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

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

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

The notion of a truth value has been explicitly introduced intologic and philosophy by Gottlob Frege—for the first time in Frege1891, and most notably in his seminal paper (Frege 1892). Although it was Frege who made the notion of a truth value to one of the central concepts of semantics, the idea of special semantical values, however, was anticipated by Boole and Peirce already, see the survey article on a “history of truth values” by Béziau (2012). According to Kneale and Kneale (1962, p. 413), Boole’s system contains all that is needed for its interpretation “in terms of truth values of propositions”, and as Church (1956, p. 17) remarks, the “explicit use of two truth-values appears for the first time in a paper by C. S. Peirce in theAmerican Journal of Mathematics, 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 (das Wahre)andthe False (das Falsche):

A sentence proper is a proper name, and its Bedeutung, ifit has one, is a truth-value: the True or the False (Beaney 1997,297).

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 itscomponents. On this basis one can also discriminate betweenextensional and intensional contexts and advance further to theconception of intensional logics. Moreover, the idea of truth valueshas induced a radical rethinking of some central issues in thephilosophy of logic, including: the categorial status of truth, thetheory of abstract objects, the subject-matter of logic and itsontological foundations, the concept of a logical system, the natureof 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
  • 1.2 Truth as a property versus truth as an object
  • 1.3 The ontology of truth values
• 2. Truth values as logical values
  • 2.1 Logic as the science of logical values
  • 2.2 Many-valued logics, truth degrees, and valuation systems
  • 2.3 Truth values, truth degrees, and vague concepts
  • 2.4 Suszko’s thesis and anti-designated values
  • Supplementary Document: Suszko’s Thesis
• 3. Ordering relations between truth-values
  • 3.1 The notion of a logical order
  • 3.2 Truth values as structured entities. Generalized truth values
  • Supplementary Document: Generalized Truth Values and Multilattices
• 4. Concluding remarks
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

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 restsessentially on the idea of a strict discrimination between two mainkinds of expressions: proper names (singular terms) and functionalexpressions. Proper names designate (signify, denote, or refer to)singular objects, and functional expressions designate (signify,denote, or refer to) functions. [Note: In the literature, theexpressions ‘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 the Ukraine to Kyiv (Kiev). 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 calledthearguments 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 ofKyiv. Note that Frege distinguishes between ann-placefunctionf as an unsaturated entity that can be completed byand applied toargumentsa₁,…,a_n anditscourse of values, which can be seen as the set-theoreticrepresentation of this function: the set{⟨a₁,…,a_n,a⟩|a=f(a₁,…,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 andfunctions? Andsecond, how—from a functional point ofview—should one deal withpredicate expressions such as‘is a city’, ‘is tall’, ‘runs’,‘is bigger than’, ‘loves’, etc., which areused to denote classes of objects, properties of objects, or relationsbetween them and which can be combined with (applied to) singularterms to obtain sentences? If one considers predicates to be a kind offunctional expressions, what sort of names are generated by applyingpredicates to their arguments, and what can serve as referents ofthese 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 truthvalues:the True andthe False. In this way one alsoobtains a solution of the second problem. Predicates are to beinterpreted as some kind of functional expressions, which beingapplied to these or those names generate sentences referring to one ofthe two truth values. For example, if the predicate ‘is acity’ is applied to the name ‘Kyiv’, one gets thesentence ‘Kyiv is a city’, which designatestheTrue (i.e., ‘Kyiv is a city’is true). Onthe other hand, by using the name ‘Mount Everest’, oneobtains the sentence ‘Mount Everest is a city’ whichclearly designatesthe False, since ‘Mount Everest is acity’is false.

Functions whose values are truth values arecalledpropositional functions. Frege also referred to themas 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 and viceversa, and conjunction is a binary function that returnstheTrue as a value when both its argument positions are filled inbythe True, etc. Propositional functionsmappingn-tuples of truth values into truth values are alsocalledtruth-value functions.

Frege thus in a first step extended the familiar notion of anumerical function to functions on singular objects in general and,moreover, introduced a new kind of singular objects that can serve asarguments and values of functions on singular objects, the truthvalues. In a further step, he considered propositional functionstaking functions as their arguments. The quantifier phrase‘every city’, for example, can be applied to the predicate‘is a capital’ to produce a sentence. The argument ofthesecond-order function denoted by ‘everycity’ is thefirst-order propositional function onsingular objects denoted by ‘is a capital’. The functionalvalue denoted by the sentence ‘Every city is a capital’ isa truth value,the False.

Truth values thus prove to be an extremely effective instrument fora logical and semantical analysis of language. Moreover, Fregeprovides truth values (as proper referents of sentences) not merelywith a pragmatical motivation but also with a strong theoreticaljustification. The idea of such justification, that can be found inFrege 1892, employs the principle ofsubstitutivity ofco-referential terms, according to which the reference of a complexsingular term must remain unchanged when any of its sub-terms isreplaced by an expression having the same reference. This is actuallyjust an instance of the compositionality principle mentioned above. Ifsentences are treated as a kind of singular terms which must havedesignations, then assuming the principle of substitutivity one“almost inevitably” (as Kurt Gödel (1944, 129) explaines) isforced to recognize truth values as the most suitable entities forsuch designations. Accordingly, Frege asks:

What else but the truth value could be found, that belongsquite generally to every sentence if the reference of its componentsis relevant, 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 four sentences:

C1.	Sir Walter Scott is the author ofWaverley.
C2.	Sir Walter Scott is the man who wrote 29Waverley Novels altogether.
C3.	The number, such that Sir Walter Scott is the man who wrote thatmanyWaverley Novels altogether is 29.
C4.	The numberof 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 authorofWaverley’ and ‘the man who wrote29Waverley Novels altogether’ designate one and thesame object, namely Walter Scott. And so must C3 and C4, because thenumber, such that Sir Walter Scott is the man who wrote thatmanyWaverley 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 truthvalue. As Church remarks, a parallel example involving false sentencescan be constructed in the same way (by considering, e.g., ‘SirWalter Scott 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 in(Barwise and Perry 1981, 395), who stressed thus an extraordinarysimplicity of the argument and the minimality of presuppositionsinvolved. Stated generally, the pattern of the argument goes asfollows (cf. (Perry 1996)). One starts with a certain sentence, andthen moves, step by step, to a completely different sentence. Everytwo sentences in any step designate presumably one and the samething. Hence, the starting and the concluding sentences of theargument must have the same designation as well. But the onlysemantically significant thing they have in common seems to be theirtruth value. Thus, what any sentence designates is just its truthvalue.

A formal version of this argument, employing the term-forming,variable-binding class abstraction (or property abstraction)operator λx (“the class of allx suchthat” or “the property of being such anxthat”), was first formulated by Church (1943) in his review ofCarnap’sIntroduction to Semantics. Quine (1954), too,presents a variant of the slingshot using class abstraction, see also(Shramko and Wansing 2009). Other remarkable variations of theargument are those by Kurt Gödel (1944) and Donald Davidson(1967), (1969), which make use of the formal apparatus of a theory ofdefinite descriptions dealing with the description-forming,variable-binding iota-operator (ιx,“thex such that”). It is worth noticing that theformal versions of the slingshot show how to move—using stepsthat ultimately preserve reference—fromany true(false) sentence toany other such sentence. In view of thisresult, it is hard to avoid the conclusion that what the sentencesrefer 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 entities conceivedas alternative candidates for denotations of declarativesentences.

1.2 Truth as a property versus truth as an object

Truth values evidently have something to do with a general conceptof truth. 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 values intologic to a large extent is just due to its philosophical neutralitywith respect to theories of truth. It does not commit one to anyspecific metaphysical doctrine of truth. In one significant respect,however, the idea of truth values contravenes traditional approachesto truth by bringing to the forefront the problem of its categorialclassification.

In most of the established conceptions, truth is usually treated asa property. 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’. By contrast with this apparently quite naturalattitude, the suggestion to interpret truth as an object may seem veryconfusing, to say the least. Nevertheless this suggestion is alsoequipped with a profound and strong motivation demonstrating that itis far from being just an oddity and has to be taken seriously(cf. (Burge 1986)).

First, it should be noted that the view of truth as a property isnot as natural as it appears on the face of it. Frege brought intoplay an argument to the effect that characterizing a sentenceastrue adds nothing new to its content, for ‘It istrue that 5 is a prime number’ says exactly the same as just‘5 is a prime number’. That is, the adjective‘true’ is in a senseredundant and thus is not areal predicate expressing a real property such as the predicates‘white’ or ‘prime’ which, on the contrary,cannot simply be eliminated from a sentence without an essential lossfor its content. In this case a superficial grammatical analogy ismisleading. This idea gave an impetus to the deflationary conceptionof truth (advocated by Ramsey, Ayer, Quine, Horwich, and others, seethe entry onthe deflationary theory of truth).

However, even admitting the redundancy of truth as a property,Frege emphasizes 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 to us”, is to claim truth for a thought. Theobject, in the sense of the point orobjective, of sentenceuse was truth. It is illuminating therefore to see truth as an object(Burge 1986, 120).

As it has been observed repeatedly in the literature (cf., e.g.,(Burge 1986, Ruffino 2003)), the stress Frege laid on the notion of atruth value was, to a great extent, pragmatically motivated. Besidesan intended gain for his system of “Basic Laws” (Frege1962) reflected in enhanced technical clarity, simplicity, and unity,Frege also sought to substantiate in this way his view on logic as atheoretical 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 linked upwith a value-theoretical tradition in German philosophy of the secondhalf of the 19th century; see also the recent (Gabriel 2013) on therelation between Frege’s value-theoretically inspired conception oftruth values and his theory of judgement. More specifically, WilhelmWindelband, the founder and the principal representative of theSouthwest school of Neo-Kantianism, was actually the first whoemployed 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 treatinga truth 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 in(1918) when he defined the subject-matter of logic (seebelow). Gabriel points out (1984, 374) that this connection betweenlogic and a value theory can be traced back to Hermann Lotze, whoseseminars in Göttingen were attended by both Windelband andFrege.

The decisive move made by Frege was to bring together aphilosophical and a mathematical understanding of values on the basisof a generalization of the notion of a function on numbers. WhileFrege may have 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 Trueandthe False), then it turns out that truth values areindeed objects, and it seems quite reasonable to generally explicatetruth and falsity as objects and not as properties. As Fregeexplains:

A statement contains no empty place, and therefore we musttake itsBedeutung as an object. But thisBedeutungis a truth-value. Thus the two truth-values are objects (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 themisbegotten doctrine that sentences are a species of complex singularterms, which dominated Frege’s later period: to rob him of the insightthat sentences play a unique role, and that the role of almost everyother linguistic 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”.

1.3 The ontology of truth values

If truth values are accepted and taken seriously as a special kindof objects, 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 asabstractobjects. Note that Frege himself never used the word‘abstract’ when describing truth values. Instead, he has aconception of so called “logical objects”, truth valuesbeing the most fundamental (and primary) of them (Frege 1976,121). Among the other logical objects Frege pays particular attentionto are sets and numbers, emphasizing thus their logical nature (inaccordance with his logicist 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 on abstract objects). Obviously, the universe of abstractobjects is much broader than the universe of logical objects as Fregeconceives them. The latter are construed as constituting anontological foundation for logic, and hence for mathematics (pursuantto Frege’s logicist program). Generally, the classofabstracta includes a wide diversity of platonic universals(such as redness, youngness, or geometrical forms) and not only thoseof them which are logically necessary. Nevertheless, it may safely besaid that logical objects can be considered as paradigmatic cases ofabstract entities, or abstract 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 obviouslyare abstract as they clearly have nothing to do with physicalspacetime. In a similar fashion truth values fulfill anotherrequirement often imposed upon abstract objects, namely the one of acausal inefficacy (see, e.g., (Grossmann 1992, 7)). Here again, truthvalues are very much like numbers and geometrical figures: they haveno causal power and make 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 symbol a is to designate an object for us, then wemust have a criterion that decides in all cases whether b is the sameas a, even if it is not always in our power to apply this criterion(Beaney 1997, 109).

More precisely, one obtains a new object by abstracting it fromsome given kind of entities, in virtue of certain criteria of identityfor this new (abstract) object. This abstraction is performed in terms of anequivalence relation defined on the given entities (see (Wrigley 2006,161)). 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 (Andersonand Zalta 2004, 2), stating that for any two sentencespandq, the truth value ofp is identical with thetruth value ofq if and only ifp is (non-logically)equivalent withq (cf. also (Dummett 1959, 141)). This ideacan be formally explicated following the style of presentation in(Lowe 1997, 620):

∀p∀q[(Sentence(p)&Sentence(q)) ⇒(tv(p)=tv(q) ⇔(p↔q))],

where &, ⇒, ⇔, ∀ stand correspondingly for‘and’, ‘if… then’, ‘if and onlyif’ and ‘for all’ in themetalanguage, and↔ stands for someobject language equivalence connective(biconditional).

Incidentally, Carnap (1947, 26), when introducing truth-values asextensions of sentences, is guided by essentially the sameidea. 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 someothers. Applying the well-known technique of interpreting sentences aspredicators of degree 0, he generalizes the fact that two predicatorsof degreen (say,P andQ) have the sameextension if and only if∀x₁∀x₂…∀x_n(Px₁x₂…x_n↔Qx₁x₂…x_n)holds. Then, analogously, two sentences (say,pandq), being interpreted as predicators of degree 0, musthave the same extension if and only ifp↔qholds, that is if and only if they are equivalent. And then, Carnapremarks, it seems quite natural to take truth values as extensions forsentences.

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 valueof a sentence (or proposition, or thelike). The criterion of identity for truth values is formulated thenthrough the logical relation of equivalence holding between theseother objects—sentences, propositions, or the like (with anexplicit quantification over them).

It should also be remarked that the properties of the objectlanguage biconditional depend on the logical system in which thebiconditional is employed. Biconditionals of different logics may havedifferent logical properties, and it surely matters what kind of theequivalence connective is used for defining truth values. This meansthat the concept of a truth value introduced by means of the identitycriterion that involves a biconditional between sentences is alsologic-relative. Thus, if ‘↔’ stands for materialequivalence, 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 from(Zalta 1983), propose the following definition of ‘the truthvalue of propositionp’(‘tv(p)’ [notationadjusted]):

tv(p) =_dfιx(A!x ∧ ∀F(xF↔ ∃q(q↔p∧F=[λy q]))),

whereA!stands for a primitive theoretical predicate ‘beingabstract’,xF is to be read as “xencodesF” and [λy q] is apropositional property (“being such aythatq”). That is, according to this definition,“the extension ofp is the abstract object that encodesall and only the properties of the form [λy q]which are constructed out of propositionsq materiallyequivalent top” (Anderson and Zalta 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 ∃p(x =tv(p)).

Using this apparatus, it is possible to explicitly define theFregean truth valuesthe True (⊤) andtheFalse (⊥):

⊤ =_df ιx(A!x ∧ ∀F(xF ↔ ∃p(p ∧F=[λy p])));
⊥ =_df ιx(A!x ∧ ∀F(xF ↔ ∃p(¬p ∧F=[λy p]))).

Anderson and Zalta prove then that ⊤ and ⊥ are indeedtruth values and, moreover, that there are exactly two suchobjects. The latter result is expected, if one bears in mind that whatthe definitions above actually introduce are theclassicaltruth values (as the underlying logic isclassical). Indeed,p↔q is classicallyequivalent to(p∧q)∨(¬p∧¬q),and ¬(p↔q) is classically equivalent to(p∧¬q)∨(¬p∧q). Thatis, the connective of material equivalence divides sentences into twodistinct 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 exactlythe same way as the word ‘beautiful’ does for aestheticsand the word ‘good’ for ethics. Thus, according to such aview, the proper task of logic consists, ultimately, in investigating“the laws of being true” (Sluga 2002, 86). By doing so,logic is interested in truth as such, understood objectively, and notin what is 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 JanŁukasiewicz. As he paradigmatically put it:

All true propositions denote one and the same object,namely truth, and all false propositions denote one and the sameobject, namely falsehood. I consider truth and falsehood tobesingular objects in the same sense as the number 2 or 4is. … Ontologically, truth has its analogue in being, andfalsehood, in non-being. The objects denoted by propositions arecalledlogical values. Truth is the positive, and falsehoodis the negative logical value. … Logic is the science ofobjects of a special kind, namely a science oflogical values(Łukasiewicz 1970, 90).

This definition may seem rather unconventional, for logic isusually treated as the science of correct reasoning and validinference. The latter understanding, however, calls for furtherjustification. This becomes evident, as soon as one asks,on whatgrounds one should qualify this or that pattern of reasoning ascorrect or incorrect.

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 “third realm”(the realm of the objective content of thoughts, and generallyabstract objects of various kinds, see Frege 1918, cf. Popper1972 and also Burge 1992, 634). Among the subdomains of this thirdrealm 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, see(Łukasiewicz 1918, Łukasiewicz 1920). Independently ofŁukasiewicz, Emil Post in his dissertation from 1920, publishedas (Post 1921), introducedm-valued truth tables,wherem is any positive integer. Whereas Post’s interestinmany-valued logic (where “many” means“more than two”) was almost exclusively mathematical,Łukasiewicz’s motivation was philosophical (see the entry onmany-valued logic). He contemplated the semantical value of sentencesabout the contingent 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, Dummett 2000, Ryan and Sadler 1992).

Consider a propositional languageLbuilt upon a set of atomic sentencesPand a set of propositional connectivesC(the set of sentences ofL being thesmallest set containingP and beingclosed under the connectivesfromC). Then avaluationsystemV for the languageLis a triple⟨V,D,F⟩,whereV is a non-empty set with at leasttwo elements,D is a non-empty propersubset ofV,andF ={f_c1,…,f_cm}is a set of functions such thatf_ci isann-place function onVifc_i is ann-placeconnective. Intuitively,V is the set oftruth values,D is the setofdesignated truth values,andF is the set of truth-valuefunctions interpreting the elementsofC. If the set of truth values of avaluation systemV hasn elements,V is said toben-valued. Any valuation system can be equipped with anassignment function which maps the set of atomic sentencesintoV. Each assignmentarelative to a valuation systemV can be extended to allsentences ofL by means of a valuationfunctionv_a defined in accordance with thefollowing conditions:

∀p ∈P,va(p) =a(p);	(1)
∀ci ∈C,va(ci(A1,…,An)) =fci(va(A1),…,va(An))	(2)

It is interesting to observe that the elementsofV are sometimes referred toasquasi truth values. Siegfried Gottwald (1989, 2) explainsthat one reason for using the term ‘quasi truth value’ isthat there is no convincing and uniform interpretation of the truthvalues that in many-valued logic are assumed in addition to theclassical truth valuesthe True andthe False, anunderstanding that, according to Gottwald, associates the additionalvalues with the naive understanding of being true, respectively thenaive understanding ofdegrees of being true (cf. also theremark by Font (2009, 383) that “[o]ne of the main problems inmany-valued logic, at least in its initial stages, was theinterpretation of the ”intermediate“ or”non-classical“ values”, et seq.). In laterpublications, Gottwald has changed his terminology and states that“[t]o avoid any confusion with the case of classical logic oneprefers in many-valued logic to speak oftruth degrees andto use the word ”truth value“ only for classicallogic” (Gottwald 2001, 4). Nevertheless in what follows the term‘truth values’ will be used even in the context ofmany-valued logics, without any commitment to a philosophicalconception of truth as a graded notion or a specific understanding ofsemantical values in addition to the classical truth values.

Since the cardinality ofV may begreater than 2, the notion of a valuation system provides a naturalfoundational framework for the very idea of a many-valued logic. ThesetD of designated values is of centralimportance for the notion of a valuation system. This set can be seenas a generalization of the classical truth valuethe True inthe sense that it determines many central logical notions and therebygeneralizes some of the important roles played by Frege’stheTrue (cf. the introductory remarks about uses of truthvalues). For example, the set of tautologies (logical laws) isdirectly specified by the given set of designated truth values: asentenceA is atautology in a valuationsystemV iff for every assignmenta relativetoV,v_a(A)∈D. Another fundamental logicalnotion—that of an entailment relation—can also be definedby referring to the setD. For a givenvaluation systemV a corresponding entailment relation(⊨_V) is usually defined by postulating thepreservation of designated values from the premises to theconclusion:

Δ⊨VA iff ∀a[(∀B ∈ Δ:va(B) ∈D) ⇒va(A) ∈D].

(3)

A pairM =⟨V,v_a⟩, whereV is an(n-valued) valuation system andv_a avaluation inV, may be called an(n-valued)model based onV. EverymodelM =⟨V,v_a⟩ comes with acorresponding entailment relation⊨_M by definingΔ⊨_M Aiff (∀B ∈ Δ:v_a(B)∈D)⇒v_a(A)∈D.

SupposeL is a syntactically defined logical systemL with a consequence relation ⊦_L, specified as a relation between the power-set ofLandL. Then a valuationalsystemV is said to bestrictly characteristic forL just in case Δ⊨_VA iff Δ⊦_LA (see (Dummett 1981, 431)). Conversely, one says thatLis characterizedbyV. Thus, if a valuation system is said todeterminea logic, the valuation systemby itself is, properlyspeaking,not a logic, but only serves as a semantic basisfor some logical system. Valuation systems are often referred toas (logical)matrices.

In this way Fregean, i.e. classical, logic can be presented asdetermined by a particular valuation system based on exactly twoelements:V_cl =⟨{T,F}, {T},{f_∧,f_∨,f_→,f_∼}⟩,wheref_∧,f_∨,f_→,f_∼are given by the classical truth tables for conjunction, disjunction,material implication, and negation.

As an example for a valuation system based on more that twoelements, consider two well-known valuation systems which determineKleene’s (strong) “logic ofindeterminacy”K₃ and Priest’s “logicof paradox”P₃. In a propositional languagewithout implication,K₃ is specified bytheKleene matrixK₃ =⟨{T,I,F}, {T},{f_c:c ∈ {∼, ∧, ∨}}⟩, where the functionsf_c are defined asfollows:

f∼			f∧	T	I	F		f∨	T	I	F
T	F		T	T	I	F		T	T	T	T
I	I		I	I	I	F		I	T	I	I
F	T		F	F	F	F		F	T	I	F

ThePriest matrixP₃ differsfromK₃ only in thatD= {T,I}. Entailment inK₃ as wellas inP₃ is defined by means of(3).

There are natural intuitive interpretations ofIinK₃ and inP₃ astheunderdetermined and theoverdetermined valuerespectively—a truth-value gap and a truth-valueglut. Formally these interpretations can be modeled by presenting thevalues as certain subsets of the set of classical truth values{T,F}. ThenT turns intoT ={T} (understood as “true only”),FintoF = {F} (“false only”),I isinterpreted inK₃ asN = {} = ∅(“neither true nor false”), andinP₃ asB = {T,F}(“both true and false”). (Note that also Asenjo (1966) considers the same truth-tables with an interpretation of the third value as “antinomic”.) The designatedness of atruth value can be understood in both cases as containment of theclassicalT as a member.

If one combines all these new values into a joint framework, oneobtains the four-valued logicB₄ introduced byDunn and Belnap (Dunn 1976, Belnap 1977a, Belnap 1977b). AGentzen-style formulation can be found in (Font 1997, 7)). This logicis determined by theBelnap matrixB₄ =⟨{N,T,F,B}, {T,B},{f_c:c ∈ {∼, ∧, ∨}}⟩,where the functionsf_c are defined as follows:

f∼

f∧

T

B

N

F

f∨

T

B

N

F

T

F

T

T

B

N

F

T

T

T

T

T

B

B

B

B

B

F

F

B

T

B

T

B

N

N

N

N

F

N

F

N

T

T

N

N

F

T

F

F

F

F

F

F

T

B

N

F

Definition (3) applied to the Belnapmatrix determines the entailment relationofB₄. This entailment relation is formalized as thewell-known logic of “first-degree entailment”(E_fde) introduced in (Anderson and Belnap1975).

The syntactic notion of a single-conclusion consequence relationhas been extensively studied by representatives of the Polish schoolof logic, most notably by Alfred Tarski, who in fact initiated thisline of research (see Tarski 1930, Tarski 1930a, cf. alsoWójcicki 1988). In view of certain key features of a standardconsequence relation it is quite remarkable—as well asimportant—that any entailment relation⊨_V defined as above has the followingstructural properties (see Ryan and Sadler 1992, 34):

Δ∪{A}⊨VA	(Reflexivity)	(4)
If Δ⊨VA thenΔ∪Γ⊨VA	(Monotonicity)	(5)
If Δ⊨VA and Γ∪{A}⊨VB, then Δ∪Γ⊨VB	(Cut)	(6)

Moreover, for everyA∈L, every Δ⊆L, and every uniform substitutionfunctionσ onL thefollowingSubstitution property holds (σ(Δ)stands for {σ(B) |B ∈ Δ}):

Δ⊨VA implies σ(Δ)⊨Vσ(A).

(7)

(The function of uniform substitution σ is defined asfollows. LetB be a formulainL, letp₁,…,p_n be all the propositional variables occurringinB, and let σ(p₁)=A₁,…, σ(p_n)=A_n for someformulasA₁,…,A_n. Thenσ(B) is the formula that results from B by substitutingsimultaneouslyA₁,…,A_nfor all occurrences ofp₁,…, p_n,respectively.)

If ⊨_V in the conditions(4)–(6) is replaced by ⊦_L, then one obtains what is often calledaTarskian consequence relation. If additionally aconsequence relation has the substitution property(7), then it iscalledstructural. Thus, any entailment relation defined fora given valuation systemV presents an important example of aconsequence relation, in thatV is strictly characteristic forsome logical systemL with a structural Tarskian consequencerelation.

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 manyother authors, suggests that truth comes by degrees, and these degreesmay be seen as truth values in an extended sense. The idea of truth asa graded notion has been applied to model vague predicates and toobtain a solution to the Sorites Paradox, the Paradox of the Heap (seethe entry on the Sorites Paradox). However, the success of applyingmany-valued logic to the problem of vagueness is highlycontroversial. Timothy Williamson (1994, 97), for example, holds thatthe phenomenon of higher-order vagueness “makes most work onmany-valued logic irrelevant to the problem of vagueness”.

In any case, the vagueness of concepts has been much debated inphilosophy (see the entry onvagueness) and it was one of the majormotivations for the development offuzzy logic (see the entryonfuzzy logic). In the 1960s, Lotfi Zadeh (1965) introduced the notion ofafuzzy set. A characteristic function of a setX isa mapping which is defined on a supersetY ofX andwhich indicates membership of an element inX. The range ofthe characteristic function of a classical setX is thetwo-element set {0,1} (which may be seen as the set of classical truthvalues). The function assigns the value 1 to elements ofXand the value 0 to all elements ofY not inX. Afuzzy set has a membership function ranging over the real interval[0,1]. A vague predicate such as ‘is much earlier than March20th, 1963’, ‘is beautiful’, or ‘is aheap’ may then be regarded as denoting a fuzzy set. Themembership functiong of the fuzzy set denoted by ‘ismuch earlier than March 20th, 1963’ thus assigns values (seen astruth degrees) from the interval [0, 1] to moments in time, forexampleg(1 p.m., August 1st, 2006) = 0,g(3 a.m.,March 19th, 1963) = 0,g(9:16 a.m., April 9th, 1960) =0.005,g(2 p.m., August 13th, 1943) = 0.05,g(7:02,a.m., December 2nd, 1278) = 1.

The application of continuum-valued logics to the Sorites Paradoxhas been suggested by Joseph Goguen (1969). The Sorites Paradox in itsso-called conditional form is obtained by repeatedlyapplyingmodus 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 thefirst premise 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 Sorites Paradoxcan be avoided. The truth-functionf_→ ofŁukasiewicz’s implication → is defined by stipulating thatifx ≤y,thenf_→(x,y) = 1, andotherwisef_→(x,y) = 1− (x −y). If, say, the truth value ofthe sentence ‘A collection of 500 grains of sand is aheap’ is 0.8 and the truth value of ‘A collection of 499grains of sand is a heap’ is 0.7, then the truth value of theimplication ‘If a collection of 500 grains of sand is a heap,then a collection 499 grains of sand is a heap.’ is0.9. Moreover, if the acceptability of a statement is defined ashaving a value greater thanj for 0 <j < 1and all the conditional premises of the Sorites Paradox do not fallbelow the valuej, thenmodus ponens does notpreserve acceptability, because the conclusion of the SoritesArgument, being evaluated as 0, is unacceptable.

Alasdair Urquhart (1986, 108) stresses “the extremelyartificial nature of the attaching of precise numerical values tosentences like … ‘Picasso’sGuernica isbeautiful’”. To overcome the problem of assigning precisevalues to predications of vague concepts, Zadeh (1975)introducedfuzzy truth values as distinct from the numericaltruth values in [0, 1], the former being fuzzy subsets of the set [0,1], understood astrue,very true,not verytrue, 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 latter sensehas been thoroughly criticized from a philosophical point of view byHaack (1996) for its “methodological extravagances” andits linguistic incorrectness. Haack emphasizes that her criticisms offuzzy logic do not apply to the base logics. Moreover, it should bepointed out that mathematical fuzzy logics are nowadays studied not inthe first place as continuum-valued logics, but as many-valued logicsrelated to residuated lattices, see (Hajek 1998, Cignolietal 2000, Gottwald 2001, Galatoset al. 2007), whereasfuzzy logic in the broad sense is to a large extent concerned withcertain 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 generalizestruth-functionality”. According to Williamson, the degree oftruth of a conjunction, a disjunction, or a conditional just fails tobe a function of the degrees of truth of vague componentsentences. The sentences ‘John is awake’ and ‘Johnis asleep’, for example, may have the same degree of truth. Bytruth-functionality the sentences ‘If John is awake, then Johnis awake’ and ‘If John is awake, then John isasleep’ are alike in truth degree, indicating for Williamson thefailure of degree-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 vaguepredicationsare either true or false, e.g., sentences of theform ‘IfA, thenA’. Supervaluationalsemantics provides a solution to this problem. A three-valuedassignmenta into {T,I,F} mayassign a truth-value gap (or rather the valueI) to the vaguesentence ‘Picasso’sGuernica is beautiful’. Anyclassical assignmenta′ that agrees withawhenevera assignsT orF may be seen as aprecisification (or superassignment) ofa. A sentence maythan be said to be supertrue under assignmenta if it is trueunder every precisificationa′ ofa. Thus,ifa is a three-valued assignment into{T,I,F} anda′ is atwo-valued assignment into {T,F} suchthata(p) =a′(p)ifa(p) ∈ {T,F},thena′ is said to be asuperassignmentofa. It turns out that ifa is an assignmentextended to a valuation functionv_a for the KleenematrixK₃, then for every formulaA in thelanguage ofK₃,v_a(A)=v_a′(A)ifv_a(A) ∈{T,F}. Therefore, thefunctionv_a′ may be calledasupervaluation ofv_a. A formula is thensaid to besupertrue under a valuationfunctionv_a forK₃ if it is trueunder every supervaluationv_a′ofv_a, i.e.,ifv_a′(A) =T forevery supervaluationv_a′ofv_a. The property of beingsuperfalseis defined analogously.

Since every supervaluation is a classical valuation, everyclassical tautology is supertrue under every valuation functioninK₃. Supervaluationism is, however, nottruth-functional with respect to supervalues. The supervalue of adisjunction, for example, does not depend on the supervalue of thedisjuncts. Supposea(p)=I. Thena(¬p) =Iandv_a′(p∨¬p) =T for everysupervaluationv_a′ofv_a. Whereas (p∨¬p) isthus supertrue underv_a,p∨pisnot, because there are superassignmentsa′ofa witha′(p) =F. Anargument against the charge that supervaluationism requires anon-truth-functional semantics of the connectives can be found in(MacFarlane 2008) (cf. also other references given there).

Although the possession of supertruth is preserved from thepremises to the conclusion(s) of valid inferences insupervaluationism, and although it might be tempting to considersupertruth an abstract object on its own, it seems that it has neverbeen suggested to hypostatize supertruth in this way, comparable toFrege’sthe True. A sentence supertrue under a three-valuedvaluationv just takes the Fregean valuethe Trueunder every supervaluation ofv. The advice not to confusesupertruth with “real truth” can be found in (Belnap2009).

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-valued logicto be “a magnificent conceptual deceit”. Suszko actuallyclaimed that “there are but two logical values, true andfalse” (Caleiroet al. 2005, 169), a statement nowcalledSuszko’s Thesis. For Suszko, the set of truth valuesassumed 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 logicalvalue,false, is preserved in the opposite direction: If the(every) conclusion is false, then so is at least one of thepremises. The logical values are thus represented by a bi-partition ofthe 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(1959) in his influential paper, where he asks “what point theremay be in distinguishing between different ways in which a statementmay be true or between different ways in which it may be false, or, aswe might say, between degrees of truth and falsity” (Dummett1959, 153). Dummett observes that, first, “the sense of asentence is determined wholly by knowing the case in which it has adesignated value and the cases in which it has an undesignatedone”, and moreover, “finer distinctions between differentdesignated values or different undesignated ones, however naturallythey come to us, are justified only if they are needed in order togive a truth-functional account of the formation of complex statementsby means of operators” (Dummett 1959, 155). Suszko’s claimevidently 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 valueT in some ofits significant roles, it would not always be adequate to interpretthe set of non-designated values as a generalization of the classicaltruth valueF. The point is that in a many-valued logic,unlike in classical logic, “not true” does not always mean“false” (cf., e.g., the above interpretation of Kleene’slogic, where sentences can be neither true nor false).

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 thatareneither designatednor antidesignated and evenfor values that areboth designatedandantidesignated.

Grzegorz Malinowski (1990), (1994) takes advantage of this proposalto give a counterexample to Suszko’s Thesis. He defines the notion ofa single-conclusionquasi-consequence(q-consequence) relation. The semantic counterpartofq-consequence iscalledq-entailment. Single-conclusionq-entailmentis defined by requiring that if no premise is antidesignated, theconclusion is designated. Malinowski (1990) proved that for everystructuralq-consequence relation, there exists acharacterizing class ofq-matrices, matrices which inaddition to a non-emptysubsetD⁺ ofdesignated values comprise a disjoint non-emptysubsetD^- ofantidesignated values. Not everyq-consequence relation has abivalent semantics.

In the supplementary documentSuszko’s Thesis, Suszko’s reduction is introduced, Malinowski’s counterexample toSuszko’s Thesis is outlined, and a short analysis of these results ispresented.

Can one provide evidence for a multiplicity of logical values? Moreconcretely,is there more than one logical value, each ofwhich may be taken to determine its own (independent) entailmentrelation? A positive answer to this question emerges fromconsiderations on truth values as structured entities which, by virtueof their internal structure, give rise to natural partial orderings onthe set of values.

3. Ordering relations between truth-values

3.1 The notion of a logical order

As soon as one admits that truth values come withvaluationsystems, it is quite natural to assume that theelements of such a system are somehowinterrelated. Andindeed, already the valuation system for classical logic constitutes awell-known algebraic structure, namely the two-element Boolean algebrawith ∩ and ∪ as meet and join operators (see theentry on themathematics of Boolean algebra). In its turn, this Boolean algebra forms a lattice with apartialorder defined bya≤_tbiffa∩b =a. This lattice may bereferred to asTWO. It is easy to see that the elementsofTWO are ordered asfollows:F≤_tT. This orderingis sometimes called thetruth order (as indicated by thecorresponding subscript), for intuitively it expresses an increase intruth:F is “less true” thanT. It canbe schematically 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 ≤_t ordersT,IandF so that the intermediate valueI is“more true” thanF, but “less true”thanT.

The relation ≤_t is also called alogicalorder, because it can be used to determine key logical notions:logical connectives and an entailment relation. Namely, if theelements of the given valuation systemV form a lattice, thenthe operations of meet and join with respect to≤_t are usually seen as the functions forconjunction and disjunction, whereas negation can be represented bythe inversion of this order. Moreover, one can consider an entailmentrelation forV as expressing agreement with the truth order,that is, the conclusion should be at least as true as the premisestaken together:

Δ⊨B iff ∀va[t{va(A)|A ∈Δ} ≤tva(B)],

(8)

where_t is thelattice meet in the corresponding lattice.

The Belnap matrixB₄ considered above also can berepresented as a partially ordered valuation system. The set of truthvalues {N,T,F,B}fromB₄ constitutes a specific algebraic structure– thebilatticeFOUR₂ presented inFigure 3 (see, e.g., (Ginsberg 1988, Arieli and Avron 1996, Fitting2006)).

Figure 3: The bilatticeFOUR₂

This bilattice is equipped withtwo partialorderings; in addition to a truth order, there is an information order(≤_i) which is said to order the values underconsideration according to the information they give concerning aformula to which they are assigned. Lattice meet and join with respectto ≤_t coincide with thefunctionsf_∧ andf_∨ inthe Belnap matrixB₄,f_~ turnsout to be the truth order inversion, and an entailment relation, whichhappens to coincide with the matrix entailment, is defined by(8).FOUR₂ arisesas a combination of two structures: the approximationlatticeA₄ and the logicallatticeL₄ which are discussed in Belnap 1977aand 1977b (see also, Anderson, Belnap and Dunn 1992,510–518)).

3.2 Truth values as structured entities. Generalized truth values

Frege (1892, 30) points out the possibility of “distinctionsof parts 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 innerstructure. It is not quite clear how serious Frege is about this view,but it seems to suggest that truth values may well be interpreted ascomplex, structured entities that can be divided into parts.

There exist several approaches to semantic constructions wheretruth values 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 toposC and for aC-object Ω one candefine a truth value ofC as an arrow 1 → Ω(“a subobject classifier forC”), where 1 is aterminal object inC (cf. Goldblatt 2006, 81, 94). The setof truth values so defined plays a special role in the logicalstructure ofC, since arrows of the form 1 → Ωdetermine central semantical notions for the given topos. And again,these truth values evidently have some inner structure.

One can also mention in this respect the so-called “factorsemantics” for many-valued logic, where truth values are definedas orderedn-tuples of classical truth values(T-F sequences, see Karpenko 1983). Then thevalue³/₅, for example, can be interpreted asaT-F sequence of length 5 with exactly 3occurrences ofT. Here the classical valuesTandF are used as “building blocks” fornon-classical truth values.

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 Dunn 2000, 7). By developing this idea, one arrives at theconcept of ageneralized truth value function, which is afunction from sentences into thesubsets of somebasicset of truth values (see Shramko and Wansing 2005). The valuesof generalized truth value functions can be calledgeneralizedtruth values.

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 {∅}taken as the basic set subject to a further generalizationprocedure. At the first stage ∅ comes out with no specificintuitive interpretation, it is only important to take it as somedistinctunit. Consider then the power-set of {∅}consisting of exactly two elements: {{∅}, ∅}. Now, theseelements can be interpreted as Frege’sthe True andtheFalse, and thus it is possible to construct a valuation systemfor classical logic,V^∅_cl= ⟨{{∅}, ∅}, {{∅}},{f_∧,f_∨,f_→,f_∼}⟩,where thefunctionsf_∧,f_∨,f_→,f_∼ are defined asfollows (forX,Y ∈ {{∅},∅}:f_∧(X,Y)=X∩Y;f_∨(X,Y)=X∪Y;f_→(X,Y)= ({{∅},∅}−X)∪Y;f_∼(X)= {{∅}, ∅}−X. It is not difficult to see that forany assignmenta relativetoV^∅_cl, and for anyformulasA andB, the followingholds:v_a(A∧B) ={∅}⇔v_a(A) = {∅}andv_a(B) ={∅};v_a(A∨B) ={∅}⇔v_a(A) = {∅}orv_a(B) ={∅};v_a(A→B) ={∅}⇔v_a(A) = ∅orv_a(B) ={∅};v_a(∼A) ={∅}⇔v_a(A) = ∅. Thisshows thatf_∧,f_∨,f_→andf_∼ determine exactly the propositionalconnectives of classical logic. One can conveniently mark the elements{∅} and ∅ in the valuationsystemV^∅_cl by theclassical labelsT andF. Note thatwithinV^∅_cl it is fullyjustifiable to associate ∅ with falsity, taking into account thevirtualmonism of truth characteristic for classical logic,which treats falsity not as an independent entity but merely as theabsence of truth.

Then, by taking the set2 = {F,T} ofthese classical values as the basic set for the next valuation system,one obtains the four truth values of Belnap’s logic as the power-setof the set of classicalvaluesP(2) =4:N= ∅,F = {F} (= {∅}),T ={T} (= {{∅}}) andB = {F,T}(= {∅, {∅}}). In this way, Belnap’s four-valued logicemerges as a certain generalization of classical logic with its twoFregean truth values. In Belnap’s logic truth and falsity areconsidered to be full-fledged, self-sufficient entities, and therefore∅ is now to be interpreted not as falsity, but as a realtruth-value gap (neither truenor false). Thedissimilarity of Belnap’s truth and falsity from their classicalanalogues is naturally expressed by passing from the correspondingclassical values to their singleton-sets, indicating thus their newinterpretations asfalse only andtrueonly. Belnap’s interpretation of the four truth values has beencritically discussed in Lewis 1982 and Dubois 2008 (see also the replyto Dubois in Wansing and Belnap 2009).

Generalized truth values have a strong intuitive background,especially as a tool for the rational explication of incomplete andinconsistent information states. In particular, Belnap’s heuristicinterpretation of truth values as information that “has beentold to a computer” (see Belnap 1977a, 1977b; alsoreproduced in Anderson, Belnap and Dunn 1992, §81) has beenwidely acknowledged. As Belnap points out, a computer may receive datafromvarious (maybe independent) sources. Belnap’s computershave to take into account various kinds of information concerning agiven sentence. Besides the standard (classical) cases, when acomputer obtains information either that the sentence is (1) true orthat 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 valuesfromB₄ naturally correspond to these foursituations: there is no information that the sentence is false and noinformation that it is true (N), there ismerelyinformation that the sentence is false (F), thereismerely information that the sentence is true (T),and there is information that the sentence is false, but there is alsoinformation that it is true (B).

Joseph Camp in (2002, 125–160) provides Belnap’s four valueswith quite a different intuitive motivation by developing what hecalls a “semantics of confused thought”. Consider arational agent, who happens to mix up two very similar objects(say,a andb) and ambiguously uses one name (say,‘C’) for both of them. Now let such an agentassert some statement, saying, for instance, thatC has someproperty. How should one evaluate this statement ifa has theproperty in question whereasb lacks it? Camp argues againstascribing truth values to such statements and puts forward an“epistemic semantics” in terms of“profitability” and “costliness” as suitablecharacterizations of sentences. A sentenceS is said to be“profitable” if one would profit from acting on the beliefthatS, and it is said to be “costly” if actingon the belief thatS would generate costs, for example asmeasured by failure to achieve an intended goal. If our“confused agent” asks some external observerswhetherC has the discussed property, the following fouranswers are possible: ‘yes’ (mark the correspondingsentence withY), ‘no’ (mark it withN),‘cannot say’ (mark it with?), ‘yes’and ‘no’ (mark it withY&N). Note that theexternal observers, who provide answers, are“non-confused” and have different objects in mind as tothe referent of ‘C’, in view of all the factsthat may be relevant here. Camp conceives these four possible answersconcerning epistemic properties of sentences as a kind of“semantic values”, interpreting them as follows: thevalueY is an indicator of profitability, the valueN isan indicator of costliness, the value? is no indicator eitherway, and the valueY&N is both an indicator ofprofitability and an indicator of costliness. A strict analogy betweenthis “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 entailmentsystemE_fde. In Zaitsev and Shramko 2013 it isdemonstrated how ontological and epistemic aspects of truth values canbe combined within a joint semantical framework.

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 truth values,then one obtains further valuation systems which can be represented byvariousmultilattices. One arrives, in particular,atSIXTEEN₃ – thetrilattice of 16truth-values, which can be viewed as a basis for a logic of computernetworks, see (Shramko and Wansing 2005), (Shramko and Wansing 2006),(Kamide and Wansing 2009), (Odintsov 2009), (Wansing 2010), (Odintsovand Wansing 2013), cf. also (Shramko, Dunn, Takenaka 2001). The notionof a multilattice andSIXTEEN₃ are discussedfurther in the supplementary documentGeneralized 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 presententry. Moreover, considerations on truth values give rise to deepontological questions concerning their own nature, the feasibility offact ontologies, and the role of truth values in such ontologicaltheories. Furthermore, there exist well-motivated theories ofgeneralized truth values that lead far beyond Frege’s classicalvaluesthe True andthe False. (For variousdirections of recent logical and philosophical investigations in thearea of truth values see (Truth Values I 2009) and (Truth Values II2009).)

Bibliography

• Anderson, Alan R. and Belnap Nuel D., 1975,Entailment: TheLogic of Relevance and Necessity, Vol. I, Princeton, NJ:Princeton University Press.
• Anderson, Alan R., Belnap, Nuel D. and Dunn, J. Michael,1992,Entailment: The Logic of Relevance and Necessity,Vol. II, Princeton, NJ: Princeton University Press.
• Anderson, David and Zalta, Edward, 2004, “Frege,Boolos, and logical objects”,Journal of PhilosophicalLogic, 33: 1–26.
• Arieli, Ofer and Avron, Arnon, 1996, “Reasoning with logicalbilattices”,Journal of Logic, Language andInformation, 5: 25–63.
• Asenjo, Florencio G., 1966, “A calculus of antinomies”,Notre Dame Journal of Formal Logic, 7: 103–105.
• Barwise, Jon and Perry, John, 1981, “Semantic innocenceand uncompromising situations”,Midwest Studies in thePhilosophy of Language, VI: 387–403.
• Beaney, Michael (ed. and transl.), 1997,The FregeReader, Oxford: Wiley-Blackwell.
• Belnap, Nuel D., 1977a, “How a computer should think”,in G. Ryle (ed.),Contemporary Aspects of Philosophy,Stocksfield: Oriel Press Ltd., 30–55.
• –––, 1977b, “A useful four-valued logic”,in: J.M. Dunn and G. Epstein (eds.),Modern Uses ofMultiple-Valued Logic, Dordrecht: D. Reidel Publishing Co.,8–37.
• –––, 2009, “Truth values,neither-true-nor-false, and kupervaluations”,StudiaLogica, 91: 305–334.
• Béziau, Jean-Yves, 2012, “A History of Truth-Values”,in D. Gabbay et al. (eds.),Handbook of the History of Logic. Vol. 11, Logic: A History of its Central Concepts,Amsterdam: North-Holland, 235–307.
• Brown, Bryson and Schotch, Peter, 1999, “Logic andaggregation”,Journal of Philosophical Logic, 28:265–287.
• Burge, Tyler, 1986, “Frege on truth”, in:L. Haaparanta and J. Hintikka (eds.),Frege Synthesized,Dordrecht: D. Reidel Publishing Co., 97–154.
• –––, 1992, “Frege on knowing the ThirdRealm”,Mind, 101: 633–650.
• Caleiro, Carlos, Carnielli, Walter, Coniglio, Marcelo, and Marcos,João, 2005, “Two’s company: ‘The humbug of manylogical values’”, in: J.-Y. Beziau (ed.),LogicaUniversalis, Basel: Birkhäuser Verlag, 169–189.
• Camp, Joseph L., 2002,Confusion: A Study in the Theory ofKnowledge, Cambridge, MA: Harvard University Press.
• Carnap, Rudolf, 1942,Introduction to Semantics,Cambridge, MA: Harvard University Press.
• –––, 1947,Meaning and Necessity. A Study inSemantics and Modal Logic, Chicago: University of Chicago Press.
• Church, Alonzo, 1943, “Review of RudolfCarnap,Introduction to Semantics”,ThePhilosophical Review, 52: 298–304.
• –––, 1956,Introduction to Mathematical Logic,Vol. I, Princeton: Princeton UniversityPress.
• Cignoli, Roberto, D’Ottaviano, Itala, and Mundici, Daniele,2000,Algebraic Foundations of Many-valued Reasoning,Dordrecht: Kluwer Academic Publishers.
• da Costa, Newton, Béziau, Jean-Yves, and Bueno, Otávio,1996, “Malinowski and Suszko on many-valued logics: on thereduction of many-valuedness to two-valuedness”,ModernLogic, 6: 272–299.
• Czelakowski, Janusz, 2001,Protoalgebraic Logics,Dordrecht: Kluwer Academic Publishers.
• Davidson, David, 1967, “Truth andmeaning”,Synthese, 17: 304–323.
• –––, 1969, “True to thefacts”,Journal of Philosophy, 66: 748–764.
• Dubois, Didier, 2008, “On ignorance and contradictionconsidered as truth-values”,Logic Journal of the IGPL,16: 195–216.
• Dummett, Michael, 1959, “Truth”, in:Proceedingsof the Aristotelian Society, 59: 141–162 (Reprinted in:Truth and Other Enigmas, Cambridge, MA: Harvard UniversityPress, 1978, 1–24).
• –––, 1981,Frege: Philosophy of Language,2nd ed., London: Duckworth Publishers.
• –––, 1991,Frege and Other Philosophers,Oxford: Oxford University Press.
• –––, 2000,Elements of Intuitionism, 2nded., Oxford: Clarendon Press.
• Dunn, J. Michael, 1976, “Intuitive semantics forfirst-degree entailment and ‘coupledtrees’”,Philosophical Studies, 29: 149–168.
• –––, 2000, “Partiality and itsdual”,Studia Logica, 66: 5–40.
• Fine, Kit, 1975, “Vagueness, truth andlogic”,Synthese, 30: 265–300.
• Fitting, Melvin, 2006, “Bilattices are nice things”,in: T. Bolander, V. Hendricks, and S.A. Pedersen(eds.),Self-Reference, Stanford: CSLI Publications,53–77.
• Font, Josep Maria, 1997, “Belnap’s four-valued logic and DeMorgan lattices”,Logic Journal of IGPL, 5: 1–29.
• –––, 2009, “Taking degrees of truthseriously”,Studia Logica, 91: 383–406.
• van Fraassen, Bas, 1966, “Singular terms, truth-value gaps,and free logic”,Journal of Philosophy, 63:481–495.
• Frankowski, Szymon, 2004, “Formalization of a plausibleinference”,Bulletin of the Section of Logic, 33:41–52.
• Frege, Gottlob, 1891, “Function und Begriff. Vortrag,gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaftfür Medicin und Naturwissenschaft”, Jena: H. Pohle(Reprinted in (Frege 1986).)
• –––, 1892, “Über Sinn undBedeutung”,Zeitschrift für Philosophie undphilosophische Kritik, 100: 25–50. (Reprinted in (Frege1986).)
• –––, 1918, “Der Gedanke”,Beiträgezur Philosophie des deutschen Idealismus 1: 58–77. (Reprinted in(Frege 1967).)
• –––, 1962,Grundgesetze der Arithmetik, Bde. I undII, Darmstadt: Wissenschaftliche Buchgesellschaft.
• –––, 1967,Kleine Schriften, Ignacio Angelli(ed.), Darmstadt: Wissenschaftliche Buchgesellschaft.
• –––, 1976,Wissenschaftlicher Briefwechsel,G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, andA. Veraart (eds.), Hamburg: Felix Meiner Verlag.
• –––, 1986,Funktion, Begriff, Bedeutung. Fünflogische Studien, G. Patzig (ed.), Göttingen: Vandenhoeck& Ruprecht.
• –––, 1988,Grundlagen der Arithmetik. Einelogisch-mathematische Untersuchung über den Begriff derZahl, Hamburg: Felix Meiner Verlag.
• –––, 1990, “Einleitung in die Logik”, in:Frege, G.,Schriften zur Logik und Sprachphilosophie,Hamburg: Felix Meiner Verlag, 74–91.
• Gabriel, Gottfried, 1984, “Fregean connection: Bedeutung,value and truth-value”,The Philosophical Quarterly,34: 372–376.
• –––, 1986, “Frege alsNeukantianer”,Kant-Studien, 77: 84–101.
• –––, 2013, “Truth, value, and truth value. Frege’s theory of judgement and its historical background”, in: M. Textor (ed.),Judgement and Truth in Early Analytic Philosophy and Phenomenology,Basingstoke: Palgrave Macmillan, 36–51.
• Galatos, Nikolaos, Jipsen, Peter, Kowalski, Tomasz and Ono,Hiroakira, 2007,Residuated Lattices: An Algebraic Glimpse atSubstructural Logics, Amsterdam: Elsevier.
• Geach, Peter and Black, Max (eds.), 1952,Translations fromthe Philosophical Writings of Gottlob Frege, New York:Philosophical Library.
• Ginsberg, Matthew, 1988, “Multivalued logics: a uniformapproach to reasoning in AI”,Computer Intelligence, 4:256–316.
• Gödel, Kurt, 1944, “Russell’s mathematicallogic”, in: P.A. Schilpp (ed.),The Philosophy of BertrandRussell, Evanston and Chicago: Northwestern University Press, 125–53.
• Goldblatt, Robert, 2006,Topoi: The Categorial Analysis ofLogic, Mineola, NY: Dover Publications.
• Gottwald, Siegfried, 1989,Mehrwertige Logik. EineEinführung in Theorie und Anwendungen, Berlin:Akademie-Verlag.
• –––, 2001,A Treatise on Many-valuedLogic, Baldock: Research Studies Press.
• Goguen, Joseph, 1969, “The logic of inexactconcepts”,Synthese, 19: 325–373.
• Grossmann, Reinhardt, 1992,The Existence of the World,London: Routledge.
• Haack, Susan, 1996,Deviant Logic, Fuzzy Logic. Beyond theFormalism, Chicago: University of Chicago Press.
• Hajek, Petr, 1998,Metamathematics of fuzzy logic,Dordrecht: Kluwer Academic Publishers.
• Jennings, Ray and Schotch, Peter, 1984, “Thepreservation of coherence”,Studia Logica, 43:89–106.
• Kamide, Norihiro and Wansing, Heinrich, 2009, “Sequentcalculi for some trilattice logics”,Review of SymbolicLogic, 2: 374–395.
• Karpenko, Alexander, 1983, “Factor semanticsforn-valued logics”,Studia Logica, 42:179–185.
• Keefe, Rosanna, 2000,Theories of Vagueness, Cambridge:Cambridge University Press.
• Kneale, William and Kneale, Martha, 1962,The Development of Logic, Oxford: Oxford University Press.
• Lewis, David, 1982, “Logic for equivocators”,Noûs, 16: 431–441.
• Lowe, Jonathan, 1995, “The metaphysics of abstractobjects”,The Journal of Philosophy, 92: 509–524.
• –––, 1997, “Objects and criteria ofidentity”, in:A Companion to the Philosophy ofLanguage, R. Hale and C. Wright (eds.), Oxford: Basil Blackwell,613–33.
• Łukasiewicz, Jan, 1918, “Farewell lecture by professorJan Łukasiewicz,” delivered in the Warsaw UniversityLecture Hall in March, 1918, in: (Łukasiewicz 1970), 87–88.
• –––, 1920, “O logicetrójwartościowej”,Ruch Filozoficny, 5:170–171. (English translation as “On three-valued logic”in: (Łukasiewicz 1970), 87–88.)
• –––, 1921, “Logikadwuwartościowa”,Przegl ad Filosoficzny, 13:189–205. (English translation as “Two-valued logic” in:(Łukasiewicz 1970), 89–109.)
• –––, 1970,Selected Works, L. Borkowski(ed.), Amsterdam: North-Holland, and Warsaw: PWN.
• MacFarlane, John, 2008, “Truth in the garden of forkingpaths”, in:Relative Truth, Max Kölbel and ManuelGarcía-Carpintero (eds.), Oxford: Oxford University Press, 81–102.
• Malinowski, Grzegorz, 1990, “Q-consequenceoperation”,Reports on Mathematical Logic, 24:49–59.
• –––, 1993,Many-Valued Logics, Oxford:Clarendon Press.
• –––, 1994, “Inferentialmany-valuedness”, in: Jan Wolenski (ed.),PhilosophicalLogic in Poland, Dordrecht: Kluwer Academic Publishers,75–84.
• Mehlberg, Henryk, 1958,The Reach of Science, Toronto:University of Toronto Press.
• Meyer, Robert K., 1978Why I Am Not a Relevantist,Research Paper No. 1, Canberra: Australian National University (LogicGroup, Research School of the Social Sciences).
• Neale, Stephen, 2001,Facing Facts, Oxford: Oxford UniversityPress.
• Odintsov, Sergei, 2009, “On axiomatizing Shramko-Wansing’slogic”,Studia Logica, 93: 407–428.
• Odintsov, Sergei and Wansing, Heinrich, 2015, “The logic of generalized truth values and the logic of bilattices”,Studia Logica, 103: 91–112.
• Perry, John, 1996, “Evading the slingshot”, in:A. Clark, J. Ezquerro, and J. Larrazabal (eds.),Philosophy andCognitive Science. Categories, Consciousness, and Reasoning,Dordrecht: Kluwer Academic Publishers.
• Popper, Karl, 1972,Objective Knowledge: An EvolutionaryApproach, Oxford: Oxford University Press.
• Post, Emil, 1921, “Introduction to a general theory ofelementary propositions”,American Journal ofMathematics, 43: 163–185.
• Priest, Graham, 1979, “Logic of Paradox”,Journalof Philosophical Logic, 8: 219–241.
• Quine, Willard Van Orman, 1954, “Reference andmodality”,Journal Symbolic Logic, 19: 137–138.
• –––, 1969,Ontological Relativity andOther Essays, New York: Columbia University Press.
• Reck, Erich, 2007, “Frege on truth, judgment, andobjectivity”,Grazer Philosophische Studien, 75:149–173.
• Rescher, Nicholas, 1969,Many-Valued Logic, New York:McGraw-Hill.
• Routley, Richard, 1975, “Universalsemantics?”,The Journal of Philosophical Logic, 4:327–356.
• Ruffino, Marco, 2003, “Wahrheit als Wert und als Gegenstandin der Logik Freges”, in: D. Greimann (ed.),Das Wahre unddas Falsche. Studien zu Freges Auffassung von Wahrheit,Hildesheim: Georg Olms Verlag, 203–221.
• Ryan, Mark and Sadler, Martin, 1992, “Valuation systemsand consequence relations”, in: S. Abramsky, D. Gabbay, andT. Maibaum (eds.),Handbook of Logic in Computer Science,Vol. 1., Oxford: Oxford University Press, 1–78.
• Shramko, Yaroslav, 2014, “The logical way of being true: Truth values and the ontological foundation of logic”,Logic and Logical Philosophy.
• Shramko, Yaroslav, Dunn, J. Michael and Takenaka, Tatsutoshi,2001, “The trilaticce of constructive truthvalues”,Journal of Logic and Computation, 11:761–788.
• Shramko, Yaroslav and Wansing, Heinrich, 2005, “Someuseful 16-valued logics: how a computer network shouldthink”,Journal of Philosophical Logic, 34:121–153.
• –––, 2006,“Hypercontradictions, generalized truth values, and logics oftruth and falsehood”,Journal of Logic, Language andInformation, 15: 403–424.
• –––, 2009, “TheSlingshot-Argument and sentential idendity”,StudiaLogica, 91: 429–455.
• Shramko, Yaroslav and Wansing, Heinrich, 2011,Truth and Falsehood: An Inquiry into Generalized Logical Values, Trends in Logic Vol. 36, Dordrecht, Heidelberg, London, New York: Springer.
• Sluga, Hans, 2002, Frege on the indefinability of truth, in:E. Reck (ed.),From Frege to Wittgenstein: Perspectives onEarly Analytic Philosophy, Oxford: Oxford University Press,75–95.
• Suszko, Roman: “The Fregean axiom and Polish mathematicallogic in the 1920’s”,Studia Logica, 36: 373–380.
• Tarski, Alfred, 1930, “Über einige fundamentaleBegriffe der Metamathematik”,Comptes Rendus desSéances de la Société des Sciences et des Lettres deVarsovie XXIII, Classe III: 22–29.
• –––, 1930, “Fundamentale Begriffe derMethodologie der deduktiven Wissenschaften, I”,Monatsheftefür Mathematik und Physik, 37: 361–404.
• Truth Values. Part I, 2009, Special issue ofStudialogica, Yaroslav Shramko and Heinrich Wansing (eds.), Vol. 91,No. 3.
• Truth Values. Part II, 2009, Special issue ofStudialogica, Yaroslav Shramko and Heinrich Wansing (eds.), Vol. 92,No. 2.
• Urquhart, Alasdair, 1986, “Many-valued logic”, in:D. Gabbay and F. Guenther (eds.),Handbook of PhilosophicalLogic, Vol. III., D. Reidel Publishing Co., Dordrecht,71–116.
• Wansing, Heinrich, 2010, “The power of Belnap. Sequentsystems forSIXTEEN₃”,Journal ofPhilosophical Logic, forthcoming.
• Wansing, Heinrich and Belnap, Nuel, 2009, “Generalized truthvalues. A reply to Dubois”,Logic Journal ofthe IGPL, forthcoming.
• Wansing, Heinrich and Shramko, Yaroslav, 2008, “Suszko’sThesis, inferential many-valuedness, and the notion of a logicalsystem”,Studia Logica, 88: 405–429, 89: 147.
• Williamson, Timothy, 1994,Vagueness, London: Routledge.
• Windelband, Wilhelm, 1915,Präludien: Aufätze undReden zur Philosophie und ihrer Geschichte, 5. Aufgabe,Bnd. 1., Tübingen.
• Wójcicki, Ryszard, 1970, “Some remarks on theconsequence operation in sentential logics”,FundamentaMathematicae, 68: 269–279.
• –––, 1988,Theory of Logical Calculi. BasicTheory of Consequence Operations, Dordrecht: Kluwer Academic Publishers.
• Wrigley, Anthony, 2006, “Abstractingpropositions”,Synthese, 151: 157–176.
• Zadeh, Lotfi, 1965, “Fuzzy sets”,Information andControl, 8: 338–53.
• Zadeh, Lotfi, 1975, “Fuzzy logic and approximatereasoning”,Synthese, 30: 407–425.
• Zaitsev, Dmitry and Shramko, Yaroslav, 2013, “Bi-facial truth: a case for generalized truth values”,Studia Logica, 101: 1299–1318.
• Zalta, Edward, 1983,Abstract Objects: An Introduction toAxiomatic Metaphysics, Dordrecht: D. Reidel Publishing Co.

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