{\rightarrow}\) ofŁukasiewicz’s implication \(\rightarrow\) is defined bystipulating that if \(x \le y\), then \(f_{\rightarrow}(x, y) = 1\),and otherwise \(f_{\rightarrow}(x, y) = 1 - (x - y)\). If, say, thetruth value of the sentence ‘A collection of 500 grains of sandis a heap’ is 0.8 and the truth value of ‘A collection of499 grains of sand is a heap’ is 0.7, then the truth value ofthe implication ‘If a collection of 500 grains of sand is aheap, then a collection 499 grains of sand is a heap.’ is 0.9.Moreover, if the acceptability of a statement is defined as having avalue greater than \(j\) for \(0 \lt j \lt 1\) and all the conditionalpremises of the Sorites Paradox do not fall below the value \(j\),thenmodus ponens does not preserve acceptability, becausethe conclusion of the Sorites Argument, being evaluated as 0, isunacceptable.

Alasdair Urquhart (1986: 108) stresses

the extremely artificial nature of the attaching of precise numericalvalues to sentences like … “Picasso’sGuernica is beautiful”.

To overcome the problem of assigning precise values to predications ofvague concepts, Zadeh (1975) introducedfuzzy truth values asdistinct from the numerical truth values in [0, 1], the former beingfuzzy subsets of the set [0, 1], understood astrue,verytrue,not very true, etc.

The interpretation of continuum-valued logics in terms of fuzzy settheory has for some time be seen as defining the field of mathematicalfuzzy logic. Susan Haack (1996) refers to such systems of mathematicalfuzzy logic as “base logics” of fuzzy logic and reservesthe term ‘fuzzy logics’ for systems in which the truthvalues themselves are fuzzy sets. Fuzzy logic in Zadeh’s lattersense has been thoroughly criticized from a philosophical point ofview by Haack (1996) for its “methodologicalextravagances” and its linguistic incorrectness. Haackemphasizes that her criticisms of fuzzy logic do not apply to the baselogics. Moreover, it should be pointed out that mathematical fuzzylogics are nowadays studied not in the first place as continuum-valuedlogics, but as many-valued logics related to residuated lattices (seeHajek 1998; Cignoliet al. 2000; Gottwald 2001; Galatoset al. 2007), whereas fuzzy logic in the broad sense is to alarge extent concerned with certain engineering methods.

A fundamental concern about the semantical treatment of vaguepredicates is whether an adequate semantics should betruth-functional, that is, whether the truth value of a complexformula should depend functionally on the truth values of itssubformulas. Whereas mathematical fuzzy logic is truth-functional,Williamson (1994: 97) holds that “the nature of vagueness is notcaptured by any approach that generalizes truth-functionality”.According to Williamson, the degree of truth of a conjunction, adisjunction, or a conditional just fails to be a function of thedegrees of truth of vague component sentences. The sentences‘John is awake’ and ‘John is asleep’, forexample, may have the same degree of truth. By truth-functionality thesentences ‘If John is awake, then John is awake’ and‘If John is awake, then John is asleep’ are alike in truthdegree, indicating for Williamson the failure ofdegree-functionality.

One way of in a certain sense non-truthfunctionally reasoning aboutvagueness is supervaluationism. The method of supervaluations has beendeveloped by Henryk Mehlberg (1958) and Bas van Fraassen (1966) andhas later been applied to vagueness by Kit Fine (1975), Rosanna Keefe(2000) and others.

Van Fraassen’s aim was to develop a semantics for sentencescontaining non-denoting singular terms. Even if one grants atomicsentences containing non-denoting singular terms and that someattributions of vague predicates are neither true nor false, itnevertheless seems natural not to preclude that compound sentences ofa certain shape containing non-denoting terms or vague predicationsare either true or false, e.g., sentences of the form‘If \(A\), then \(A\)’. Supervaluational semanticsprovides a solution to this problem. A three-valued assignment \(a\)into \(\{T, I, F\}\) may assign a truth-value gap (or rather the value\(I)\) to the vague sentence ‘Picasso’sGuernicais beautiful’. Any classical assignment \(a'\) that agrees with\(a\) whenever \(a\) assigns \(T\) or \(F\) may be seen as aprecisification (or superassignment) of \(a\). A sentence may than besaid to be supertrue under assignment \(a\) if it is true under everyprecisification \(a'\) of \(a\). Thus, if \(a\) is a three-valuedassignment into \(\{T, I, F\}\) and \(a'\) is a two-valued assignmentinto \(\{T, F\}\) such that \(a(p) = a'(p)\) if \(a(p) \in \{T, F\}\),then \(a'\) is said to be asuperassignment of \(a\). Itturns out that if \(a\) is an assignment extended to a valuationfunction \(v_a\) for the Kleene matrix \(\mathbf{K}_3\), then forevery formula \(A\) in the language of \(\mathbf{K}_3\), \(v_a (A) =v_{a'}(A)\) if \(v_a (A) \in \{T, F\}\). Therefore, the function\(v_{a'}\) may be called asupervaluation of \(v_a\). Aformula is then said to besupertrue under a valuationfunction \(v_a\) for \(\mathbf{K}_3\) if it is true under everysupervaluation \(v_{a'}\) of \(v_a\), i.e., if \(v_{a'}(A) = T\) forevery supervaluation \(v_{a'}\) of \(v_a\). The property of beingsuperfalse is defined analogously.

Since every supervaluation is a classical valuation, every classicaltautology is supertrue under every valuation function in\(\mathbf{K}_3\). Supervaluationism is, however, not truth-functionalwith respect to supervalues. The supervalue of a disjunction, forexample, does not depend on the supervalue of the disjuncts. Suppose\(a(p) = I\). Then \(a(\neg p) = I\) and \(v_{a'} (p\vee \neg p) = T\)for every supervaluation \(v_{a'}\) of \(v_a\). Whereas \((p\vee \negp)\) is thus supertrue under \(v_a,p\vee p\) isnot, becausethere are superassignments \(a'\) of \(a\) with \(a'(p) = F\). Anargument against the charge that supervaluationism requires anon-truth-functional semantics of the connectives can be found inMacFarlane (2008) (cf. also other references given there).

Although the possession of supertruth is preserved from the premisesto the conclusion(s) of valid inferences in supervaluationism, andalthough it might be tempting to consider supertruth an abstractobject on its own, it seems that it has never been suggested tohypostatize supertruth in this way, comparable to Frege’sthe True. A sentence supertrue under a three-valued valuation\(v\) just takes the Fregean valuethe True under everysupervaluation of \(v\). The advice not to confuse supertruth with“real truth” can be found in Belnap (2009).

2.4 Suszko’s thesis and anti-designated values

One might, perhaps, think that the mere existence of many-valuedlogics shows that there exist infinitely, in fact, uncountably manytruth values. However, this is not at all clear (recall the morecautious use of terminology advocated by Gottwald).

In the 1970’s Roman Suszko (1977: 377) declared many-valuedlogic to be “a magnificent conceptual deceit”. Suszkoactually claimed that “there are but two logical values, trueand false” (Caleiroet al. 2005: 169), a statement nowcalledSuszko’s Thesis. For Suszko, the set of truthvalues assumed in a logical matrix for a many-valued logic is a set of“admissible referents” (called “algebraicvalues”) of formulas but not a set of logical values. Whereasthe algebraic values are elements of an algebraic structure andreferents of formulas, the logical valuetrue is used todefine valid consequence: If every premise is true, then so is (atleast 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. Thelogical values are thus represented by a bi-partition of the set ofalgebraic values into a set of designated values (truth) and itscomplement (falsity).

Essentially the same idea has been taken up earlier by Dummett (1959)in an influential paper, where he asks

what point there may be in distinguishing between different ways inwhich a statement may be true or between different ways in which itmay be false, or, as we might say, between degrees of truth andfalsity. (Dummett 1959: 153)

Dummett observes that, first,

the sense of a sentence is determined wholly by knowing the case inwhich it has a designated value and the cases in which it has anundesignated one,

and moreover,

finer distinctions between different designated values or differentundesignated ones, however naturally they come to us, are justifiedonly if they are needed in order to give a truth-functional account ofthe formation of complex statements by means of operators. (Dummett1959: 155)

Suszko’s claim evidently echoes this observation by Dummett.

Suszko’s Thesis is substantiated by a rigorous proof (the SuszkoReduction) showing that every structural Tarskian consequence relationand therefore also every structural Tarskian many-valued propositionallogic is characterized by a bivalent semantics. (Note also thatRichard Routley (1975) has shown that every logic based on aλ-categorical language has a sound and complete bivalentpossible worlds semantics.) The dichotomy between designated valuesand values which are not designated and its use in the definition ofentailment plays a crucial role in the Suszko Reduction. Nevertheless,while it seems quite natural to construe the set of designated valuesas a generalization of the classical truth value \(T\) in some of itssignificant roles, it would not always be adequate to interpret theset of non-designated values as a generalization of the classicaltruth value \(F\). The point is that in a many-valued logic, unlike inclassical logic, “not true” does not always mean“false” (cf., e.g., the above interpretation ofKleene’s logic, where sentences can be neither true norfalse).

In the literature on many-valued logic it is sometimes proposed toconsider a set ofantidesignated values which notobligatorily constitute the complement of the set of designated values(see, e.g., Rescher 1969, Gottwald 2001). The set of antidesignatedvalues can be regarded as representing a generalized concept offalsity. This distinction leaves room for values that areneither designatednor antidesignated and even forvalues that areboth designatedandantidesignated.

Grzegorz Malinowski (1990, 1994) takes advantage of this proposal togive a counterexample to Suszko’s Thesis. He defines the notionof a single-conclusionquasi-consequence \((q\)-consequence)relation. The semantic counterpart of \(q\)-consequence is called\(q\)-entailment. Single-conclusion \(q\)-entailment is defined byrequiring that if no premise is antidesignated, the conclusion isdesignated. Malinowski (1990) proved that for every structural\(q\)-consequence relation, there exists a characterizing class of\(q\)-matrices, matrices which in addition to a subset\(\mathcal{D}^{+}\)of designated values comprise a disjoint subset\(\mathcal{D}^-\)of antidesignated values. Not every \(q\)-consequencerelation has a bivalent semantics.

In the supplementary documentSuszko’s Thesis, Suszko’s reduction is introduced, Malinowski’scounterexample to Suszko’s Thesis is outlined, and a shortanalysis of these results is presented.

Can one provide evidence for a multiplicity of logical values? Moreconcretely, \(is\) there more than one logical value, each of whichmay be taken to determine its own (independent) entailment relation? Apositive answer to this question emerges from considerations on truthvalues as structured entities which, by virtue of their internalstructure, give rise to natural partial orderings on the set ofvalues.

2.5 Some approaches to truth value functions

One way of generalizing the notion of a truth value function wasproposed by Arnon Avron and his collaborators by introducing the ideaofnon-deterministic matrices (see, e.g., Avron andKonikowska 2005, Avron and Lev 2005, Avron and Zamansky 2010, and, forearlier work, Kearns 1981, Ivlev 1988). The standard valuation systemsas described above rely on the principle of compositionality, so thatthe truth value of a (compound) sentence is uniquely determined by thetruth values of its components. However, this principle may be toostrong for some situations, and one may try to weaken it by allowing(in certain cases) the truth values of sentences to be chosennondeterministically from some non-empty set of options. Inthis way we obtain the notion of a non-deterministic matrix, whichdiffers from a standard valuation system in that each truth valuefunction from a set of functions \(\cal{F}\) \(=\) \(\{f_{c_{1}},\ldots f_{c_{m}}\}\) is now defined as a map from \({\cal V}^n\) to\({\cal P}({\cal V}) \backslash \lbrace \varnothing\rbrace\). Thus,the value \(v_{a}(c_i (A_1, \ldots , A_n))\) of the sentence \(c_i(A_1, \ldots , A_n)\) induced by an assignment \(a\) is taken(non-deterministically) from a non-empty set of values. Such asituation may occur, in particular, in cases of linguistic ambiguity,when, for example, we use the connective “or” and it isnot quite clear whether we are dealing with inclusive or exclusivedisjunction, see (Avron and Zamansky 2010). Then we simply allow thedisjunction to take some value from the set \(\{T, F\}\) if bothdisjuncts are true, and which value is chosen in this or that case maydepend on additional information, namely whether “or” isused inclusively or exclusively:

\[\begin{array}{c|c|c} f_\vee& T & F \\ \hline T & \{T, F\} & \{T\} \\ F &\{T\} & \{F\} \\ \end{array}\]

Non-deterministic matrices have proven to be very useful in variousareas, such as automated deduction, approximating logics,paraconsistent logics, fuzzy logics and others. There is now a bulk ofwork on non-deterministic semantics, including especiallynon-deterministic semantics for modal logics (see, for example Omoriand Skurt 2016, 2020, Coniglio, del Cerro, and Peron 2019, Pawlowskiand Skurt 2024). In (Omori and Skurt 2020) it is shown that thereexist modal logics that possess a non-deterministic semantics but nostandard Kripke semantics.

Usually, in a context, sentences are taken to be connected by variousrelations. From a logical perspective, it is important to focus onthose relations between sentences that are determined by the relationsbetween their truth-values. Such relations are associated with truthfunctions. On the other side, every function is a relation.Humberstone (2023) discusses, as one way of connecting a relation witha given function, the association of an \(n\)-place Boolean truthfunction \({f \!\!:}\) \({\{T,F\}^n {\longrightarrow} \{T,F\}}\) withthe \(n\)-place de-characteristic relation\({\stackrel{\leftarrow}{\chi}}(f)\) of \(f\) that is defined asfollows:

\[{\stackrel{\leftarrow}{\chi}}(f) := \{\langle a_1, \ldots , a_n\rangle \mid f( a_1, \ldots , a_n) = T\}.\]

Since \({\stackrel{\leftarrow}{\chi}}({\stackrel{}{\chi}}(R)) = R\)for any relation \(R\) and its characteristic function\({\stackrel{}{\chi}}(R)\) and \({\stackrel{}{\chi}}({\stackrel{\leftarrow}{\chi}}(f)) = f\) for any function into\(\{T,F\}\), the functional and relational perspectives areintertranslatable. Humberstone shows that the relational point of viewnevertheless can be useful in practice, for instance, when theconsideration of relations among truth values is lifted to relationsamong formulas of propositional languages that obtain in virtue of thetruth-value relations among the truth-values assigned to the formulas.Given a class \(\cal V\) of valuation functions taking values in\(\{T,F\}\) and \(v \in {\cal V}\), and restricting attention tobinary connectives \(\sharp\), two kinds of such relations amongformulas can be defined by setting:

\[ \begin{array}{lcl}R^{\sharp}_{{v}}(A , B) & \mbox{iff} & v(A \sharp B) = T \\ R^{\sharp}_{{\cal V}}(A , B) & \mbox{iff} & \mbox{for all } v \in {\cal V}\!: R^{\sharp}_{{v}}(A , B), \end{array} \]

where in general \(\sharp\) is not required to denote a truth functionover \(\cal V\) such that there is a function \({g\!:}\) \(\{T,F\}^2\longrightarrow \{T,F\}\) and for all \(v \in {\cal V}\) and formulas\(A\), \(B\), \(v(A \sharp B) = g((v(A), v(B))\). If such a function\(g\) exists, then the first item can be stated as\(R^{\sharp}_{{v}}(A ,B)\) iff \(\langle v(A), v(B)\rangle \in{\stackrel{\leftarrow}{\chi}} (g)\). One can then compare propertiesof the local relations \(R^{\sharp}_{{v}}\) with properties of theglobal relation \(R^{\sharp}_{{\cal V}}\). If the semantic consequencerelation \(\models _{{\cal V}}\) with respect to \(\cal V\) is asusually defined by requiring that for any set \(\Gamma \cup \{A\}\) offormulas and \(v \in {\cal V}\), \(\Gamma \models _{{\cal V}} A\) iff\(v(A) = T\) whenever \(v(B) = T\) for every \(B \in \Gamma\), one canobserve, e.g., that in general the transitivity of \(R^{\sharp}_{{\calV}}\) does not imply the transitivity of the relations\(R^{\sharp}_{{v}}\) and that also the symmetry of \(R^{\sharp}_{{\calV}}\) does not imply the symmetry of the relations\(R^{\sharp}_{{v}}\). Humberstone's motivation for taking thetruth-value relational perspective is its explanatory value, forexample in making certain proofs more evident by invoking facts suchas that any reflexive relation on a two-element set is transitive.

3. Ordering relations between truth-values

3.1 The notion of a logical order

As soon as one admits that truth values come with valuationsystems, it is quite natural to assume that the elements ofsuch a system are somehowinterrelated. And indeed, alreadythe valuation system for classical logic constitutes a well-knownalgebraic structure, namely the two-element Boolean algebra with\(\cap\) and \(\cup\) as meet and join operators (see the entry on themathematics of Boolean algebra). In its turn, this Boolean algebra forms a lattice with apartialorder defined by \(a\le_t b \textrm{ iff } a\cap b = a\). Thislattice may be referred to asTWO. It is easy to see that theelements ofTWO are ordered as follows: \(F\le_t T\). Thisordering is sometimes called thetruth order (as indicated bythe corresponding subscript), for intuitively it expresses an increasein truth: \(F\) is “less true” than \(T\). It can beschematically presented by means of a so-called Hasse-diagram as inFigure 1.

Figure 1: LatticeTWO

It is also well-known that the truth values of both Kleene’s andPriest’s logic can be ordered to form a lattice(THREE), which is diagrammed in Figure 2.

Figure 2: LatticeTHREE

Here \(\le_t\) orders \(T, I\) and \(F\) so that the intermediatevalue \(I\) is “more true” than \(F\), but “lesstrue” than \(T\).

The relation \(\le_t\) is also called alogical order,because it can be used to determine key logical notions: logicalconnectives and an entailment relation. Namely, if the elements of thegiven valuation system \(\mathbf{V}\) form a lattice, then theoperations of meet and join with respect to \(\le_t\) are usually seenas the functions for conjunction and disjunction, whereas negation canbe represented by the inversion of this order. Moreover, one canconsider an entailment relation for \(\mathbf{V}\) as expressingagreement with the truth order, that is, the conclusion should be atleast as true as the premises taken together:

\[ \tag{8} \Delta\vDash B\textrm{ iff }\forall v_a[\Pi_t\{ v_a (A) \mid A \in \Delta\} \le_t v_a (B)], \]

where \(\Pi_t\) is the lattice meet in the corresponding lattice.

The Belnap matrix \(\mathbf{B}_4\) considered above also can berepresented as a partially ordered valuation system. The set of truthvalues \(\{\mathbf{N}, \mathbf{T}, \mathbf{F}, \mathbf{B}\}\) from\(\mathbf{B}_4\) constitutes a specific algebraic structure –thebilattice FOUR\(_2\) presented in Figure 3 (see, e.g.,Ginsberg 1988, Arieli and Avron 1996, Fitting 2006).

Figure 3: The bilatticeFOUR\(_2\)

This bilattice is equipped withtwo partial orderings; inaddition to a truth order, there is an information order \((\le_i )\)which is said to order the values under consideration according to theinformation they give concerning a formula to which they are assigned.Lattice meet and join with respect to \(\le_t\) coincide with thefunctions \(f_{\wedge}\) and \(f_{\vee}\) in the Belnap matrix\(\mathbf{B}_4\), \(f_{{\sim}}\) turns out to be the truth orderinversion, and an entailment relation, which happens to coincide withthe matrix entailment, is defined by (8).FOUR\(_2\) arises as a combination of two structures: theapproximation lattice \(A_4\) and the logical lattice \(L_4\) whichare discussed in Belnap 1977a and 1977b (see also, Anderson, Belnapand Dunn 1992: 510–518)).

3.2 Truth values as structured entities. Generalized truth values

Frege (1892: 30) points out the possibility of “distinctions ofparts within truth values”. Although he immediately specifiesthat the word ‘part’ is used here “in a specialsense”, the basic idea seems nevertheless to be that truthvalues are not something amorphous, but possess some inner structure.It is not quite clear how serious Frege is about this view, but itseems 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 truthvalues are represented as being made up from some primitivecomponents. For example, in some explications of Kripke models forintuitionistic logic propositions (identified with sets of“worlds” in a model structure) can be understood as truthvalues of a certain kind. Then the empty proposition is interpreted asthe valuefalse, and the maximal proposition (the set of allworlds in a structure) as the valuetrue. Moreover, one canconsider non-empty subsets of the maximal proposition as intermediatetruth values. Clearly, the intuitionistic truth values so conceivedare composed from some simpler elements and as such they turn out tobe complex entities.

Another prominent example of structured truth values are the“truth-value objects” in topos models from category theory(see the entry oncategory theory). For any topos \(C\) and for a \(C\)-object Ω one can define atruth value of \(C\) as an arrow \(1 \rightarrow \Omega\) (“asubobject classifier for \(C\)”), where 1 is a terminal objectin \(C\) (cf. Goldblatt 2006: 81, 94). The set of truth values sodefined plays a special role in the logical structure of \(C\), sincearrows of the form \(1 \rightarrow \Omega\) determine central semanticalnotions for the given topos. And again, these truth values evidentlyhave some inner structure.

One can also mention in this respect the so-called “factorsemantics” for many-valued logic, where truth values are definedas ordered \(n\)-tuples of classical truth values \((T\)-\(F\)sequences, see Karpenko 1983). Then the value \(3/5\), for example,can be interpreted as a \(T\)-\(F\) sequence of length 5 with exactly3 occurrences of \(T\). Here the classical values \(T\) and \(F\) areused as “building blocks” for non-classical truthvalues.

Moreover, the idea of truth values as compound entities nicelyconforms with the modeling of truth values considered above inthree-valued (Kleene, Priest) and four-valued (Belnap) logics ascertain subsets of the set of classical truth values. The latterapproach stems essentially from Dunn (1976), where a generalization ofthe notion of a classical truth-value function has been proposed toobtain so-called “underdetermined” and“overdetermined” valuations. Namely, Dunn considers avaluation to be a function not from sentences to elements of the set\(\{T, F\}\) but from sentences to subsets of this set (see also Dunn2000: 7). By developing this idea, one arrives at the concept of ageneralized truth value function, which is a function fromsentences into thesubsets of somebasic set of truthvalues (see Shramko and Wansing 2005). The values of generalizedtruth value functions can be calledgeneralized truthvalues.

By employing the idea of generalized truth value functions, one canobtain a hierarchy of valuation systems starting with a certainset-theoretic representation of the valuation system for classicallogic. The representation in question is built on a single initialvalue which serves then as the designated value of the resultingvaluation system. More specifically, consider the singleton\(\{\varnothing \}\) taken as the basic set subject to a furthergeneralization procedure. Set-theoretically the basic set can serve asthe universal set (the complement of the empty set) for the valuationsystem \(\mathbf{V}^{\varnothing}_{cl}\) introduced below. At thefirst stage \(\varnothing\) comes out with no specific intuitiveinterpretation, it is only important to take it as some distinctunit. Consider then the power-set of \(\{\varnothing \}\)consisting of exactly two elements: \(\{\{\varnothing \}, \varnothing\}\). Now, these elements can be interpreted as Frege’stheTrue andthe False, and thus it is possible to constructa valuation system for classical logic,\(\mathbf{V}^{\varnothing}_{cl} = \langle \{\{\varnothing \},\varnothing \}, \{\{\varnothing \}\}, \{f_{\wedge}, f_{\vee},f_{\rightarrow}, f_{\sim}\}\rangle\), where the functions\(f_{\wedge}, f_{\vee}, f_{\rightarrow}, f_{\sim}\) are defined asfollows (for \[ \begin{align}X, Y \in \{\{\varnothing \}, \varnothing \}:\quad & f_{\wedge}(X, Y) = X\cap Y; \\ & f_{\vee}(X, Y) = X\cup Y; \\ & f_{\rightarrow}(X, Y) = X^{c}\cup Y; \\ & f_{\sim}(X) = X^{c}. \end{align} \] It is not difficult to see that for anyassignment \(a\) relative to \(\mathbf{V}^{\varnothing}_{cl}\), andfor any formulas \(A\) and \(B\), the following holds:

\[\begin{align}v_a (A\wedge B) = \{\varnothing \}&\Leftrightarrow v_a (A) = \{\varnothing \} \text{ and } v_a (B) = \{\varnothing \}; \\ v_a (A\vee B) = \{\varnothing \}&\Leftrightarrow v_a (A) = \{\varnothing \} \text{ or } v_a (B) = \{\varnothing \}; \\ v_a (A\rightarrow B) = \{\varnothing \}&\Leftrightarrow v_a (A) = \varnothing \text{ or } v_a (B) = \{\varnothing \}; \\ v_a (\sim A) = \{\varnothing \}&\Leftrightarrow v_a (A) = \varnothing. \end{align}\]

This shows that \(f_{\wedge}, f_{\vee}, f_{\rightarrow}\) and\(f_{\sim}\) determine exactly the propositional connectives ofclassical logic. One can conveniently mark the elements\(\{\varnothing \}\) and \(\varnothing\) in the valuation system\(\mathbf{V}^{\varnothing}_{cl}\) by the classical labels \(T\) and\(F\). Note that within \(\mathbf{V}^{\varnothing}_{cl}\) it is fullyjustifiable to associate \(\varnothing\) with falsity, taking intoaccount the virtualmonism of truth characteristic forclassical logic, which treats falsity not as an independent entity butmerely as the absence of truth.

Then, by taking the set \(\mathbf{2} = \{F, T\}\) of these classicalvalues as the basic set for the next valuation system, one obtains thefour truth values of Belnap’s logic as the power-set of the setof classical values \(\mathcal{P}(\mathbf{2}) = \mathbf{4}: \mathbf{N}= \varnothing\), \(\mathbf{F} = \{F\} (= \{\varnothing \})\),\(\mathbf{T} = \{T\} (= \{\{\varnothing \}\})\) and \(\mathbf{B} =\{F, T\} (= \{\varnothing, \{\varnothing \}\})\). In this way,Belnap’s four-valued logic emerges as a certain generalizationof classical logic with its two Fregean truth values. InBelnap’s logic truth and falsity are considered to befull-fledged, self-sufficient entities, and therefore \(\varnothing\)is now to be interpreted not as falsity, but as a real truth-value gap(neither truenor false). The dissimilarity ofBelnap’s truth and falsity from their classical analogues isnaturally expressed by passing from the corresponding classical valuesto their singleton-sets, indicating thus their new interpretations asfalse only andtrue only. Belnap’sinterpretation of the four truth values has been critically discussedin (Lewis 1982) and (Dubois 2008) (see also the reply to Dubois inWansing and Belnap 2010).

Generalized truth values have a strong intuitive background,especially as a tool for the rational explication of incomplete andinconsistent information states. In particular, Belnap’sheuristic interpretation of truth values as information that“has been told to a computer” (see Belnap 1977a,b; alsoreproduced in Anderson, Belnap and Dunn 1992, §81) has beenwidely acknowledged. As Belnap points out, a computer may receive datafromvarious (maybe independent) sources. Belnap’scomputers have to take into account various kinds of informationconcerning a given sentence. 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 arepossible: (3) nothing is told about the sentence or (4) the sourcessupply inconsistent information, information that the sentence is trueand information that it is false. And the four truth values from\(\mathbf{B}_4\) naturally correspond to these four situations: thereis no information that the sentence is false and no information thatit is true \((\mathbf{N})\), there ismerely information thatthe sentence is false \((\mathbf{F})\), there ismerelyinformation that the sentence is true \((\mathbf{T})\), and there isinformation that the sentence is false, but there is also informationthat it is true \((\mathbf{B})\).

Joseph Camp in (2002: 125–160) provides Belnap’s fourvalues with quite a different intuitive motivation by developing whathe calls a “semantics of confused thought”. Consider arational agent, who happens to mix up two very similar objects (say,\(a\) and \(b)\) and ambiguously uses one name (say,‘\(C\)’) for both of them. Now let such an agent assertsome statement, saying, for instance, that \(C\) has some property.How should one evaluate this statement if \(a\) has the property inquestion whereas \(b\) lacks it? Camp argues against ascribing truthvalues to such statements and puts forward an “epistemicsemantics” in terms of “profitability” and“costliness” as suitable characterizations of sentences. Asentence \(S\) is said to be “profitable” if one wouldprofit from acting on the belief that \(S\), and it is said to be“costly” if acting on the belief that \(S\) would generatecosts, for example as measured by failure to achieve an intended goal.If our “confused agent” asks some external observerswhether \(C\) has the discussed property, the following four answersare possible: ‘yes’ (mark the corresponding sentence with\(\mathbf{Y})\), ‘no’ (mark it with \(\mathbf{N})\),‘cannot say’ (mark it with?),‘yes’ and ‘no’ (mark it withY&N). Note that the external observers, whoprovide answers, are “non-confused” and have differentobjects in mind as to the referent of ‘\(C\)’, in view ofall the facts that may be relevant here. Camp conceives these fourpossible answers concerning epistemic properties of sentences as akind of “semantic values”, interpreting them as follows:the value \(\mathbf{Y}\) is an indicator of profitability, the value\(\mathbf{N}\) is an indicator of costliness, the value? is no indicator either way, and the valueY&N is both an indicator of profitability and anindicator of costliness. A strict analogy between this“semantics of confused reasoning” and Belnap’s fourvalued logic is straightforward. And indeed, as Camp (2002: 157)observes, the set of implications valid according to his semantics isexactly the set of implications of the entailment system \(E_{fde}\).In (Zaitsev and Shramko 2013) it is demonstrated how ontological andepistemic aspects of truth values can be combined within a jointsemantical framework. Kapsner (2019) extends Belnap’s frameworkby two additional values “Contestedly-True” and“Contestedly-False” which allows for new outcomes fordisjunctions and conjunctions between statements with values\(\mathbf{B}\) and \(\mathbf{N}\).

The conception of generalized truth values has its purely logicalimport as well. If one continues the construction and applies the ideaof generalized truth value functions to Belnap’s four truthvalues, then one obtains further valuation systems which can berepresented by variousmultilattices. One arrives, inparticular, atSIXTEEN\(_3\) – thetrilatticeof 16 truth-values, which can be viewed as a basis for a logic ofcomputer networks (see Shramko and Wansing 2005, 2006; Kamide andWansing 2009; Odintsov 2009; Wansing 2010; Odintsov and Wansing 2015;cf. also Shramko, Dunn, Takenaka 2001). The notion of a multilatticeandSIXTEEN\(_3\) are discussed further in the supplementarydocumentGeneralized truth values and multilattices. A comprehensive study of the conception of generalized logical valuescan be found in (Shramko and Wansing 2011).

4. Concluding remarks

Gottlob Frege’s notion of a truth value has become part of thestandard philosophical and logical terminology. The notion of a truthvalue is an indispensable instrument of realistic, model-theoreticapproaches to semantics. Indeed, truth values play an essential rolein applications of model-theoretic semantics in areas such as, forexample, knowledge representation and theorem proving based onsemantic tableaux, which could not be treated in the present entry.Moreover, considerations on truth values give rise to deep ontologicalquestions concerning their own nature, the feasibility of factontologies, and the role of truth values in such ontological theories.There also exist well-motivated theories of generalized truth valuesthat lead far beyond Frege’s classical valuesthe Trueandthe False. (For various directions of further logical andphilosophical investigations in the area of truth values see Shramko& Wansing 2009b, 2009c.)

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We'd like to thank an anonymous reviewer for comments about thematerial in Section 1.2, which led to an improved discussion.

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