The problem of arti cial precision demonstrates the inadequacy of nave fuzzy semantics for vagueness. This problem is, nevertheless, satisfactorily remedied by fuzzy plurivaluationism; i.e., by taking a class of fuzzy models (a fuzzy plurivaluation), instead of a single fuzzy model, for the semantics of a vague concept. Such a fuzzy plurivaluation in turn represents the class of models of a formal theory, preferably formulated in rst- or higher-order fuzzy logic, which formalizes the meaning postulates of the vague concepts involved. The consequence relation of formal fuzzy logic then corresponds to the (super)truth of propositions involving these vague concepts. An adequate formal treatment of vague propositions by means of fuzzy logic thus consists in derivations in the formal calculus of a suitable fuzzy logic, while the particular truth degrees found in engineering applications actually pertain to arti cially precisi ed (so no longer vague) gradual notions.
? This paper is an elaboration of a part of the short comment [1]. from L accounts for the graduality of vague notions and thereby solves the sorites paradox for gradable predicates in formal semantics. From the practical point of view, the graduality of L provides a feedback mechanism in fuzzy control, which is responsible for its successful applicability.
Nevertheless, fuzzy set theory and fuzzy logic have been criticized by philosophers and linguists as an inadequate model of vagueness (e.g., [11, 20, 12]). Perhaps the most convincing argument against the fuzzy semantics of vagueness is the problem of arti cial precision. As discussed below in Section 2, due to this problem, nave fuzzy models are ruled out as an adequate semantics of vague predicates. Is then the claim of fuzzy logic's suitability for dealing with vagueness to be revoked, or can the claim be salvaged through some re nement? In this paper I argue for the latter, and sketch the way in which fuzzy logic can be viewed as a viable apparatus for dealing with vagueness, albeit in a di erent manner than employed in the prevailing engineering applications.
In particular, as has been demonstrated in detail by Smith [19], fuzzy plurivaluationism, which takes a set of fuzzy models (rather than a single fuzzy model) for the fuzzy semantics of a vague predicate, provides a satisfactory solution to the problem of arti cial precision; and since most philosophical objections against the fuzzy semantics of vagueness turn out to be reducible to some aspect of this central problem, it actually answers a large portion of the criticisms of the degree-theoretical approach to vagueness. Here I will show that fuzzy plurivaluations can be regarded as the classes of models of theories over fuzzy logic, and consequently that the consequence relation of fuzzy logic captures the notion of truth for vague concepts. An adequate knowledge representation of vagueness should therefore be based on deductions in formal systems of mathematical fuzzy logic, rather than calculations of particular membership degrees. On that account, even if traditional fuzzy methods work well in fuzzy control, strictly speaking they only work with technical precisi cations of vague notions rather than vague notions themselves|accommodating the graduality, but neglecting the indeterminacy aspect of vagueness.
These theses will be explained and illustrated in the following sections on the example of the vague predicate tall. Due to space restrictions, only a sketch of the full argument is given here. 2
In the traditional fuzzy approach to vagueness, the semantics of a given set L = fPi(ki)gi2I of (possibly vague) predicates Pi of arities ki > 0, for i 2 I, is given by a fuzzy model M = hM; fkPikMgi2I i, where kPikM : M ki ! L is a fuzzy set (if ki = 1) or a ki-ary fuzzy relation (if ki > 1) valued in a suitable xed structure L of degrees (usually on the real unit interval [0; 1]). The function kPikM is called the membership function of the predicate Pi, and the value kPikM(x) for x 2 M is called the membership degree of the individual x in the fuzzy set kPikM (and similarly for fuzzy relations).
In a fuzzy model, vague predicates are represented by particular membership functions; particular individuals are thus assigned particular degrees of vague properties. For instance, if we choose the membership function T~ : h 7! min(1; max(0; (h 150)=50)), assigning the degree T~(h) 2 [0; 1] to each height h 2 R+ measured in centimeters, to represent the semantics of the predicate tall, then a man of height 186 cm is alleged to be tall to degree 0.72.
However, it can be observed that the choice of the membership function T~ has been completely arbitrary: there is no convincing reason why this particular linear function (rather than, e.g., some S-shaped function) should have been chosen, or why the man of height 186 cm should be assigned the degree 0.72 rather than, e.g., 0.74. Instead of capturing a vague property, for which it is indeterminate whether it applies to borderline cases or not, a fuzzy model yields something much more precise: a particular real number, such as 0.3428. In view of the arbitrariness of the choice of membership function, this precision, not present in the original vague predicate, is arti cial. Consequently, the fuzzy model does not capture the vagueness of the predicate in a satisfactory manner. This is the problem of arti cial precision of fuzzy semantics of vagueness, and a serious objection to its adequacy.
In engineering applications, the choice of a particular membership function is guided by pragmatic considerations: e.g., a linear function may be preferred for the reason of its e cient computability. The choice is legitimate i the intended real-world application works well. From the point of view of formal semantics, however, the fuzzy model is not an adequate semantic representation of the original vague notion, but is rather its technical precisi cation, suitable for the particular engineering application in question. If we want to know which propositions are true about the original vague notion (e.g., whether all tall people from a given sample are skinny ), the use of the technical precisi cation may well yield an answer which is an artifact of the choice of the membership function. Thus, while fuzzy models are admittedly useful for applied methods in fuzzy control, they are hardly an accurate representation of vague knowledge. Eventually, the meaning of a vague predicate is determined by its usage in natural language; however, one can hardly nd any linguistic fact that would determine whether a man of height 186 cm should be considered 0.72- or rather 0.73-tall. 3
A remedy to the problem of arti cial precision is o ered by fuzzy plurivaluationism [19]. Fuzzy plurivaluationism acknowledges that the meaning-determining facts of natural language do not uniquely determine the values of membership functions, and so do not narrow the set of admissible fuzzy models for a given predicate down to a singleton set. Therefore, instead of a single fuzzy model, fuzzy plurivaluationism assigns a set P of fuzzy models (as de ned in Section 2) as the semantics for a given list L of (possibly vague) predicates; let us call the set P of fuzzy models a fuzzy plurivaluation for the language L.
The set of admissible fuzzy models for a given language L does not consist of all fuzzy models: rather, it has to re ect the meanings of the predicates in L. Even though the meaning-determining facts do not pinpoint the exact values of membership degrees, they do determine certain properties of membership functions and their relationships. For instance, the membership function of the predicate tall must in each admissible model be non-decreasing: it is part of the meaning of the word tall that people of larger heights are not less tall than those of lesser heights; those who do not acknowledge this property do not understand the meaning of the word tall. These constraints on admissible fuzzy models can be regarded as the meaning postulates of the vague predicates involved, i.e., their semantic properties and relationships that would be acknowledged by competent speakers, and which therefore have to be satis ed in each fuzzy model. In other words, fuzzy plurivaluatistic semantics of a given vague language L is the set P of those fuzzy models for the language L that satisfy the meaning postulates of the predicates in L.1
In a formal semantics of natural language, these meaning postulates are expected to be expressible in the rigorous language of mathematics and logic, thereby comprising a formal theory T that constrains the set P of admissible models. In typical cases, this is indeed so; let us illustrate it on the example of the predicate tall. 4
Let us present a (simpli ed) analysis of the meaning of the vague predicate tall. One condition that the intended usage of the term tall (in any given context) has to satisfy is the one mentioned earlier:
x,
then a person of height y is tall as well. (
In the degree-theoretical framework of fuzzy plurivaluationism, these
meaning postulates for the predicate tall can be reformulated as conditions on its
membership function. Let the membership function of the vague unary
predicate tall, assigning degrees from [0; 1] to heights in centimeters, be denoted by
1 Since typical meaning postulates of vague concepts do not refer to truth degrees
(which are just part of our semantic modeling rather than linguistic usage), it is
no wonder that the membership functions are underdetermined by the meaning
postulates and that P is usually a non-singleton class of models.
T~ : R+ ! [0; 1]. The meaning postulates (
T~(x))
(
The fuzzy plurivaluation representing the meaning of the predicate tall
according to our analysis would therefore be the set P of all fuzzy models in which
the membership function T~ of the predicate tall satis es the constraints (
The conditions (
Membership functions of predicates from L in a particular model M 2 P are (fuzzy) precisi cations of the vague concepts from L. The condition that M j= T for all M 2 P ensures that the meaning postulates of all vague concepts Pi 2 L are jointly satis ed by their fuzzy precisi cations in each model M 2 P. 5
As mentioned in Section 2, a proposition ' about the predicates in the language
L that is true in only some, but not all, of the models M 2 P, can be an
artifact of the choice of membership functions in these particular models. Only such
propositions that are true in all models M 2 P, i.e., in all models satisfying
the meaning postulates of the predicates in L, are true independently of
particular choices of membership functions in these models, and so can be regarded
as truths about the vague concepts from L. Since fuzzy plurivaluationism is a
2 In reality, the meaning postulates for the predicate tall are more complex than the
simpli ed version (
Thus, those and only those propositions ' which are supertrue represent truths about the vague concepts from L regardless of a chosen precisi cation. Like in supervaluationism, it is only supertruth that matters, rather than truth in a particular model.
Mathematically, ' is supertrue i M j= ' for all models M 2 P, i.e., for all models M such that M j= T ; that is, i T j= '. Thus, supertruths about vague predicates are exactly the logical consequences of their meaning postulates.4 If T is formulated in classical rst-order logic, then due to the strong completeness theorem, supertruths are also exactly those ' that can be formally derived from T , i.e., ' is supertrue i T ` '.
It can be observed, however, that the formalization of the meaning postulates
in rst-order classical logic is not completely straightforward. In particular, vague
predicates from L are represented by functions, which is a type mismatch that has
to be handled by the formalization: consequently, logical connectives occurring
in the meaning postulates have to be represented by certain operations on truth
degrees. For instance, in our toy example of Section 4, compare the meaning
postulate (
Fuzzy logics form a well-developed family of many-valued and substructural logics [3]. For a detailed information on mathematical fuzzy logic we refer the reader to [2]. Here we shall just brie y introduce the perhaps best known system of formal fuzzy logic, namely (in nite-valued) Lukasiewicz logic.
The syntax of Lukasiewicz rst-order logic is the same as that of classical rst-order logic. Let us take : and ! for the primitive connectives. The standard semantics of Lukasiewicz logic employs the real unit interval [0; 1] as the set of truth degrees. Let L be a non-empty set of predicate symbols and individual constants.5 The value ktkM;v of a term (i.e., a variable or constant) t under 4 Let us note that the meaning postulates correspond to what in supervaluationism is called the penumbral connections; see, e.g., [4, 20, 13, 19]. 5 Further on we expand the de nition of fuzzy models (see Section 2) by interpretations kckM 2 M of individual constants c 2 L. Function symbols can be added to Lukasiewicz logic as well (see [2]); we leave them aside here, as they are not needed in the present paper. a valuation v of individual variables in a fuzzy model M for L is the element kckM 2 M if t is a constant c 2 L or the element v(x) 2 M if t is a variable x. The truth degree k'kM;v of a formula ' in the language L under a valuation v in a fuzzy model M for L is de ned by the following recursive Tarski conditions: kP (t1; : : : ; tn)kM;v = kP kM(kt1kM;v; : : : ; ktnkM;v) k:'kM;v = 1
k'kM;v k' ! kM;v = min(1
k'kM;v + k kM;v; 1) k(8x)'kM;v = infa2M k'kM;vx:a k(9x)'kM;v = supa2M k'kM;vx:a ; where vx:a(x) = a and vx:a(y) = v(y) for each variable y di erent from x. Further propositional connectives of Lukasiewicz logic are de ned as follows:6 Consequently we obtain k'& kM;v = max(k'kM;v+k kM;v 1; 0), and similarly for other de ned connectives.
We say that a formula ' is true in M, written M j= ', if k'kM;v = 1 for all valuations v in M. A fuzzy model M is a model of a theory (i.e., a set of formulae) T , written M j= T , if M j= ' for each ' 2 T . Let the class fM j M j= T g of models of T be denoted by Mod(T ). A formula ' is a logical consequence of T , written T j= ', if M j= ' for all M 2 Mod(T ).
The axiomatic system for rst-order Lukasiewicz logic consists of the following axiom schemata, where the term t is substitutable for x in ' and does not contain free x: ) ! (( ! ) ! (' !
)) (' ! ' ! (
! ') (: ! :') ! (' ! ((' ! ) ! ) ! (( (8x)'(x) ! '(t) (8x)( ! ') ! ( ) ! ') ! ') ! (8x)'); plus the derivation rules of modus ponens (from ' and ' ! derive ) and generalization (from ' derive (8x)'). The notions of proof and provability in a theory (written T ` ') are de ned as in classical logic. 6 Usually (see [2]), an additional conjunctive connective ' ^ df ' & (' ! ), an additional disjunctive connective ' df :(' ! : ), and optionally, the unary connective 4' (\determinately '", with the standard semantics k4'kM;v = 1 if k'kM;v = 1, otherwise k4'kM;v = 0) are introduced in Lukasiewicz logic. However, we will not need these connectives in the present exposition.
The completeness of the above axiomatic system with respect to Lukasiewicz rst-order logic is somewhat tricky. To obtain the completeness theorem (T j= ' i T ` ') for the standard [0; 1]-valued semantics described above, an additional in nitary derivation rule has to be added [10]. Nevertheless, the axiomatic system is sound and complete if more general algebras of truth degrees (besides the standard algebra on [0; 1]) are admitted in fuzzy models.7
A similar situation is found in other systems of formal fuzzy logic, which di er
from Lukasiewicz logic mainly in the choice of algebraic operations in the Tarski
conditions (
Let us return to our toy example of Section 4. The straightforward formalization
of the meaning postulates (
T x & (y
: T a
T b:
x) ! T y
(
The semantics of Lukasiewicz fuzzy logic thus captures the plurivaluationistic
meaning of tall if its meaning postulates (
A similar situation is found in the plurivaluationistic semantics for other vague predicates besides our example of tall: typically, the class of admissible fuzzy models P can be de ned as the class Mod(T ) of models of a theory T in Lukasiewicz (or another suitable) rst-order fuzzy logic, where T is a straightforward formalization of the meaning postulates in this fuzzy logic. This is, of course, no coincidence: Lukasiewicz and other fuzzy logics have been designed to straightforwardly formalize typical (i.e., relational and quanti cational) conditions between truth degrees of gradual notions. 8
The utility of formal fuzzy logic in fuzzy plurivaluationistic semantic of vagueness goes beyond the delimitation of fuzzy plurivaluations by means of theories over fuzzy logic. Recall from Section 5 that genuine truths about vague predicates (i.e., truths which are not artifacts of particular choices of membership functions) are those propositions which are true in all admissible fuzzy models for the vague language L; i.e., those propositions which are supertrue in the fuzzy plurivaluation P = Mod(T ), where T is the theory over fuzzy logic formalizing the meaning postulates for L. Now observe that by de nition, a formula ' of Lukasiewicz fuzzy logic in the language L is true in all models M 2 P = Mod(T ) i T j= ' in Lukasiewicz fuzzy logic; in other words, if ' is entailed by the meaning postulates of L in Lukasiewicz logic (or another fuzzy logic chosen for formalization of the meaning postulates for L).
Supertruths about vague predicates thus coincide with fuzzy-logical consequences of their meaning postulates. The completeness theorems for fuzzy logics further translate10 the latter into the provability from T in fuzzy logic. Supertruths thus can be found by axiomatic derivation in fuzzy logic.
This is in fact the way which formal fuzzy logicians (unlike most designers of applied fuzzy methods, who tend to employ single fuzzy models) have implicitly used for more than a decade in modeling vagueness: cf., e.g., the way in which the vague predicate small is treated in [9]|namely by rst axiomatizing its basic properties and, subsequently, deriving theorems from these axioms in formal fuzzy logic. The same approach has been employed in the formal modeling of vague linguistic hedges [7, 16], vague evaluative linguistic expressions [16], and vague quanti ers [17]. 10 Modulo the provisos of Section 6 on the preconditions of the completeness theorems|namely, adding in nitary derivation rules or admitting more general systems of truth degrees than [0; 1]|for some fuzzy logics (including Lukasiewicz logic; for the fundamental fuzzy logic MTL mentioned below, however, these preconditions are not needed).
On top of that, mathematical fuzzy logic applies the same fuzzy
plurivaluationistic approach to propositional connectives, too. In Section 6 the reader might
rightly have wondered what justi ed the particular truth functions in the Tarski
conditions (
Indeed it is. The standard account of Lukasiewicz logic is actually somewhat simpli ed: mathematical fuzzy logic in fact takes an axiomatic approach even to propositional connectives, constraining them only by certain properties which can be viewed as embodying their meaning postulates. The most common approach (originating in [6]) posits that conjunction is commutative, associative, monotone, neutral w.r.t. determinate truth, and continuous in degrees. These constraints represent the meaning postulates on and, namely that ' and is as true as and ', etc. Similar constraints are imposed on other connectives and their mutual interplay. Within these constraints, the truth functions of connectives are allowed to vary across admissible models, analogously to the indeterminacy of membership functions in fuzzy plurivaluations|precisely because their (accepted, reasonable) meaning postulates fail to narrow them down to single truth functions.
Particular systems of fuzzy logic (of which there are many|see [2]) di er from each other by making di erent choices regarding the requisite properties of propositional connectives, i.e., their stipulated meaning postulates. E.g., the above constraints on conjunction lead to the fuzzy logic BL, while slightly weaker requirements produce the (arguably more fundamental) fuzzy logic MTL. In these logics, the set of admissible truth functions for conjunction is only narrowed down to the (still rather broad) class of so-called (left-)continuous t-norms.
In some contexts, additional constraints on connectives are appropriate,
leading to stronger fuzzy logics: for instance, adding the law of double negation
(which formalizes a meaning postulate for not, namely: \not not ' is as true
as '") to BL yields Lukasiewicz logic. Some of such more specialized fuzzy logic
(including Lukasiewicz) can be shown to be sound and complete with respect
to fuzzy models with a xed standard choice of truth functions for connectives;
e.g., in Lukasiewicz logic, the functions 1 x for negation and min(1 x + y; 1)
for implication. In such logics, even if the constraints on connectives admit truth
functions other than the single standard ones, fuzzy models with non-standard
truth functions happen to be irrelevant for the consequence relation. In such
logics, then, logical consequence is fully determined by evaluating by means of
the single standard truth functions, and so we can pretend that the standard
truth functions are the only admissible ones (as we did in (
Fuzzy plurivaluationism de facto characterizes vagueness as a combination of two phenomena: the graduality of vague concepts, by which we are forced to use fuzzy, rather than classical, models (to avoid the sorites paradox), and the indeterminacy of meaning, by which we are forced to use sets of fuzzy models rather than single fuzzy models (to avoid the problem of arti cial precision).
The graduality component of fuzzy plurivaluationism acknowledges that most vague predicates apply to some individuals more than to others; this is modeled by means of truth degrees (the \fuzzy" aspect). The indeterminacy component acknowledges that the meaning-determining facts do not narrow down the meanings of vague concepts to single fuzzy models, and leave the membership functions of vague predicates (as well as the truth functions of fuzzy connectives) underdetermined; this is modeled by letting the membership and truth functions vary across admissible fuzzy models, constrained only by the meaning postulates of the vague concepts involved (the \plurivaluationistic" aspect).
Application-oriented fuzzy methods do re ect the graduality component of vagueness, by using fuzzy models; however, they neglect the semantic indeterminacy of vague predicates and rather work with their fuzzy precisi cations in a single fuzzy model. As a result, they are able to compute particular membership degrees of these gradual precisi cations (and act accordingly to the degree, e.g., in fuzzy control). However, the results can be artifacts of the choice of the particular fuzzy model; their correctness is assessed on pragmatic grounds (namely, whether the application works well or not).
On the other hand, supervaluationism|a rival theory of vagueness to degreetheoretical ones|can be cast as a bivalent version of fuzzy plurivaluationism, representing vague predicates by sets of classical, bivalent models.12 This re ects the indeterminacy aspect of vagueness, but neglects the graduality of most vague notions. The supervaluationistic representation of vagueness is therefore subject 11 The admittance of di erent sets of truth degrees across fuzzy models (cf. footnote 7) is an e ect of the same principle: the system of truth degrees is only determined by the postulates on connectives, rather than given in advance, and the postulates do not narrow it down to algebras with isomorphic universes. In only a few fuzzy logics (such as MTL or Godel logic, see [2]), the real-valued completeness theorem holds, entailing that we can ignore all the non-[0; 1]-valued algebras. 12 Smith [19] makes a subtle distinction between two variants of supervaluationism that occur in the literature, using the terms `supervaluationism' and `plurivaluationism'. The distinction is not important in our context, therefore we use the more established term `supervaluationism' for both variants. to several problems exposing its insu ciency (most prominently, the `jolt problem', instantiated, e.g., by the fact that it is supertrue that there is the least large natural number; see [19]).
Fuzzy plurivaluationism combines both detected aspects of vagueness; and as we have seen in Section 8, the consequence relation of fuzzy logic describes its notion of supertruth, identifying it with fuzzy-logical consequences of the meaning postulates for the vague notions. In this sense, fuzzy logic is the logic underlying the fuzzy plurivaluationistic semantics of vagueness. Consequently, fuzzy logic as the consequence relation can be regarded as the logic of vagueness (fuzzy-plurivaluationistically modeled).
On the other hand, particular semantic models of fuzzy logic (with xed membership functions) are insu cient for capturing vagueness, as exposed by the problem of arti cial precision described in Section 2, as well as various further objections raised by philosophers of vagueness (summarized, e.g., in [20, 12, 19]).13 This answers the question of whether or not, or in which sense, fuzzy logic can be regarded as a logic suitable for the formal semantics of vagueness: the answer is yes for fuzzy logic in the sense of the consequence relation (provided we accept the fuzzy plurivaluationistic model of vagueness), and no for fuzzy logic in the sense of particular fuzzy models.
The practical problem with this answer is that fuzzy plurivaluationistic semantics, or fuzzy logic qua consequence relation, does not make it possible to calculate any truth degrees or membership functions, as they vary across the admissible fuzzy models. Thus it is hardly imaginable that we could construct, for instance, a fuzzy controller based on the fuzzy-logical consequence: for such applications, the method of arti cial precisi cation (retaining graduality to ensure a feedback mechanism for the controlled process, while eliminating indeterminacy in order to be able to calculate values for controlling) is obviously superior. Nevertheless, the more adequate representation of vagueness by fuzzy plurivaluations and consequence-based fuzzy logic may open the way to another kind of applications in which the indeterminacy aspect of vagueness matters, such as in linguistic modeling, knowledge representation, or logical inference (e.g., in answering database queries) under vagueness. Imaginably, for instance, logical programming aimed at accommodating genuinely vague notions (as opposed to the current practice of fuzzy logical programming employing particular truth degrees) might be achieved by replacing the inference rules of classical logic by suitable derivation rules of formal fuzzy logic. An area with practical applications where steps towards employing consequence-based fuzzy logic has already been made (starting with [8]) is fuzzy description logic. However, the applicability of consequence-based formal fuzzy logic to practical problems (rather than just as 13 It is worth noting that most philosophical criticisms of fuzzy logic as the logic of vagueness assume xed membership functions, and thus do not apply to fuzzy plurivaluationism nor fuzzy logic as consequence relation. The fuzzy-plurivaluationistic analysis agrees with these objections against the nave fuzzy semantics of xed membership functions, and explains the problems as stemming from its failure to address the indeterminacy aspect of vagueness. a theoretical background of fuzzy methods with xed membership functions) is still a matter of future research.
Acknowledgments. This work was supported by the European Regional Development Fund in the project \Centre of Excellence IT4Innovations" (CZ.1.05/ 1.1.00/02.0070) and the project \Strengthening research teams at the University of Ostrava" (CZ.1.07/2.3.00/30.0010). Thanks are due to two anonymous referees for detailed comments.