## 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 essentially
on the idea of a strict discrimination between two main kinds of
expressions: proper names (singular terms) and functional
expressions. Proper names designate (signify, denote, or refer to)
singular objects, and functional expressions designate (signify,
denote, or refer to) functions. [Note: In the literature, the
expressions 'designation', 'signification',
'denotation', and 'reference' are usually
taken to be synonymous. This practice is used throughout the present
entry.] The name 'Ukraine', for example, refers to a
certain 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 (Kiev). Whereas names are
"saturated" (complete) expressions, functional expressions
are "unsaturated" (incomplete) and may be saturated by
applying them to names, producing in this way new names. Similarly,
the objects to which singular terms refer are saturated and the
functions denoted by functional expression are unsaturated. Names to
which a functional expression can be applied are called
the *arguments* of this functional expression, and entities to
which a function can be applied are called the *arguments* of
this function. The object which serves as the reference for the name
generated by an application of a functional expression to its
arguments is called the *value* of the function for these
arguments. Particularly, the above mentioned functional expression
'the capital of' remains incomplete until applied to some
name. An application of the function denoted by 'the capital
of' to Ukraine (as an argument) returns Kyiv as the object
denoted by the compound expression 'the capital of
Ukraine' which, according to Frege, is a proper name of
Kyiv. Note that Frege distinguishes between an \(n\)-place function
\(f\) as an unsaturated entity that can be completed by and applied to
arguments \(a\_1\),..., \(a\_n\) and its *course of values*,
which can be seen as the set-theoretic representation of this
function: 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 with
two intricate problems. *First*, how should one treat
declarative *sentences*? Should one perhaps separate them into
a specific linguistic category distinct from the ones of names and
function symbols? And *second*, how--from a functional
point of view--should one deal with *predicate
expressions* such as 'is a city', 'is
tall', 'runs', 'is bigger than',
'loves', etc., which are used to denote classes of
objects, properties of objects, or relations between them and which
can 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 their
arguments, and what can serve as referents of these names,
respectively values of these functions?



A uniform solution of both problems is obtained by introducing the
notion of a *truth value*. Namely, by applying the criterion of
"saturatedness" Frege provides a negative answer to the
first of the above problems. Since sentences are a kind of complete
entities, they should be treated as a sort of proper names, but names
destined to denote some specific objects, namely the truth values:
*the True* and *the False*. In this way one also obtains
a solution of the second problem. Predicates are to be interpreted as
some kind of functional expressions, which being applied to these or
those names generate sentences referring to one of the two truth
values. For example, if the predicate 'is a city' is
applied to the name 'Kyiv', one gets the sentence
'Kyiv is a city', which designates *the True*
(i.e., 'Kyiv is a city' *is true*). On the other
hand, by using the name 'Mount Everest', one obtains the
sentence 'Mount Everest is a city' which clearly
designates *the False*, since 'Mount Everest is a
city' *is false*.



Functions whose values are truth values are called *propositional
functions*. Frege also referred to them as concepts
(*Begriffe*). A typical kind of such functions (besides the
ones denoted by predicates) are the functions denoted by propositional
connectives. Negation, for example, can be interpreted as a unary
function converting *the* True into *the False* and
*vice versa*, and conjunction is a binary function that returns
*the True* as a value when both its argument positions are
filled in by *the True*, etc. Propositional functions mapping
\(n\)-tuples of truth values into truth values are also called
*truth-value functions*.



Frege thus in a first step extended the familiar notion of a numerical
function to functions on singular objects in general and, moreover,
introduced a new kind of singular objects that can serve as arguments
and values of functions on singular objects, the truth values. In a
further step, he considered propositional functions taking 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 of the
*second-order* function denoted by 'every city' is
the *first-order* propositional function on singular objects
denoted by 'is a capital'. The functional value denoted by
the sentence 'Every city is a capital' is a truth value,
*the False*.



Truth values thus prove to be an extremely effective instrument for a
logical and semantical analysis of 
 language.[1]
 Moreover, Frege provides truth values (as proper referents of
sentences) not merely with a pragmatical motivation but also with a
strong theoretical justification. The idea of such justification, that
can be found in Frege 1892, employs the principle
of *substitutivity* of co-referential terms, according to which
the reference of a complex singular term must remain unchanged when
any of its sub-terms is replaced by an expression having the same
reference. This is actually just an instance of the compositionality
principle mentioned above. If sentences are treated as a kind of
singular terms which must have designations, then assuming the
principle of substitutivity one "almost inevitably" (as
Kurt Godel (1944: 129) explains) is forced to recognize truth
values as the most suitable entities for such
designations. Accordingly, Frege asks:



> 
> 
> What else but the truth value could be found, that belongs quite
> generally to every sentence if the reference of its components is
> relevant, and remains unchanged by substitutions of the kind in
> question? (Geach and Black 1952: 64) 
> 
> 
> 



The idea underlying this question has been neatly reconstructed by
Alonzo Church in his *Introduction to Mathematical Logic*
(1956: 24-25) by considering the following sequence of four
sentences:



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


C1-C4 present a number of conversion steps each producing
co-referential sentences. It is claimed that C1 and C2 must have the
same designation by substitutivity, for the terms 'the author of
*Waverley*' and 'the man who wrote 29
*Waverley* Novels altogether' designate one and the same
object, namely Walter Scott. And so must C3 and C4, because the
number, such that Sir Walter Scott is the man who wrote that many
*Waverley* Novels altogether is the same as the number of
counties in Utah, namely 29. Next, Church argues, it is plausible to
suppose that C2, even if not completely synonymous with C3, is at
least so close to C3 "so as to ensure its having the same
denotation". If this is indeed the case, then C1 and C4 must
have the same denotation (designation) as well. But it seems that the
only (semantically relevant) thing these sentences have in common is
that both are true. Thus, taken that there must be something what the
sentences designate, one concludes that it is just their truth value.
As Church remarks, a parallel example involving false sentences can be
constructed in the same way (by considering, e.g., 'Sir Walter
Scott is not the author of *Waverley*').



This line of reasoning is now widely known as the "slingshot
argument", a term coined by Jon Barwise and John Perry (in
Barwise and Perry 1981: 395), who stressed thus an extraordinary
simplicity of the argument and the minimality of presuppositions
involved. Stated generally, the pattern of the argument goes as
follows (cf. Perry 1996). One starts with a certain sentence, and then
moves, step by step, to a completely different sentence. Every two
sentences in any step designate presumably one and the same thing.
Hence, the starting and the concluding sentences of the argument must
have the same designation as well. But the only semantically
significant thing they have in common seems to be their truth value.
Thus, what any sentence designates is just its truth value.



A formal version of this argument, employing the term-forming,
variable-binding class abstraction (or property abstraction) operator
l\(x\) ("the class of all \(x\) such that" or
"the property of being such an \(x\) that"), was first
formulated by Church (1943) in his review of Carnap's
*Introduction to Semantics*. Quine (1953), too, presents a
variant of the slingshot using class abstraction, see also (Shramko
and Wansing 2009a). Other remarkable variations of the argument are
those by Kurt Godel (1944) and Donald Davidson (1967, 1969),
which make use of the formal apparatus of a theory of definite
descriptions dealing with the description-forming, variable-binding
iota-operator (i\(x\), "the \(x\) such that"). It is
worth noticing that the formal versions of the slingshot show how to
move--using steps that ultimately preserve reference--from
*any* true (false) sentence to *any* other such
sentence. In view of this result, it is hard to avoid the conclusion
that 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 the
part of fact-theorists, i.e., adherents of facts, situations,
propositions, states of affairs, and other fact-like entities
conceived as alternative candidates for denotations of declarative
sentences. Also see the
 supplement 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 of
truth. Therefore it may seem rather tempting to try to incorporate
considerations on truth values into the broader context of traditional
truth-theories, such as correspondence, coherence, anti-realistic, or
pragmatist conceptions of truth. Yet, it is unlikely that such
attempts can give rise to any considerable success. Indeed, the
immense fruitfulness of Frege's introduction of truth values
into logic to a large extent is just due to its philosophical
neutrality with respect to theories of truth. It does not commit one
to any specific metaphysical doctrine of truth. In one significant
respect, however, the idea of truth values contravenes traditional
approaches to truth by bringing to the forefront the problem of its
categorial classification.



In most of the established conceptions, truth is usually treated as a
property. It is customary to talk about a "truth
predicate" and its attribution to sentences, propositions,
beliefs or the like. Such an understanding corresponds also to a
routine linguistic practice, when one operates with the adjective
'true' and asserts, e.g., 'That 5 is a prime number
is true'. By contrast with this apparently quite natural
attitude, the suggestion to interpret truth as an object may seem very
confusing, to say the least. Nevertheless this suggestion is also
equipped with a profound and strong motivation demonstrating that it
is 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 is not
as natural as it appears on the face of it. Frege brought into play an
argument to the effect that characterizing a sentence as *true*
adds nothing new to its content, for 'It is true 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 sense
*redundant* and thus is not a real 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 loss for its content.
In this case a superficial grammatical analogy is misleading. This
idea gave an impetus to the deflationary conception of truth
(advocated by Ramsey, Ayer, Quine, Horwich, and others, see the entry
on
 the deflationary theory of truth).



However, even admitting the redundancy of truth as a property, Frege
emphasizes its importance and indispensable role in some other
respect. Namely, truth, accompanying every act of judgment as its
ultimate goal, secures an objective *value of cognition* by
arranging for every assertive sentence a transition from the level of
sense (the thought expressed by a sentence) to the level of denotation
(its truth value). This circumstance specifies the significance of
taking 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. The object, in the sense
> of the point or *objective*, of sentence use 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 a truth
value was, to a great extent, pragmatically motivated. Besides an
intended gain for his system of "Basic Laws" (Frege
1893/1903) reflected in enhanced technical clarity, simplicity, and
unity, Frege also sought to substantiate in this way his view on logic
as a theoretical discipline with truth as its main goal and primary
subject-matter. Incidentally, Gottfried Gabriel (1986) demonstrated
that in the latter respect Frege's ideas can be naturally linked
up with a value-theoretical tradition in German philosophy of the
second half of the 19th century; see also (Gabriel 2013) on
the relation between Frege's value-theoretically inspired
conception of truth values and his theory of judgement. More
specifically, Wilhelm Windelband, the founder and the principal
representative of the Southwest school of Neo-Kantianism, was actually
the first who employed the term "truth value"
("*Wahrheitswert*") in his essay "What is
Philosophy?" published in 1882 (see Windelband 1915: 32), i.e.,
nine years before Frege 1891, even if he was very far from treating a
truth value as a value of a function.



Windelband defined philosophy as a "critical science about
universal values". He considered philosophical statements to be
not mere judgements but rather *assessments*, dealing with some
fundamental values, *the value of truth* being one of the most
important among them. This latter value is to be studied by logic as a
special philosophical discipline. Thus, from a value-theoretical
standpoint, the main task of philosophy, taken generally, is to
establish the principles of logical, ethical and aesthetical
assessments, and Windelband accordingly highlighted the triad of basic
values: "true", "good" and
"beautiful". Later this triad was taken up by Frege in
1918 when he defined the subject-matter of logic (see below). Gabriel
points out (1984: 374) that this connection between logic and a value
theory can be traced back to Hermann Lotze, whose seminars in
Gottingen were attended by both Windelband and Frege.



The decisive move made by Frege was to bring together a philosophical
and a mathematical understanding of values on the basis of a
generalization of the notion of a function on numbers. While Frege may
have been inspired by Windelband's use of the word
'value' (and even more concretely - 'truth
value'), it is clear that he uses the word in its mathematical
sense. If predicates are construed as a kind of functional expressions
which, being applied to singular terms as arguments, produce
sentences, then the values of the corresponding functions must be
references of sentences. Taking into account that the range of any
function typically consists of objects, it is natural to conclude that
references of sentences must be objects as well. And if one now just
takes it that sentences refer to truth values (*the True* and
*the False*), then it turns out that truth values are indeed
objects, and it seems quite reasonable to generally explicate truth
and falsity as objects and not as properties. As Frege explains:



> 
> 
> A statement contains no empty place, and therefore we must take its
> *Bedeutung* as an object. But this *Bedeutung* is a
> truth-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 been
criticized, for example, by Michael Dummett who stated rather
dramatically:



> 
> 
> This was the most disastrous of the effects of the misbegotten
> doctrine that sentences are a species of complex singular terms, which
> dominated Frege's later period: to rob him of the insight that
> sentences play a unique role, and that the role of almost every other
> linguistic expression ... consists in its part in forming
> sentences. (Dummett 1981: 196) 
> 
> 
> 



But even Dummett (1991: 242) concedes that "to deny that
truth-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 kind of
objects, the obvious question as to the nature of these entities
arises. The above characterization of truth values as objects is far
too general and requires further specification. One way of such
specification is to qualify truth values as *abstract* objects.
Note that Frege himself never used the word 'abstract'
when describing truth values. Instead, he has a conception of so
called "logical objects", truth values being primary and
the most fundamental of them (Frege 1976: 121). Among the other
logical objects Frege pays particular attention to are sets and
numbers, emphasizing thus their logical nature (in accordance with his
logicist view).



Church (1956: 25), when considering truth values, explicitly
attributes to them the property of being abstract. Since then it is
customary to label truth values as abstract objects, thus allocating
them into the same category of entities as mathematical objects
(numbers, classes, geometrical figures) and propositions. One may pose
here an interesting question about the correlation between Fregean
logical objects and abstract objects in the modern sense (see the
entry on
 abstract objects). 
Obviously, the universe of abstract objects is much broader than the
universe of logical objects as Frege conceives them. The latter are
construed as constituting an ontological foundation for logic, and
hence for mathematics (pursuant to Frege's logicist program).
Generally, the class of *abstracta* includes a wide diversity
of platonic universals (such as redness, youngness, justice or
triangularity) and not only those of them which are logically
necessary. Nevertheless, it may safely be said that logical objects
can be considered as paradigmatic cases of abstract entities, or
abstract objects in their purest form.



It should be noted that finding an adequate definition of abstract
objects is a matter of considerable controversy. According to a common
view, abstract entities lack spatio-temporal properties and relations,
as opposed to concrete objects which exist in space and time (Lowe
1995: 515). In this respect truth values obviously *are*
abstract as they clearly have nothing to do with physical spacetime.
In a similar fashion truth values fulfill another requirement often
imposed upon abstract objects, namely the one of a causal inefficacy
(see, e.g., Grossmann 1992: 7). Here again, truth values are very much
like numbers and geometrical figures: they have no causal power and
make nothing happen.



Finally, it is of interest to consider how truth values can be
introduced by applying so-called *abstraction principles*,
which are used for supplying abstract objects with *criteria of
identity*. The idea of this method of characterizing abstract
objects is also largely due to Frege, who wrote:



> 
> 
> If the symbol *a* is to designate an object for us, then we must
> have a criterion that decides in all cases whether *b* is the
> same as *a*, even if it is not always in our power to apply this
> criterion. (Frege 1884, trans. Beaney 1997: 109) 
> 
> 
> 



More precisely, one obtains a new object by abstracting it from some
given kind of entities, in virtue of certain criteria of identity for
this new (abstract) object. This abstraction is performed in terms of
an equivalence relation defined on the given entities (see Wrigley
2006: 161). The celebrated slogan by Quine (1969: 23) "No entity
without identity" is intended to express essentially the same
understanding of an (abstract) object as an "item falling under
a sortal concept which supplies a well-defined criterion of identity
for its instances" (Lowe 1997: 619).



For truth values such a criterion has been suggested in Anderson and
Zalta (2004: 2), stating that for any two sentences \(p\) and \(q\),
the truth value of \(p\) is identical with the truth value of \(q\) if
and only if \(p\) is (non-logically) equivalent with \(q\) (cf. also
Dummett 1959: 141). This idea can be formally explicated following the
style 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 &, \(\Rightarrow, \Leftrightarrow, \forall\) stand
correspondingly for 'and', 'if... then',
'if and only if' and 'for all' in the
*metalanguage*, and \(\leftrightarrow\) stands for some
*object language* equivalence connective (biconditional).



Incidentally, Carnap (1947: 26), when introducing truth-values as
extensions of sentences, is guided by essentially the same idea.
Namely, he points out a strong analogy between extensions of
predicators and truth values of sentences. Carnap considers a wide
class of designating expressions ("designators") among
which there are predicate expressions ("predicators"),
functional expressions ("functors"), and some others.
Applying the well-known technique of interpreting sentences as
predicators of degree 0, he generalizes the fact that two predicators
of degree \(n\) (say, \(P\) and \(Q)\) have the same extension if and
only 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, two
sentences (say, \(p\) and \(q)\), being interpreted as predicators of
degree 0, must have the same extension if and only if
\(p\leftrightarrow q\) holds, that is if and only if they are
equivalent. And then, Carnap remarks, it seems quite natural to take
truth values as extensions for sentences.



Note that this criterion employs a *functional dependency*
between an introduced abstract object (in this case a truth value) and
some other objects (sentences). More specifically, what is considered
is the truth value of a sentence (or proposition, or the like).
The criterion of identity for truth values is formulated then through
the logical relation of equivalence holding between these other
objects--sentences, propositions, or the like (with an explicit
quantification over them).



It should also be remarked that the properties of the object language
biconditional depend on the logical system in which the biconditional
is employed. Biconditionals of different logics may have different
logical properties, and it surely matters what kind of the equivalence
connective is used for defining truth values. This means that the
concept of a truth value introduced by means of the identity criterion
that involves a biconditional between sentences is also
logic-relative. Thus, if '\(\leftrightarrow\)' stands for
material equivalence, one obtains classical truth values, but if the
intuitionistic biconditional is employed, one gets truth values of
intuitionistic logic, etc. Taking into account the role truth values
play 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 truth
value of proposition \(p\)' ('\(tv(p)\)' [notation
adjusted]):

 

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


where \(A\)! stands for a primitive theoretical predicate 'being
abstract', \(xF\) is to be read as "\(x\) encodes
\(F\)" and [l*y q*] is a propositional
property ("being such a \(y\) that \(q\)"). That is,
according to this definition, "the extension of \(p\) is the
abstract object that encodes all and only the properties of the form
[l*y q*] which are constructed out of
propositions \(q\) materially equivalent to \(p\)" (Anderson and
Zalta 2004: 14).



The notion of a truth value in general is then defined as an object
which 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 Fregean
truth values *the True* \((\top)\) and *the False*
\((\bot)\):

 

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


Anderson and Zalta prove then that \(\top\) and \(\bot\) are indeed
truth values and, moreover, that there are exactly two such objects.
The latter result is expected, if one bears in mind that what the
definitions above actually introduce are the *classical* truth
values (as the underlying logic is classical). Indeed,
\(p\leftrightarrow q\) is classically equivalent to \((p\wedge
q)\vee(\neg p\wedge \neg q)\), and \(\neg(p\leftrightarrow q)\) is
classically equivalent to \((p\wedge \neg q)\vee(\neg p\wedge q)\).
That is, the connective of material equivalence divides sentences into
two distinct collections. Due to the law of excluded middle these
collections are exhaustive, and by virtue of the law of
non-contradiction they are exclusive. Thus, we get exactly two
equivalence classes of sentences, each being a hypostatized
representative 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 the
word 'beautiful' does for aesthetics and the word
'good' for ethics. Thus, according to such a view, 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 not in what
is merely taken to be true. Now, if one admits that truth is a
specific abstract object (the corresponding truth value), then logic
in the first place has to explore the features of this object and its
interrelations to other entities of various other kinds.



A prominent adherent of this conception was Jan Lukasiewicz. As
he paradigmatically put it:



> 
> 
> All true propositions denote one and the same object, namely truth,
> and all false propositions denote one and the same object, namely
> falsehood. I consider truth and falsehood to be *singular*
> objects in the same sense as the number 2 or 4 is. ...
> Ontologically, truth has its analogue in being, and falsehood, in
> non-being. The objects denoted by propositions are called *logical
> values*. Truth is the positive, and falsehood is the negative
> logical value. ... Logic is the science of objects of a special
> kind, namely a science of *logical values*. (Lukasiewicz
> 1970: 90) 
> 
> 
> 



This definition may seem rather unconventional, for logic is usually
treated as the science of correct reasoning and valid inference. The
latter understanding, however, calls for further justification. This
becomes evident, as soon as one asks, *on what grounds* one
should qualify this or that pattern of reasoning as correct or
incorrect.



In answering this question, one has to take into account that any
valid inference should be based on logical rules which, according to a
commonly accepted view, should at least guarantee that in a valid
inference the conclusion(s) is (are) true if all the premises are
true. Translating this demand into the Fregean terminology, it would
mean that in the course of a correct inference the possession of the
truth value *The True* should be *preserved* from the
premises to the conclusion(s). Thus, granting the realistic treatment
of truth values adopted by Frege, the understanding of logic as the
science of truth values in fact provides logical rules with an
ontological justification placing the roots of logic in a certain kind
of ideal entities (see Shramko 2014).



These entities constitute a certain uniform domain, which can be
viewed as a subdomain of Frege's so-called "third
realm" (the realm of the objective content of thoughts, and
generally abstract objects of various kinds, see Frege 1918, cf.
Popper 1972 and also Burge 1992: 634). Among the subdomains of this
third realm one finds, e.g., the collection of mathematical objects
(numbers, classes, etc.). The set of truth values may be regarded as
forming another such subdomain, namely the one of *logical
values*, and logic as a branch of science rests essentially on
this *logical domain* and on exploring its features and
regularities.


### 2.2 Many-valued logics, truth degrees and valuation systems



According to Frege, there are exactly two truth values, *the
True* and *the False*. This opinion appears to be rather
restrictive, and one may ask whether it is really indispensable for
the concept of a truth value. One should observe that in elaborating
this conception, Frege assumed specific requirements of his system of
the *Begriffsschrift*, especially the principle of bivalence
taken as a metatheoretical principle, viz. that there exist only two
distinct logical values. On the object-language level this principle
finds its expression in the famous classical laws of excluded middle
and non-contradiction. The further development of modern logic,
however, has clearly demonstrated that classical logic is only one
particular theory (although maybe a very distinctive one) among the
vast variety of logical systems. In fact, the Fregean ontological
interpretation of truth values depicts logical principles as a kind of
ontological postulations, and as such they may well be modified or
even abandoned. For example, by giving up the principle of bivalence,
one is naturally led to the idea of postulating *many truth
values*.



It was Lukasiewicz, who as early as 1918 proposed to take
seriously other logical values different from truth and falsehood (see
Lukasiewicz 1918, 1920). Independently of Lukasiewicz, Emil
Post in his dissertation from 1920, published as Post 1921, introduced
\(m\)-valued truth tables, where \(m\) is any positive integer.
Whereas Post's interest in *many-valued logic* (where
"many" means "more than two") was almost
exclusively mathematical, Lukasiewicz's motivation was
philosophical (see the entry on
 many-valued logic).
 He contemplated the semantical value of sentences about the
contingent future, as discussed in Aristotle's *De
interpretatione*. Lukasiewicz introduced a third truth value
and interpreted it as "possible". By generalizing this
idea and also adopting the above understanding of the subject-matter
of logic, one naturally arrives at the representation of particular
logical systems as a certain kind of *valuation systems* (see,
e.g., Dummett 1981, 2000; Ryan and Sadler 1992).



Consider a propositional language \(\mathcal{L}\) built upon a set of
atomic sentences \(\mathcal{P}\) and a set of propositional
connectives \(\mathcal{C}\) (the set of sentences of \(\mathcal{L}\)
being the smallest set containing \(\mathcal{P}\) and being closed
under the connectives from \(\mathcal{C})\). Then a *valuation
system* \(\mathbf{V}\) for the language \(\mathcal{L}\) is a
triple \(\langle \mathcal{V}, \mathcal{D}, \mathcal{F}\rangle\), 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 of truth values, \(\mathcal{D}\) is the set of
*designated* truth values, and \(\mathcal{F}\) is the set of
truth-value functions interpreting the elements of \(\mathcal{C}\). If
the set of truth values of a valuation system \(\mathbf{V}\) has \(n\)
elements, \(\mathbf{V}\) is said to be \(n\)-valued. Any valuation
system can be equipped with an assignment function which maps the set
of atomic sentences into \(\mathcal{V}\). Each assignment \(a\)
relative to a valuation system \(\mathbf{V}\) can be extended to all
sentences of \(\mathcal{L}\) by means of a valuation function \(v\_a\)
defined in accordance 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 \(\mathcal{V}\) are
sometimes referred to as *quasi truth values*. Siegfried
Gottwald (1989: 2) explains that one reason for using the term
'quasi truth value' is that there is no convincing and
uniform interpretation of the truth values that in many-valued logic
are assumed in addition to the classical truth values *the
True* and *the False*, an understanding that, according to
Gottwald, associates the additional values with the naive
understanding of being true, respectively the naive understanding of
*degrees of being true* (cf. also the remark by Font (2009:
383) that "[o]ne of the main problems in many-valued logic, at
least in its initial stages, was the interpretation of the
'intermediate' or 'non-classical'
values", et seq.). In later publications, Gottwald has changed
his terminology and states that 



> 
> 
> [t]o avoid any confusion with the case of classical logic one prefers
> in many-valued logic to speak of *truth degrees* and to use the
> word "truth value" only for classical logic. (Gottwald
> 2001: 4) 
> 
> 
> 



Nevertheless in what follows the term 'truth values' will
be used even in the context of many-valued logics, without any
commitment to a philosophical conception of truth as a graded notion
or a specific understanding of semantical values in addition to the
classical truth values.



Since the cardinality of \(\mathcal{V}\) may be greater than 2, the
notion of a valuation system provides a natural foundational framework
for the very idea of a many-valued logic. The set \(\mathcal{D}\) of
designated values is of central importance for the notion of a
valuation system. This set can be seen as a generalization of the
classical truth value *the True* in the sense that it
determines many central logical notions and thereby generalizes some
of the important roles played by Frege's *the True* (cf.
the introductory remarks about uses of truth values). For example, the
set of tautologies (logical laws) is directly specified by the given
set of designated truth values: a sentence \(A\) is a
*tautology* in a valuation system \(\mathbf{V}\) iff for every
assignment \(a\) relative to \(\mathbf{V}\), \(v\_a(A) \in
\mathcal{D}\). Another fundamental logical notion--that of an
entailment relation--can also be defined by referring to the set
\(\mathcal{D}\). For a given valuation system \(\mathbf{V}\) a
corresponding entailment relation \((\vDash\_V)\) is usually defined by
postulating the preservation of designated values from the premises to
the conclusion:




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



A pair \(\mathcal{M} = \langle \mathbf{V}, v\_a\rangle\), where
\(\mathbf{V}\) is an \((n\)-valued) valuation system and \(v\_a\) a
valuation 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 entailment
relation \(\vDash\_{\mathcal{M}}\) by defining
\(D\vDash\_{\mathcal{M} }A\textrm{ iff }(\forall B \in D:
v\_a (B) \in \mathcal{D}) \Rightarrow v\_a(A) \in \mathcal{D}\).



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 said
to be *strictly characteristic* for \(\mathfrak{L}\) just in
case \(D\vDash\_V A \textrm{ iff } D\vdash\_{ \mathfrak{L}
}A\) (see Dummett 1981: 431). Conversely, one says that
\(\mathfrak{L}\) *is characterized* by \(\mathbf{V}\). Thus, if
a valuation system is said to *determine* a logic, the
valuation system *by itself* is, properly speaking,
*not* a logic, but only serves as a semantic basis for some
logical 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 required to
be non-empty, and in Dunn & Hardegree 2001, \(\mathcal{D}\) is
required to be a non-empty proper subset of \(\mathbf{V}\). With a
view on semantically defining a many-valued logic, these restrictions
are very natural and have been taken up in Shramko & Wansing 2011
and elsewhere. For the characterization of consequence relations (see
the supplementary document
 Suszko's Thesis), however, the restrictions do not apply. 



In this way Fregean, i.e., classical, logic can be presented as
determined by a particular valuation system based on exactly two
elements: \(\mathbf{V}\_{cl} = \langle \{T, F\}, \{T\}, \{ f\_{\wedge},
f\_{\vee}, f\_{\rightarrow}, f\_{\sim}\}\rangle\), where \(f\_{\wedge},
f\_{\vee}, f\_{\rightarrow},f\_{\sim}\) are given by the classical truth
tables for conjunction, disjunction, material implication, and
negation.



As an example for a valuation system based on more that two elements,
consider two well-known valuation systems which determine
Kleene's (strong) "logic of indeterminacy" \(K\_3\)
and Priest's "logic of paradox" \(P\_3\). In a
propositional language without implication, \(K\_3\) is specified by
the *Kleene matrix* \(\mathbf{K}\_3 = \langle \{T, I, F\},
\{T\}, \{ f\_c: c \in \{\sim , \wedge , \vee \}\} \rangle\), where the
functions \(f\_c\) are defined as follows:

 

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


The *Priest matrix* \(\mathbf{P}\_3\) differs from
\(\mathbf{K}\_3\) only in that \(\mathcal{D} = \{T, I\}\). Entailment
in \(\mathbf{K}\_3\) as well as in \(\mathbf{P}\_3\) is defined by means
of
 (3).



There are natural intuitive interpretations of \(I\) in
\(\mathbf{K}\_3\) and in \(\mathbf{P}\_3\) as the
*underdetermined* and the *overdetermined* value
respectively--a truth-value gap and a truth-value glut. Formally
these interpretations can be modeled by presenting the values as
certain 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 nor
false"), 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 the
third value as "antinomic".) The designatedness of a truth
value 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, one
obtains the four-valued logic \(B\_4\) introduced by Dunn and Belnap
(Dunn 1976; Belnap 1977a,b). A Gentzen-style formulation can be found
in Font (1997: 7)). This logic is determined by the *Belnap
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
defined as 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 the well-known logic of
 "first-degree entailment" (\(E\_{fde}\)) introduced in
 Anderson & Belnap 1975 (see also Omori and Wansing 2017).



The syntactic notion of a single-conclusion consequence relation has
been extensively studied by representatives of the Polish school of
logic, most notably by Alfred Tarski, who in fact initiated this line
of research (see Tarski 1930a,b; cf. also Wojcicki 1988). In
view of certain key features of a standard consequence relation it is
quite remarkable--as well as important--that any entailment
relation \(\vDash\_V\) defined as above has the following structural
properties (see Ryan and Sadler 1992: 34):




\[\begin{align} 
\tag{4} D\cup \{A\}&\vDash\_V A &\textrm{(Reflexivity)} \\ 
\tag{5} \textrm{If } D\vDash\_V A \textrm{ then } D\cup G &\vDash\_V A &\textrm{(Monotonicity)}\\ 
\tag{6} \textrm{If } D\vDash\_V A \textrm{ for every } A \in G &\\
 \textrm{ and } G\cup D \vDash\_V B, \textrm{ then } D &\vDash\_VB & \textrm{(Cut)}
\end{align}\]



Moreover, for every \(A \in \mathcal{L}\), every \(D \subseteq
\mathcal{L}\), and every uniform substitution function \(s\) on
\(\mathcal{L}\) the following *Substitution* property holds
(\(s(D)\) stands for \(\{ s(B) \mid B \in
D\})\):




\[
\tag{7}
D\vDash\_V A \textrm{ implies } s(D)\vDash\_Vs(A). 
\]



(The function of uniform substitution \(s\) is defined as follows.
Let \(B\) be a formula in \(\mathcal{L}\), let \(p\_1,\ldots, p\_n\) be
all the propositional variables occurring in \(B\), and let
\(s(p\_1) = A\_1,\ldots , s(p\_n) = A\_n\) for some formulas
\(A\_1 ,\ldots ,A\_n\). Then \(s(B)\) is the formula that results
from B by substituting simultaneously \(A\_1\),..., \(A\_n\) 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 is
often called a *Tarskian consequence relation*. If additionally
a consequence relation has the substitution property
 (7),
 then it is called *structural*. Thus, any entailment relation
defined for a given valuation system \(\mathbf{V}\) presents an
important example of a consequence relation, in that \(\mathbf{V}\) is
strictly characteristic for some logical system \(\mathfrak{L}\) with
a structural Tarskian consequence relation.



Generally speaking, the framework of valuation systems not only
perfectly suits the conception of logic as the science of truth
values, but also turns out to be an effective technical tool for
resolving various sophisticated and important problems in modern
logic, 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 other
authors, suggests that truth comes by degrees, and these degrees may
be seen as truth values in an extended sense. The idea of truth as a
graded notion has been applied to model vague predicates and to obtain
a solution to the Sorites Paradox, the Paradox of the Heap (see the
entry on the
 Sorites Paradox).
 However, the success of applying many-valued logic to the problem of
vagueness is highly controversial. Timothy Williamson (1994: 97), for
example, holds that the phenomenon of higher-order vagueness
"makes most work on many-valued logic irrelevant to the problem
of vagueness".



In any case, the vagueness of concepts has been much debated in
philosophy (see the entry on
 vagueness)
 and it was one of the major motivations for the development of
*fuzzy logic* (see the entry on
 fuzzy logic).
 In the 1960s, Lotfi Zadeh (1965) introduced the notion of a *fuzzy
set*. A characteristic function of a set \(X\) is a mapping which
is defined on a superset \(Y\) of \(X\) and which indicates membership
of an element in \(X\). The range of the characteristic function of a
classical set \(X\) is the two-element set \(\{0,1\}\) (which may be
seen as the set of classical truth values). The function assigns the
value 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 the
real interval [0,1]. A vague predicate such as 'is much earlier
than March 20th, 1963', 'is beautiful',
or 'is a heap' may then be regarded as denoting a fuzzy
set. The membership function \(g\) of the fuzzy set denoted by
'is much earlier than March 20th, 1963' thus
assigns values (seen as truth degrees) from the interval [0, 1] to
moments in time, for example \(g\)(1p.m., August 1st, 2006)
\(= 0\), \(g\)(3a.m., March 19th, 1963) \(= 0\),
\(g\)(9:16a.m., April 9th, 1960) \(= 0.005\), \(g\)(2p.m.,
August 13th, 1943) \(= 0.05\), \(g\)(7:02a.m., December
2nd, 1278) \(= 1\).



The application of continuum-valued logics to the Sorites Paradox has
been suggested by Joseph Goguen (1969). The Sorites Paradox in its
so-called conditional form is obtained by repeatedly applying
*modus 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 a
collection 99,999 grains of sand is a heap.
* If a collection of 99,999 grains of sand is a heap, then a
collection 99,998 grains of sand is a heap.
* ...
* If a collection of 2 grains of sand is a heap, then a collection
of 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 first
premise is true and one grain does not make a difference to a
collection of grains being a heap or not, the conclusion is, of
course, unacceptable. If the predicate 'is a heap' denotes
a fuzzy set and the conditional is interpreted as implication in
Lukasiewicz's continuum-valued logic, then the Sorites
Paradox can be avoided. The truth-function \(f\_{\rightarrow}\) of
Lukasiewicz's implication \(\rightarrow\) is defined by
stipulating that if \(x \le y\), then \(f\_{\rightarrow}(x, y) = 1\),
and otherwise \(f\_{\rightarrow}(x, y) = 1 - (x - y)\). If, say, the
truth value of the sentence 'A collection of 500 grains of sand
is a heap' is 0.8 and the truth value of 'A collection of
499 grains of sand is a heap' is 0.7, then the truth value of
the implication 'If a collection of 500 grains of sand is a
heap, then a collection 499 grains of sand is a heap.' is 0.9.
Moreover, if the acceptability of a statement is defined as having a
value greater than \(j\) for \(0 \lt j \lt 1\) and all the conditional
premises of the Sorites Paradox do not fall below the value \(j\),
then *modus ponens* does not preserve acceptability, because
the conclusion of the Sorites Argument, being evaluated as 0, is
unacceptable.



Alasdair Urquhart (1986: 108) stresses 



> 
> 
> the extremely artificial nature of the attaching of precise numerical
> values to sentences like ... "Picasso's
> *Guernica* is beautiful". 
> 
> 
> 



To overcome the problem of assigning precise values to predications of
vague concepts, Zadeh (1975) introduced *fuzzy truth values* as
distinct from the numerical truth values in [0, 1], the former being
fuzzy subsets of the set [0, 1], understood as *true*, *very
true*, *not very true*, etc.



The interpretation of continuum-valued logics in terms of fuzzy set
theory has for some time be seen as defining the field of mathematical
fuzzy logic. Susan Haack (1996) refers to such systems of mathematical
fuzzy logic as "base logics" of fuzzy logic and reserves
the term 'fuzzy logics' for systems in which the truth
values themselves are fuzzy sets. Fuzzy logic in Zadeh's latter
sense has been thoroughly criticized from a philosophical point of
view by Haack (1996) for its "methodological
extravagances" and its linguistic incorrectness. Haack
emphasizes that her criticisms of fuzzy logic do not apply to the base
logics. Moreover, it should be pointed out that mathematical fuzzy
logics are nowadays studied not in the first place as continuum-valued
logics, but as many-valued logics related to residuated lattices (see
Hajek 1998; Cignoli *et al.* 2000; Gottwald 2001; Galatos
*et al.* 2007), whereas fuzzy logic in the broad sense is to a
large extent concerned with certain engineering methods.



A fundamental concern about the semantical treatment of vague
predicates is whether an adequate semantics should be
truth-functional, that is, whether the truth value of a complex
formula should depend functionally on the truth values of its
subformulas. Whereas mathematical fuzzy logic is truth-functional,
Williamson (1994: 97) holds that "the nature of vagueness is not
captured by any approach that generalizes truth-functionality".
According to Williamson, the degree of truth of a conjunction, a
disjunction, or a conditional just fails to be a function of the
degrees of truth of vague component sentences. The sentences
'John is awake' and 'John is asleep', for
example, may have the same degree of truth. By truth-functionality the
sentences 'If John is awake, then John is awake' and
'If John is awake, then John is asleep' are alike in truth
degree, indicating for Williamson the failure of
degree-functionality.



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



Van Fraassen's aim was to develop a semantics for sentences
containing non-denoting singular terms. Even if one grants atomic
sentences containing non-denoting singular terms and that some
attributions of vague predicates are neither true nor false, it
nevertheless seems natural not to preclude that compound sentences of
a certain shape containing non-denoting terms or vague predications
*are* either true or false, e.g., sentences of the form
'If \(A\), then \(A\)'. Supervaluational semantics
provides 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's *Guernica*
is beautiful'. Any classical assignment \(a'\) that agrees with
\(a\) whenever \(a\) assigns \(T\) or \(F\) may be seen as a
precisification (or superassignment) of \(a\). A sentence may than be
said to be supertrue under assignment \(a\) if it is true under every
precisification \(a'\) of \(a\). Thus, if \(a\) is a three-valued
assignment into \(\{T, I, F\}\) and \(a'\) is a two-valued assignment
into \(\{T, F\}\) such that \(a(p) = a'(p)\) if \(a(p) \in \{T, F\}\),
then \(a'\) is said to be a *superassignment* of \(a\). It
turns out that if \(a\) is an assignment extended to a valuation
function \(v\_a\) for the Kleene matrix \(\mathbf{K}\_3\), then for
every 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 a *supervaluation* of \(v\_a\). A
formula is then said to be *supertrue* under a valuation
function \(v\_a\) for \(\mathbf{K}\_3\) if it is true under every
supervaluation \(v\_{a'}\) of \(v\_a\), i.e., if \(v\_{a'}(A) = T\) for
every supervaluation \(v\_{a'}\) of \(v\_a\). The property of being
*superfalse* is defined analogously.



Since every supervaluation is a classical valuation, every classical
tautology is supertrue under every valuation function in
\(\mathbf{K}\_3\). Supervaluationism is, however, not truth-functional
with respect to supervalues. The supervalue of a disjunction, for
example, 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 \((p\vee \neg
p)\) is thus supertrue under \(v\_a,p\vee p\) is *not*, because
there are superassignments \(a'\) of \(a\) with \(a'(p) = F\). An
argument against the charge that supervaluationism requires a
non-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 the premises
to the conclusion(s) of valid inferences in supervaluationism, and
although it might be tempting to consider supertruth an abstract
object on its own, it seems that it has never been suggested to
hypostatize supertruth in this way, comparable to Frege's
*the True*. A sentence supertrue under a three-valued valuation
\(v\) just takes the Fregean value *the True* under every
supervaluation 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-valued
logics shows that there exist infinitely, in fact, uncountably many
truth values. However, this is not at all clear (recall the more
cautious use of terminology advocated by Gottwald).



In the 1970's Roman Suszko (1977: 377) declared many-valued
logic to be "a magnificent conceptual deceit". Suszko
actually claimed that "there are but two logical values, true
and false" (Caleiro *et al*. 2005: 169), a statement now
called *Suszko's Thesis*. For Suszko, the set of truth
values assumed in a logical matrix for a many-valued logic is a set of
"admissible referents" (called "algebraic
values") of formulas but not a set of logical values. Whereas
the algebraic values are elements of an algebraic structure and
referents of formulas, the logical value *true* is used to
define valid consequence: If every premise is true, then so is (at
least one of) the conclusion(s). The other logical value,
*false*, is preserved in the opposite direction: If the (every)
conclusion is false, then so is at least one of the premises. The
logical values are thus represented by a bi-partition of the set of
algebraic values into a set of designated values (truth) and its
complement (falsity).



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



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



Dummett observes that, first, 



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



and moreover, 



> 
> 
> finer distinctions between different designated values or different
> undesignated ones, however naturally they come to us, are justified
> only if they are needed in order to give a truth-functional account of
> the formation of complex statements by means of operators. (Dummett
> 1959: 155) 
> 
> 
> 



Suszko's claim evidently echoes this observation by Dummett.



Suszko's Thesis is substantiated by a rigorous proof (the Suszko
Reduction) showing that every structural Tarskian consequence relation
and therefore also every structural Tarskian many-valued propositional
logic is characterized by a bivalent semantics. (Note also that
Richard Routley (1975) has shown that every logic based on a
l-categorical language has a sound and complete bivalent
possible worlds semantics.) The dichotomy between designated values
and values which are not designated and its use in the definition of
entailment plays a crucial role in the Suszko Reduction. Nevertheless,
while it seems quite natural to construe the set of designated values
as a generalization of the classical truth value \(T\) in some of its
significant roles, it would not always be adequate to interpret the
set of non-designated values as a generalization of the classical
truth value \(F\). 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's logic, where sentences can be neither true nor
false).



In the literature on many-valued logic it is sometimes proposed to
consider a set of *antidesignated* values which not
obligatorily constitute the complement of the set of designated values
(see, e.g., Rescher 1969, Gottwald 2001). The set of antidesignated
values can be regarded as representing a generalized concept of
falsity. This distinction leaves room for values that are
*neither* designated *nor* antidesignated and even for
values that are *both* designated *and*
antidesignated.



Grzegorz Malinowski (1990, 1994) takes advantage of this proposal to
give a counterexample to Suszko's Thesis. He defines the notion
of a single-conclusion *quasi*-consequence \((q\)-consequence)
relation. The semantic counterpart of \(q\)-consequence is called
\(q\)-entailment. Single-conclusion \(q\)-entailment is defined by
requiring that if no premise is antidesignated, the conclusion is
designated. 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
\(q\)-consequence relation has a bivalent semantics.



In the supplementary document
 Suszko's Thesis,
 Suszko's reduction is introduced, Malinowski's
counterexample to Suszko's Thesis is outlined, and a short
analysis of these results is presented.



Can one provide evidence for a multiplicity of logical values? More
concretely, \(is\) there more than one logical value, each of which
may be taken to determine its own (independent) entailment relation? A
positive answer to this question emerges from considerations on truth
values as structured entities which, by virtue of their internal
structure, give rise to natural partial orderings on the set of
values.


## 3. Ordering relations between truth-values


### 3.1 The notion of a logical order



As soon as one admits that truth values come with valuation
*systems*, it is quite natural to assume that the elements of
such a system are somehow *interrelated*. And indeed, already
the valuation system for classical logic constitutes a well-known
algebraic structure, namely the two-element Boolean algebra with
\(\cap\) and \(\cup\) as meet and join operators (see the entry on the
 mathematics of Boolean algebra).
 In its turn, this Boolean algebra forms a lattice with a *partial
order* defined by \(a\le\_t b \textrm{ iff } a\cap b = a\). This
lattice may be referred to as *TWO*. It is easy to see that the
elements of *TWO* are ordered as follows: \(F\le\_t T\). This
ordering is sometimes called the *truth order* (as indicated by
the corresponding subscript), for intuitively it expresses an increase
in truth: \(F\) is "less true" than \(T\). It can be
schematically presented by means of a so-called Hasse-diagram as in
Figure 1.



![[a horizontal line segment with the left endpoint labeled 'F' and the right endpoint labeled 'T', below an arrow goes from left to right with the arrowhead labeled 't'.]](Figure1.png)

Figure 1: Lattice *TWO*




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



![[The same as figure 1 except the line segment has a point near the middle labeled 'I'.]](Figure2.png)

Figure 2: Lattice *THREE*




Here \(\le\_t\) orders \(T, I\) and \(F\) so that the intermediate
value \(I\) is "more true" than \(F\), but "less
true" than \(T\).



The relation \(\le\_t\) is also called a *logical order*,
because it can be used to determine key logical notions: logical
connectives and an entailment relation. Namely, if the elements of the
given valuation system \(\mathbf{V}\) form a lattice, then the
operations of meet and join with respect to \(\le\_t\) are usually seen
as the functions for conjunction and disjunction, whereas negation can
be represented by the inversion of this order. Moreover, one can
consider an entailment relation for \(\mathbf{V}\) as expressing
agreement with the truth order, that is, the conclusion should be at
least as true as the premises taken together:




\[
\tag{8}
 D\vDash B\textrm{ iff }\forall v\_a[\Pi\_t\{ v\_a (A) \mid A \in D\} \le\_t v\_a (B)],
\]



where \(\Pi\_t\) is the lattice meet in the corresponding lattice.



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



![[a graph with the y axis labeled 'i' and the x axis labeled 't'. A square with the corners labeled 'B' (top), 'T' (right), 'N' (bottom), and 'F' (left).]](Figure3.png)

Figure 3: The bilattice
*FOUR*\(\_2\)




This bilattice is equipped with *two* partial orderings; in
addition to a truth order, there is an information order \((\le\_i )\)
which is said to order the values under consideration according to the
information they give concerning a formula to which they are assigned.
Lattice meet and join with respect to \(\le\_t\) coincide with the
functions \(f\_{\wedge}\) and \(f\_{\vee}\) in the Belnap matrix
\(\mathbf{B}\_4\), \(f\_{{\sim}}\) turns out to be the truth order
inversion, and an entailment relation, which happens to coincide with
the matrix entailment, is defined by
 (8).
 *FOUR*\(\_2\) arises as a combination of two structures: the
approximation lattice \(A\_4\) and the logical lattice \(L\_4\) which
are discussed in Belnap 1977a and 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 "distinctions of
parts within truth values". Although he immediately specifies
that the word 'part' is used here "in a special
sense", the basic idea seems nevertheless to be that truth
values are not something amorphous, but possess some inner structure.
It is not quite clear how serious Frege is about this view, but it
seems to suggest that truth values may well be interpreted as complex,
structured entities that can be divided into parts.



There exist several approaches to semantic constructions where truth
values are represented as being made up from some primitive
components. For example, in some explications of Kripke models for
intuitionistic logic propositions (identified with sets of
"worlds" in a model structure) can be understood as truth
values of a certain kind. Then the empty proposition is interpreted as
the value *false*, and the maximal proposition (the set of all
worlds in a structure) as the value *true*. Moreover, one can
consider non-empty subsets of the maximal proposition as intermediate
truth values. Clearly, the intuitionistic truth values so conceived
are composed from some simpler elements and as such they turn out to
be complex entities.



Another prominent example of structured truth values are the
"truth-value objects" in topos models from category theory
(see the entry on
 category theory).
 For any topos \(C\) and for a \(C\)-object O one can define a
truth value of \(C\) as an arrow \(1 \rightarrow O\) ("a
subobject classifier for \(C\)"), where 1 is a terminal object
in \(C\) (cf. Goldblatt 2006: 81, 94). The set of truth values so
defined plays a special role in the logical structure of \(C\), since
arrows of the form \(1 \rightarrow O\) determine 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 "factor
semantics" for many-valued logic, where truth values are defined
as 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 exactly
3 occurrences of \(T\). Here the classical values \(T\) and \(F\) are
used as "building blocks" for non-classical truth
values.



Moreover, the idea of truth values as compound entities nicely
conforms with the modeling of truth values considered above in
three-valued (Kleene, Priest) and four-valued (Belnap) logics as
certain subsets of the set of classical truth values. The latter
approach stems essentially from Dunn (1976), where a generalization of
the notion of a classical truth-value function has been proposed to
obtain so-called "underdetermined" and
"overdetermined" valuations. Namely, Dunn considers a
valuation 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 the concept of a
*generalized truth value function*, which is a function from
sentences into the *subsets* of some *basic set of truth
values* (see Shramko and Wansing 2005). The values of generalized
truth value functions can be called *generalized truth
values*.



By employing the idea of generalized truth value functions, one can
obtain a hierarchy of valuation systems starting with a certain
set-theoretic representation of the valuation system for classical
logic. The representation in question is built on a single initial
value which serves then as the designated value of the resulting
valuation system. More specifically, consider the singleton
\(\{\varnothing \}\) taken as the basic set subject to a further
generalization procedure. Set-theoretically the basic set can serve as
the universal set (the complement of the empty set) for the valuation
system \(\mathbf{V}^{\varnothing}\_{cl}\) introduced below. At the
first stage \(\varnothing\) comes out with no specific intuitive
interpretation, it is only important to take it as some
distinct *unit*. Consider then the power-set of \(\{\varnothing
\}\) consisting of exactly two elements: \(\{\{\varnothing \},
\varnothing \}\). Now, these elements can be interpreted as
Frege's *the True* and *the False*, and thus it is
possible to construct a 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 as
follows (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 any
assignment \(a\) relative to \(\mathbf{V}^{\varnothing}\_{cl}\), and
for 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 of
classical 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 fully
justifiable to associate \(\varnothing\) with falsity, taking into
account the virtual *monism of truth* characteristic for
classical logic, which treats falsity not as an independent entity but
merely as the absence of truth.



Then, by taking the set \(\mathbf{2} = \{F, T\}\) of these classical
values as the basic set for the next valuation system, one obtains the
four truth values of Belnap's logic as the power-set of the set
of 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 generalization
of classical logic with its two Fregean truth values. In
Belnap's logic truth and falsity are considered to be
full-fledged, self-sufficient entities, and therefore \(\varnothing\)
is now to be interpreted not as falsity, but as a real truth-value gap
(*neither* true *nor* false). The dissimilarity of
Belnap's truth and falsity from their classical analogues is
naturally expressed by passing from the corresponding classical values
to their singleton-sets, indicating thus their new interpretations as
*false only* and *true only*. Belnap's
interpretation of the four truth values has been critically discussed
in Lewis 1982 and Dubois 2008 (see also the reply to Dubois in Wansing
and Belnap 2010).



Generalized truth values have a strong intuitive background,
especially as a tool for the rational explication of incomplete and
inconsistent information states. In particular, Belnap's
heuristic interpretation of truth values as information that
"has been told to a computer" (see Belnap 1977a,b; also
reproduced in Anderson, Belnap and Dunn 1992, SS81) has been
widely acknowledged. As Belnap points out, a computer may receive data
from *various* (maybe independent) sources. Belnap's
computers have to take into account various kinds of information
concerning 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 are
possible: (3) nothing is told about the sentence or (4) the sources
supply inconsistent information, information that the sentence is true
and information that it is false. And the four truth values from
\(\mathbf{B}\_4\) naturally correspond to these four situations: there
is no information that the sentence is false and no information that
it is true \((\mathbf{N})\), there is *merely* information that
the sentence is false \((\mathbf{F})\), there is *merely*
information that the sentence is true \((\mathbf{T})\), and there is
information that the sentence is false, but there is also information
that it is true \((\mathbf{B})\).



Joseph Camp in 2002: 125-160 provides Belnap's four values
with quite a different intuitive motivation by developing what he
calls a "semantics of confused thought". Consider a
rational 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 assert
some statement, saying, for instance, that \(C\) has some property.
How should one evaluate this statement if \(a\) has the property in
question whereas \(b\) lacks it? Camp argues against ascribing truth
values to such statements and puts forward an "epistemic
semantics" in terms of "profitability" and
"costliness" as suitable characterizations of sentences. A
sentence \(S\) is said to be "profitable" if one would
profit from acting on the belief that \(S\), and it is said to be
"costly" if acting on the belief that \(S\) would generate
costs, for example as measured by failure to achieve an intended goal.
If our "confused agent" asks some external observers
whether \(C\) has the discussed property, the following four answers
are 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 with
**Y&N**). Note that the external observers, who
provide answers, are "non-confused" and have different
objects in mind as to the referent of '\(C\)', in view of
all the facts that may be relevant here. Camp conceives these four
possible answers concerning epistemic properties of sentences as a
kind 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 value
**Y&N** is both an indicator of profitability and an
indicator of costliness. A strict analogy between this
"semantics of confused reasoning" and Belnap's four
valued logic is straightforward. And indeed, as Camp (2002: 157)
observes, the set of implications valid according to his semantics is
exactly the set of implications of the entailment system \(E\_{fde}\).
In Zaitsev and Shramko 2013 it is demonstrated how ontological and
epistemic aspects of truth values can be combined within a joint
semantical framework. Kapsner (2019) extends Belnap's framework
by two additional values "Contestedly-True" and
"Contestedly-False" which allows for new outcomes for
disjunctions and conjunctions between statements with values
\(\mathbf{B}\) and \(\mathbf{N}\).



The conception of generalized truth values has its purely logical
import as well. If one continues the construction and applies the idea
of generalized truth value functions to Belnap's four truth
values, then one obtains further valuation systems which can be
represented by various *multilattices*. One arrives, in
particular, at *SIXTEEN*\(\_3\) - the *trilattice*
of 16 truth-values, which can be viewed as a basis for a logic of
computer networks (see Shramko and Wansing 2005, 2006; Kamide and
Wansing 2009; Odintsov 2009; Wansing 2010; Odintsov and Wansing 2015;
cf. also Shramko, Dunn, Takenaka 2001). The notion of a multilattice
and *SIXTEEN*\(\_3\) are discussed further in the supplementary
document
 Generalized truth values and multilattices.
 A comprehensive study of the conception of generalized logical values
can be found in Shramko and Wansing 2011.


## 4. Concluding remarks



Gottlob Frege's notion of a truth value has become part of the
standard philosophical and logical terminology. The notion of a truth
value is an indispensable instrument of realistic, model-theoretic
approaches to semantics. Indeed, truth values play an essential role
in applications of model-theoretic semantics in areas such as, for
example, knowledge representation and theorem proving based on
semantic tableaux, which could not be treated in the present entry.
Moreover, considerations on truth values give rise to deep ontological
questions concerning their own nature, the feasibility of fact
ontologies, and the role of truth values in such ontological theories.
There also exist well-motivated theories of generalized truth values
that lead far beyond Frege's classical values *the True*
and *the False*. (For various directions of further logical and
philosophical investigations in the area of truth values see Shramko &
Wansing 2009b, 2009c.)