Ten questions and one problem on fuzzy logic

Recently, I had a very interesting friendly e-mail discussion with Professor Parikh on vagueness and fuzzy logic. Parikh published several papers concerning the notion of vagueness. They contain critical remarks on fuzzy logic and its ability to formalize reasoning under vagueness [10,11]. On the other hand, for some years I have tried to advocate fuzzy logic (in the narrow sense, as Zadeh says, i.e. as formal logical systems formalizing reasoning under vagueness) and in particular, to show that such systems (of many-valued logic of a certain kind) offer a fully fledged and extremely interesting logic [4, 5]. But this leaves open the question of intuitive adequacy of many-valued logic as a logic of vagueness. Below I shall try to isolate eight questions Parikh asks, add two more and to comment on all of them. Finally, I formulate a problem on truth (in)definability in Łukasiewicz logic which shows, in my opinion, that fuzzy logic is not just “applied logic” but rather belongs to systems commonly called “philosophical logic” like modal logics, etc.     

Logic and Natural Selection

Is logic, feasibly, a product of natural selection? In this paper we treat this question as dependent upon the prior question of where logic is founded. After excluding other possibilities, we conclude that logic resides in our language, in the shape of inferential rules governing the logical vocabulary of the language. This means that knowledge of (the laws of) logic is inseparable from the possession of the logical constants they govern. In this sense, logic may be seen as a product of natural selection: the emergence of logic requires the development of creatures who can wield structured languages of a specific complexity, and who are capable of putting the languages to use within specific discursive practices.     

The Operators of Vector Logic

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type I and Type II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. Mathematics Subject Classification: 03B05, 03B50.     

The Individuation of Causal Powers by Events (and Consequences of the Approach)

In this paper, I explore the notion of a “causal power,” particularly as it is relevant to a theory of properties whereby properties are individuated by the causal powers they bestow on the objects that instantiate them. I take as my target certain eliminativist positions that argue that certain kinds of properties (or relations) do not exist because they fail to bestow unique causal powers on objects. In reply, I argue that the notion of causal powers is inextricably bound up with our notion of what an event is, and not only is there disagreement as to which theory of events is appropriate, but on the three prevailing theories, it can be shown that the eliminativists arguments do not follow.     

Implicational (semilinear) logics I: a new hierarchy

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field.     

Abstract Logical Constants

A possibility of defining logical constants within abstract logical frameworks is discussed, in relation to abstract definition of logical consequence. We propose using duals as a general method of applying the idea of invariance under replacement as a criterion for logicality.     

In Defense of Logical Universalism: Taking Issue with Jean van Heijenoort

Van Heijenoort??s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schr?der, and L?wenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2).     

The Concept of Relevance and the Logic Diagram Tradition

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains.     

From finitary to infinitary second‐order logic

A back and forth condition on interpretations for those second‐order languages without functional variables whose non‐logical vocabulary is finite and excludes functional constants is presented. It is shown that this condition is necessary and sufficient for the interpretations to be equivalent in the language. When applied to second‐order languages with an infinite non‐logical vocabulary, excluding functional constants, the back and forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second‐order languages whose non‐logical vocabulary is infinite for which the back and forth condition is both necessary and sufficient. It is also shown that some applications of the back and forth construction for second‐order languages can be extended to the infinitary second‐order languages. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hale on the Absoluteness of Logical Necessity

Hale (Metaphysics, 10, 93–117, 1996, 2013) has argued that logical necessities are absolute in the sense that there is no competing kind of modality under which they may be false. In this paper, I argue that there are competing kinds of modality, which I call “essentialist modalities,” under which logical necessities may be false. Since it is counter-intuitive to say that logical necessities are not absolute, my argument, if correct, shows that Hale’s characterization of absolute necessity does not adequately capture the intuitive notion of absolute necessity. Then, a qualified version of Hale’s characterization of absolute necessity is proposed. On the qualified version, the absoluteness of logical necessities is no longer defeated by essentialist possibilities.     

Logics that define their own semantics

The capability of logical systems to express their own satisfaction relation is a key issue in mathematical logic. Our notion of self definability is based on encodings of pairs of the type (structure, formula) into single structures wherein the two components can be clearly distinguished. Hence, the ambiguity between structures and formulas, forming the basis for many classical results, is avoided. We restrict ourselves to countable, regular, logics over finite vocabularies. Our main theorem states that self definability, in this framework, is equivalent to the existence of complete problems under quantifier free reductions. Whereas this holds true for arbitrary structures, we focus on examples from Finite Model Theory. Here, the theorem sheds a new light on nesting hierarchies for certain generalized quantifiers. They can be interpreted as failure of self definability in the according extensions of first order logic. As a further application we study the possibility of the existence of recursive logics for PTIME. We restate a result of Dawar concluding from recursive logics to complete problems. We show that for the model checking Turing machines associated with a recursive logic, it makes no difference whether or not they may use built in clocks. Received: 7 February 1997     

Why the Logical Hexagon?

The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the ??logical square?? (or ??square of opposition??), of which it is a much more general and powerful ??relative??. The discovery of the former did not raise interest, neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, ??sentenced to death?? by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with ??oppositions??) has appeared, ??oppositional geometry?? (also called ??n-opposition theory??, ??NOT??), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of ??logical bi-simplexes of dimension m??, itself just one term of the more general infinite series (of series) of the ??logical poly-simplexes of dimension m??. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of ??hybrid logical hexagon??, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change.     

Towards a new theory of confirmation

Any sequence of events can be “explained” by any of an infinite number of hypotheses. Popper describes the “logic of discovery” as a process of choosing from a hierarchy of hypotheses the first hypothesis which is not at variance with the observed facts. Blum and Blum formalized these hierarchies of hypotheses as hierarchies of infinite binary sequences and imposed on them certain decidability conditions. In this paper we also consider hierarchies of infinite binary sequences but we impose only the most elementary Bayesian considerations. We use the structure of such hierarchies to define “confirmation”. We then suggest a definition of probability based on the amount of confirmation a particular hypothesis (i.e. pattern) has received. We show that hypothesis confirmation alone is a sound basis for determining probabilities and in particular that Carnap’s logical and empirical criteria for determining probabilities are consequences of the confirmation criterion in appropriate limiting cases.     

Mixed logic and storage operators

In 1990 J-L. Krivine introduced the notion of storage operators. They are -terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF2 type system for storage operators using G?del translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF2 to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. Received: 7 May 1997     

Expressing and Describing Experiences. A Case of Showing Versus Saying

Experiences are interpreted as conscious mental occurrences that are of phenomenal character. There is already a kind of (weak) intentionality involved with this phenomenal interpretation. A stricter conception of experiences distinguishes between purely phenomenal experiences and intentional experiences in a narrow sense. Wittgenstein’s account of psychological (experiential) verbs is taken over: Usually, expressing mental states verbally is not describing them. According to this, “I believe” can be seen as an expression of one’s own belief, but not as an expression of a belief about one’s belief. Hence, the utterance “I believe it is raining” shows that I believe that it is raining, although it is not said by these words that I believe that it is raining. Thinking thoughts such as “I believe it is raining, but it is not raining” (a variant of Moore’s paradox) is an absurdity between what is already said by silently uttering “It is not raining” and what is shown by silently uttering “I believe it is raining.” The paper agrees with a main result of Wittgenstein’s considerations of Moore’s paradox, namely the view that logical structure, deducibility, and consistency cannot be reduced solely to propositions—besides a logic of propositions, there is, for example, a logic of assertions and of imperatives, respectively.     

Visualizations of the Square of Opposition

In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples. I would like to thank the two anonymous reviewers for helpful comments and criticisms.     

Forcing operators on MTL‐algebras

We study the forcing operators on MTL‐algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t ‐norm based logic (MTL). At logical level, they provide the notion of the forcing value of an MTL‐formula. We characterize the forcing operators in terms of some MTL‐algebras morphisms. From this result we derive the equality of the forcing value and the truth value of an MTL‐formula (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

N.A. Vasil’ev’s Logical Ideas and the Categorical Semantics of Many-Valued Logic

Here we suggest a formal using of N.A. Vasil’ev’s logical ideas in categorical logic: the idea of “accidental” assertion is formalized with topoi and the idea of the notion of nonclassical negation, that is not based on incompatibility, is formalized in special cases of monoidal categories. For these cases, the variant of the law of “excluded n-th” suggested by Vasil’ev instead of the tertium non datur is obtained in some special cases of these categories. The paraconsistent law suggested by Vasil’ev is also demonstrated with linear and tensor logics but in a form weaker than he supposed. As we have, in fact, many truth-values in linear logic and topos logic, the admissibility of the traditional notion of inference in the categorical interpretation of linear and intuitionistic proof theory is discussed.     

Relevance and Relationalism

This paper will provide support for relationalism; the claim that the identity of objects is constituted by the totality of their relations to other things in the world. I will consider how Kit Fine’s criticisms of essentialism within modal logic not only highlight the inability of modal logic to account for essential properties but also arouse suspicion surrounding the possibility of nonrelational properties. I will claim that Fine’s criticisms, together with concerns surrounding Hempel’s paradox, show that it is not possible to provide a satisfactory account of certain properties in abstraction from their place within a wider context. Next, we will shift attention to natural kinds and consider the notion that relevance plays in metaphysical accounts of identity, by examining Peter Geach’s notion of relative identity. I will argue that the intensional relation between subject and object must be included in a satisfactory account of metaphysical identity.     

Two Early Arabic Applications of Model-Theoretic Consequence

We trace two logical ideas further back than they have previously been traced. One is the idea of using diagrams to prove that certain logical premises do—or don’t—have certain logical consequences. This idea is usually credited to Venn, and before him Euler, and before him Leibniz. We find the idea correctly and vigorously used by Abū al-Barakāt in 12th century Baghdad. The second is the idea that in formal logic, P logically entails Q if and only if every model of P is a model of Q. This idea is usually credited to Tarski, and before him Bolzano. But again we find Abū al-Barakāt  already exploiting the idea for logical calculations. Abū al-Barakāt’s work follows on from related but inchoate research of Ibn Sīnā in eleventh century Persia. We briefly trace the notion of model-theoretical consequence back through Paul the Persian (sixth century) and in some form back to Aristotle himself.