From Fine and Kamp in the 70’s—through Osherson and Smith in the 80’s, Williamson, Kamp and Partee in the 90’s and Keefe in the 00’s—up to Sauerland in the present decade, the objection continues to be run that fuzzy logic based theories of vagueness are incompatible with ordinary usage of compound propositions in the presence of borderline cases. These arguments against fuzzy theories have been rebutted several times but evidently not put to rest. I attempt to do so (...) in this paper. (shrink)

The choice of connectives for many valued propositional logics suitable for theoretical and applicational interests in most cases is an open problem up to now. We will not offer a general solution here, but support the point of view of some recent developments in fuzzy set theory that the triangular norms – t-norms for short – of Schweizer/Sklar [3] and the ϕ-operators of Pedrycz [2] represent quite general classes of connectives at least for many valued logics with truth (...) value set the real interval [0,1] . Special classes of t-norms and general properties of them have been discussed in some papers just recently, e.g. by Dombi [1], Weber [4], Yager [5] to mention only a few of them. (shrink)

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is (...) an involutive negation. However, for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. (shrink)

Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies and satisfiable sentences as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.

It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general).

A new criterion is introduced for judging the suitability of various fuzzy logics for practical uncertain reasoning in a probabilistic world and the relationship of this criterion to several established criteria, and its consequences for truth functional belief, are investigated.

We look at the problem of revising fuzzy belief bases, i.e., belief base revision in which both formulas in the base as well as revision-input formulas can come attached with varying degrees. Working within a very general framework for fuzzy logic which is able to capture certain types of uncertainty calculi as well as truth-functional fuzzy logics, we show how the idea of rational change from “crisp” base revision, as embodied by the idea of partial meet (...) (base) revision, can be faithfully extended to revising fuzzy belief bases. We present and axiomatise an operation of partial meet fuzzy base revision and illustrate how the operation works in several important special instances of the framework. We also axiomatise the related operation of partial meet fuzzy base contraction. (shrink)

The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of a certain length n (...) to the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with k≥1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (Citation2020) as well as enriched with some previous results from Kostrzycka and Zaionc (Citation2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (Citation2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. (shrink)

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based (...) logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. (shrink)

Kamp and Fine presented an influential argument against the use of fuzzy logic for linguistic semantics in 1975. However, the argument assumes that contradictions of the form "A and not A" have semantic value zero. The argument has been recently criticized because sentences of this form are actually not perceived as contradictory by naive speakers. I present new experimental evidence arguing that fuzzy logic still isn't useful for linguistic semantics even if we take such naive speaker judgements at (...) face value. Specifically I show that naive speakers judge "A and not A" in the relevant cases as more true than "B and not A" even when A and B are judged to be equally true. A truth functional semantics such as fuzzy logic cannot account for these intuitions directly. (shrink)

ABSTRACT The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of a certain length (...) n to the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with $ k\ge 1 $ k ≥ 1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (2020) as well as enriched with some previous results from Kostrzycka and Zaionc (2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. (shrink)

This short paper about fuzzy set-based approximate reasoning first emphasizes the three main semantics for fuzzy sets: similarity, preference and uncertainty. The difference between truth-functional many-valued logics of vague or gradual propositions and non fully compositional calculi such as possibilistic logic or similarity logics is stressed. Then, potentials of fuzzy set-based reasoning methods are briefly outlined for various kinds of approximate reasoning: deductive reasoning about flexible constraints, reasoning under uncertainty and inconsistency, hypothetical reasoning, exception-tolerant plausible reasoning (...) using generic knowledge, interpolative reasoning, and abductive reasoning. Open problems are listed in the conclusion. (shrink)

How does metaphysical grounding interact with the truth-functions? I argue that the answer varies according to whether one has a worldly conception or a conceptual conception of grounding. I then put forward a logic of worldly grounding and give it an adequate semantic characterisation.

Quine argued that any context allowing substitution of logical equivalents and coextensive terms is truth functional. We argue that Quine's proof for this claim is flawed.

It is shown that the standard definitions of truth-functionality, though useful for their purposes, ignore some aspects of the usual informal characterisations of truth-functionality. An alternative definition is given that results in a stronger notion which pays attention to those aspects.

Benjamin Schnieder has argued that several traditional definitions of truth-functionality fail to capture a central intuition informal characterizations of the notion often capture. The intuition is that the truth-value of a sentence that employs a truth-functional operator depends upon the truth-values of the sentences upon which the operator operates. Schnieder proposes an alternative definition of truth-functionality that is designed to accommodate this intuition. We argue that one traditional definition of ‘truth-functionality’ is immune from the (...) counterexamples that Schnieder proposes and is preferable to Schnieder’s alternative. (shrink)

In this paper I argue against Michael Morris’ claim, that the Tractatus view involves holding that the possibility of truth-functional combination is prior to the possibility for sentential constituents to combine with one another. I provide an alternative interpretation in which I deny the presence of any distinction in the Tractatus between these two possibilities. I then turn to Adrian Moore’s ‘disjunctivist’ account of sentencehood, itself inspired by the Tractatus view. I argue that Moore’s account need not involve a (...) commitment to the kind of priority Morris describes, and that it need not involve a commitment to transcendental idealism. (shrink)

We consider time as a fuzzy concept. Based on this, the Fuzzy Time-Particle interpretation Of Quantum Mechanics is introduced as an interpretation of Quantum Mechanics [4],[5],[6]. Here, we show how to compute the function associated to Fuzzy time.

Many-valued logics are standardly defined by logical matrices. They are truth-functional. In this paper non truth-functional many-valued semantics are presented, in a philosophical and mathematical perspective.

Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truth-as-correspondence, advocates of truth-as-coherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he (...) adds that each of the contenders also faces a ‘scope problem’, in that none of them makes sense of all legitimate truth attributions. Some deflationists make similar observations – usually less concessive – and conclude that truth simply has no substantial, underlying nature. There is nothing for the philosopher to articulate and defend. Lynch agrees with deflationists that the realm of discourse is too diverse for any single ‘first-level’ property, such as correspondence or coherence, to fully characterize it, but against deflationism, he holds that truth does have a nature. He presents a functionalist theory. The definition is this: x is true if, and only if, x has a property that plays the truth-role. According to Lynch, the truth-role is given in terms of what he calls the ‘core truisms’ of the folk concept of truth. These are platitudes that hold of the everyday notion, and are such that anyone who denies all of them can be accused of changing the subject – of not talking about truth. A given property T plays the truth-role for a batch of propositions, just in case, for any proposition P in the batch: P is T if, and only if, where P is believed, things are as they are believed to be; other things being equal, it is a worthy goal …. (shrink)

Two of the main intuitions that underlie the phenomenon of vagueness are the truth-functional and the penumbral intuitions. After presenting and contrasting them, I will put forward Tappenden's gappy approach to vagueness (which takes into account the truth-functional intuition). I will contrast Tappenden'sview with another of the theories of vagueness that see it as a semantic phenomenon: Supervaluationism (which takes into account the penumbral intuition). Then I will analyze some objections to Tappenden's approach and some objections to Supervaluationism. (...) Finally, I will present my own worries about Tappenden's account. (shrink)

Although it is eommon to attribute to Wittgenstein in the Tractatus a treatment of general propositions as equivalent to eonjunctions and disjunctions of instance propositions, the evidence for this is not perfeetly clear. This article considers Wittgenstein’s comments in 5.521, which can be read as rejecting such a treatment. It argues that properly situating the Tractatus historically allows for a revised reading of 5.521 and other parts of the Tractatus relevant to Wittgenstein’s theory of generality. The result is that 5.521 (...) does not conflict with the view that general propositions are truth-functions of instance propositions. Common problems with such a view are to some extent obviated by the fact that Wittgenstein, following Russell and Moore, was not concerned with a syntactically defined language, but with propositions conceived as independent of a fixed language. (shrink)

Although it is eommon to attribute to Wittgenstein in the Tractatus a treatment of general propositions as equivalent to eonjunctions and disjunctions of instance propositions, the evidence for this is not perfeetly clear. This article considers Wittgenstein’s comments in 5.521, which can be read as rejecting such a treatment. It argues that properly situating the Tractatus historically allows for a revised reading of 5.521 and other parts of the Tractatus relevant to Wittgenstein’s theory of generality. The result is that 5.521 (...) does not conflict with the view that general propositions are truth-functions of instance propositions. Common problems with such a view are to some extent obviated by the fact that Wittgenstein, following Russell and Moore, was not concerned with a syntactically defined language, but with propositions conceived as independent of a fixed language. (shrink)

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This (...) completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established. (shrink)

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the (...) hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883?1884 in connection with the composition of Peirce's ?On the Algebra of Logic: A Contribution to the Philosophy of Notation? that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions (...) of the connectives in question. (shrink)

Peirce's example puts another nail in the coffin of the truth-functionality thesis. Conditionals are not truth-functional.

Motivated by aspects of reasoning in theories of physics, Robin Giles defined a characterization of infinite valued Łukasiewicz logic in terms of a game that combines Lorenzen-style dialogue rules for logical connectives with a scheme for betting on results of dispersive experiments for evaluating atomic propositions. We analyze this game and provide conditions on payoff functions that allow us to extract many-valued truth functions from dialogue rules of a quite general form. Besides finite and infinite valued Łukasiewicz logics, also (...) Meyer and Slaney’s Abelian logic and Cancellative Hoop Logic turn out to be characterizable in this manner. (shrink)

The material interpretation of conditionals is commonly recognized as involving some paradoxical results. I here argue that the truth functional approach to natural language is the reason for the inadequacy of this material interpretation, since the truth or falsity of some pair of statements ‘p’ and ‘q’ cannot per se be decisive for the truth or falsity of a conditional relation ‘if p then q’. This inadequacy also affects the ability of the overall formal system to establish (...) whether or not arguments involving conditionals are valid. I also demonstrate that the Paradox of Indicative Conditionals does not actually involve a paradox, but instead contains some paralogistic elements that make it appear to be a paradox. The discussion of the paradox in this paper further reveals that the material interpretation of conditionals adversely affects the treatment of disjunctions. -/- Much has been said about these matters in the literature that point in the same direction. However, there seems to be some reluctance against fully complying with the arguments against the truth functional account of conditionals, since many of the alternative accounts rely on the material conditional, or at least on an understanding of the conditional as a function of antecedent and consequent in a similar sense as the material conditional. My argument against truth functionality indicates that it may in general involve similar problems to treat conditionals as such functions, whether one deals with theories of truth, assertability or probability. (shrink)