In Belnap's useful 4-valued logic, the set 2 = {T, F} of classical truth values is generalized to the set 4 = ������(2) = {Ø, {T}, {F}, {T, F}}. In the present paper, we argue in favor of extending this process to the set 16 = ᵍ (4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR₂ with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN₃ with (...) an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN₃ coincide with the logic of FOUR₂, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli's and Avron's notion of a logical bilattice and state a number of open problems for future research. (shrink)

A short description of a toy theorem prover for 16-valued trilattice logics. Written for the occasion of my friend's Jan van Eijck's retirement. With a link to a swish interface to the prolog prover.

In a recent paper we have defined an analytic tableau calculus PL_16 for a functionally complete extension of Shramko and Wansing's logic based on the trilattice SIXTEEN_3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic---such as the relations |=_t, |=_f, and |=_i that each correspond to a lattice order in SIXTEEN_3; and |=, the intersection of |=_t and |=_f,. -/- It turns out that our method of characterising these semantic relations---as (...) intersections of auxiliary relations that can be captured with the help of a single calculus---lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that |=, when restricted to L_{tf}, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano. (shrink)

This is a collection of invited papers from the 1975 International Sym posium on Multiple-valued Logic. Also included is an extensive bib liography of works in the field of multiple-valued logic prior to 1975 - this supplements and extends an earlier bibliography of works prior to 1965, by Nicholas Rescher in his book Many-Valued Logic, McGraw-Hill, 1969. There are a number of possible reasons for interest in the present volume. First, the range of various uses covered in (...) this collection of papers may be taken as indicative of a breadth which occurs in the field of multiple-valued logic as a whole - the papers here can do no more than cover a small sample: question-answering systems, analysis of computer hazards, algebraic structures relating to multiple-valued logic, algebra of computer programs, fuzzy sets. Second, a large part of the interest in such uses and applications has occurred in the last twenty, even ten years. It would be too much to expect this to be reflected in Rescher's 1969 book. Third, in the 1970's a series of annual symposia have been held on multiple-valued logic, which have brought much of this into a sharp focus. * The 1971 and 1972 symposia were held at the SUNY at Buffalo, the 1973 symposium at the Uni versity of Toronto, and the 1974 symposium at West Virginia Uni versity. Papers from these symposia are included in the bibliography which may be found in an appendix of this book. (shrink)

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data is (...) asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT4. Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a–order and d–order coincide with first-degree entailment. Finally logic with two kinds of operations and consequence relation defined via a–ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN 3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data (...) is asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT 4 . Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a –order (truth order) and d –order (falsity order) coincide with first-degree entailment. Finally logic with two kinds of operations ( a –connectives and d –connectives) and consequence relation defined via a –ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

A sequential axiomatization is given for the 16-valued logic that has been proposed by Shramko-Wansing as a candidate for the basic logic of logical bilattices.

By using the notions of exact truth and exact falsity, one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the Strong Kleene schema. Besides familiar logics such as Strong Kleene logic, the Logic of Paradox and First Degree Entailment, the resulting class of all Strong Kleene generalizations of classical logic also contains a host of (...) unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus that is sound and complete with respect to all of them. Two further sequent calculi and \ calculus) will be considered, which serve the same purpose and which are obtained by applying general methods to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \ and the \ calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \ and the \ calculus, we also hint at its philosophical significance. (shrink)

This paper offers a brief analysis of the unification problem in modal transitive logics related to the logic S4 : S4 itself, K4, Grz and Gödel-Löb provability logic GL . As a result, new, but not the first, algorithms for the construction of ‘best’ unifiers in these logics are being proposed. The proposed algorithms are based on our earlier approach to solve in an algorithmic way the admissibility problem of inference rules for S4 and Grz . The first (...) algorithms for the construction of ‘best’ unifiers in the above mentioned logics have been given by S. Ghilardi in [ 16 ]. Both the algorithms in [ 16 ] and ours are not much computationally efficient. They have, however, an obvious significant theoretical value a portion of which seems to be the fact that they stem from two different methodological approaches. (shrink)

Typescripts, essays, and an authoritative edition of Knowing and the known, Dewey's collaborative work with Arthur F. Bentley. In an illuminating Introduction T. Z. Lavine defines the collaboration's three goals-the construction of a new language for behavioral inquiry, a critique of formal logicians, in defense of Dewey's Logic, and a critique of logical positivism. In Dewey's words: Largely due to Bentley, I've finally got the nerve inside of me to do what I should have done years ago. What is it (...) to be a linguistic sign or name? and Values, valuations, and social facts, ' both written in 1945, are published here for the first time. (shrink)

1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement of (...) mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)

Typescripts, essays, and an authoritative edition of _Knowing and the Known, _Dewey’s collaborative work with Arthur F. Bentley. In an illuminating Introduction T. Z. Lavine defines the collaboration's three goals—the "construction of a new language for behavioral inquiry," "a critique of formal logicians, in defense of Dewey’s _Logic,_"_ _and "a critique of logical positivism." In Dewey’s words: "Largely due to Bentley, I’ve finally got the nerve inside of me to do what I should have done years ago." "What Is It (...) to Be a Linguistic Sign or Name?" and "Values, Valuations, and Social Facts,’ both written in 1945, are published here for the first time._ _. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:A Time of Novelty: Logic, Emotion, and Intellectual Life in Early Modern India, 1500-1700 C.E. by Samuel WrightAnusha Rao (bio)A Time of Novelty: Logic, Emotion, and Intellectual Life in Early Modern India, 1500-1700 C.E. By Samuel Wright. Oxford: Oxford University Press, 2021. Pp. xxi + 278. Paper $99.00, isbn 978-0-197568-16-3Samuel Wright's A Time of Novelty examines the discipline of Nyāya, or Sanskrit logic, between 1500 and 1700 CE (...) and argues that novelty, both as an intellectual and affective category, was the structuring principle through which Nyāya intellectuals built a philosophical community in early modernity. Wright uses a set of essays and commentaries by Nyāya philosophers on a range of topics to reflect on how intellectual life in this period was shaped by critical inquiry, emotion, and manuscript culture.The book studies a very fruitful, if contentious, period within the history of Sanskrit intellectual disciplines. Sheldon Pollock's "Sanskrit Knowledge Systems on the Eve of Colonialism" (SKSEC project) was the first to draw attention to this period as early modernity in India. Pollock has pointed to both parallels with and differences from modernity in Europe. Since then, scholarly attention has been directed to how periodizing techniques were used by Nyāya intellectuals, and to the relationship that these "new" Nyāya intellectuals had with the tradition. While Wright is in general agreement with the aims of the SKSEC project, he argues against focusing exclusively on intellectual historical change without understanding the values and sentiments that underwrite intellection. For this reason, the book undertakes the challenging task of linking a set of understudied Nyāya arguments with philosophy of emotions to understand affective novelty. This offers interesting insights into the emotional concerns behind Nyāya intellection in a way that earlier scholarship has been unable to, but it has the disadvantage of imposing frameworks alien to Nyāya philosophers onto their arguments--such as Wright's arguments about how space is produced in chapter 3.The book is organized into six chapters, each of which deals with an important topic within Nyāya arguments in early modern India. While the first four chapters, titled "Doubt," "Objectivity," "Happiness," and "Dying," directly deal with these contentious issues and debates around them by Nyāya thinkers, in the last two chapters, "Space" and "Time," Wright does not take up specific arguments from primary texts, but focuses on how we can best understand the space that Nyāya intellectuals inhabited and how novelty can be conceptualised in time. [End Page 1]In the first chapter on doubt, Wright zooms in on arguments about how doubting cognitions emerge, focusing particularly on Raghunātha Śiromaṇi and how commentators present his novel position on the issue. Earlier Nyāya intellectuals argued that speech and other indirect forms of perception could not generate doubting cognitions. Instead, they advanced a theory of apperceptive doubt, where doubt arises in a subsequent, remembering cognition. Raghunātha, on the other hand, argues that doubting cognitions can arise directly from speech, and he substantially changes the definition of doubt, a change which is noted and carefully explained by his commentators. Beginning with the theory of doubt allows Wright to then examine how the new definition of doubt leads to structural changes in the crafting of Nyāya debates, particularly in genres such as the short essay (vāda), where the topic of contention is stated in the newly developed format. Wright also links this debate with the notion of trust. Careful attention to how the debate is framed by commentators also underscores the emotional underpinnings of the argument--the "old" view is not to be trusted, since it does not stand up to reasoning (p. 60). Doubting cognitions are not ignored in Nyāya, but become the beginning point of debates and discussions, demonstrating that Nyāya philosophers in early modern India were fashioning a community that engaged peers through new ways of understanding and caring about critical inquiry.The second chapter focuses on arguments around objectivity and deals at greater length with the periodizing techniques used by Nyāya thinkers in the early modern period. This... (shrink)

Do farmers in the southern Mexican highlands practice science? The anthropologist Roberto González argues that they do in his well‐written and solidly researched account of Zapotec farmers' cultivation of corn, sugarcane, and coffee. This book speaks to enduring questions concerning the nature of science through its focus on the often‐overlooked sophistication of traditional farming. Historians of science interested in agriculture, international development, and definitions of science more broadly will find in Zapotec Science a rich store of wide‐ranging questions and provocative (...) answers.González aims to move beyond the limitations of many sociological and anthropological studies of traditional societies. Many ignore international “cosmopolitan” influences, and most that do deal explicitly with these treat the traditional and the cosmopolitan as separate spheres. González finds no reason to treat these categories as nonoverlapping, as he documents how Zapotec farmers have experimented with and adopted techniques, implements, and crop varieties from many parts of the globe. And the flow goes both ways. Many so‐called cosmopolitan scientists are beginning to recognize the importance of traditional knowledge for solving problems in industrialized nations. But though the boundaries may be blurred, González argues that the conceptual bases of Zapotec agriculture are “culturally incommensurable with those predominating in industrialized societies” .In Chapter 1, González elaborates his position relative to other anthropological work on local science and traditional ecological knowledge and poses comparative questions about Zapotec and cosmopolitan science. Chapters 2 and 3 richly detail the daily practice of Zapotec agriculture and its environmental and geographical, cultural, and historical contexts. González emphasizes campesinos' craft, the refined skills that underlie every aspect of home, community, and agricultural life.González's research comes to the fore in the middle chapters on maize, sugarcane, and coffee cultivation. Here he clearly illustrates how Zapotec farming is founded on five fundamental values: household maintenance , mutual aid among families, personification of the earth and supernatural beings, an understanding of physical work as normal and desirable, and an emphasis on the quality of food over its quantity. For each crop he shows how present agricultural practices have emerged from an evolving balance between local crop development over hundreds or thousands of years and carefully weighed periodic opportunities to adopt new practices or plant varieties. Sugarcane cultivation, for example, illustrates how Zapotec farmers' decisions grow out of the five core values rather than a more narrow “economic rationality.” Zapotec communities do not consume cheap, white, processed sugar; rather, they produce and use an older, less processed, more expensive form called panela. Panela's persistence in their diet is due to its role in reciprocal social relationships and to Zapotecs' preference for high‐quality, good‐tasting food. The book concludes with a critique of Green Revolution development strategies and a lively discussion of the fertile borderlands between traditional and cosmopolitan knowledges.González's definition of science, closely related to Bronislaw Malinowski's, centers on practical knowledge. González argues that Zapotec farmers practiced science as they experimented with new techniques and implements from beyond Oaxaca's borders. Although traditional and cosmopolitan knowledge are conceptually distant, the author places great weight on what they share: both are systematic searches for practical knowledge. Traditional agriculture offers materials and methods that cosmopolitan scientists increasingly need.This book's strength lies, not surprisingly, near González's expertise: his careful description and analysis of Zapotec farmers' practices, logic, and values. As with all ambitious and worthwhile projects that pull together scientific enterprises that are culturally, geographically, and conceptually distant, González's book leaves questions unanswered. His characterization of cosmopolitan science is at times contradictory and often narrow. While Zapotec scientists are said to consider a variety of procedures and techniques and “adopt what seems to work according to their own categories, and reject what does not,” cosmopolitan scientists operate according to “one view, one procedure” . A more thorough characterization of cosmopolitan science would strengthen the comparisons with Zapotec farming and would complement the direct treatment that González provides for cosmopolitan science's exceptions, sustainable agriculture and alternative medicine. This would also help to clarify differences between natural scientists and development economists, whose values are often not distinguished. Furthermore, cosmopolitan science seems to dwell outside of Mexico. What would it look like to widen the lens to include the Mexican urban academic scientific community? How does this book speak to the wider literature on scientific practice that largely focuses on cosmopolitan science?Zapotec Science is an important contribution to our understanding of nonindustrialized agriculture outside of the United States. It will, I hope, prompt historians to delve into the historical relationships among people who employ agricultural knowledge toward divergent goals: scientists, farmers, and government leaders dwelling in different national and professional cultures. (shrink)

This book provides an incisive, basic introduction to many-valued logics and to the constructions that are "many-valued" at their origin. Using the matrix method, the author sheds light on the profound problems of many-valuedness criteria and its classical characterizations. The book also includes information concerning the main systems of many-valued logic, related axiomatic constructions, and conceptions inspired by many-valuedness. With its selective bibliography and many useful historical references, this book provides logicians, computer scientists, philosophers, and mathematicians (...) with a valuable survey of the subject. (shrink)

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.

Suszko's Thesis maintains that many-valued logics do not exist at all. In order to support it, R. Suszko offered a method for providing any structural abstract logic with a complete set of bivaluations. G. Malinowski challenged Suszko's Thesis by constructing a new class of logics (called q-logics by him) for which Suszko's method fails. He argued that the key for logical two-valuedness was the "bivalent" partition of the Lindenbaum bundle associated with all structural abstract logics, (...) while his q-logics were generated by "trivalent" matrices. This paper will show that contrary to these intuitions, logical two-valuedness has more to do with the geometrical properties of the deduction relation of a logical structure than with the algebraic properties embedded on it. (shrink)

The most natural and straightforward step beyond two‐valued logic is to introduce more logical values, thereby rejecting the principle of bivalence. Another, indirect, way consists in challenging the classical laws concerning the sentence connectives and introducing other non‐two‐valued connectives into the language. Either way, prepositional logic seems fundamental to many‐valuedness, rather than its first‐order extension. Hence, although there has been interesting research into first‐order many‐valued logics, we shall confine our discussion here to the 0‐order case.

Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question (...) whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5. (shrink)

This work presents the concepts of translation and conservative translation between logics. By using algebraic semantics we introduce several conservative translations involving the classical propositional calculus and the many-valued calculi of Post and Lukasiewicz.

The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth. Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued (...) logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions. (shrink)

Originally published in 1967. An introduction to the literature of nonstandard logic, in particular to those nonstandard logics known as many-valued logics. Part I expounds and discusses implicational calculi, modal logics and many-valued logics and their associated calculi. Part II considers the detailed development of various many-valued calculi, and some of the important metathereoms which have been proved for them. Applications of the calculi to problems in the philosophy are also surveyed. This work (...) combines criticism with exposition to form a comprehensive but concise survey of the field. (shrink)

The paper discusses functional properties of some four-valued logics which are the expansions of four-valued Belnap’s logic DM4. At first, we consider the logics with two designated values, and then logics defined by matrices having the same underlying algebra, but with a different choice of designated values, i.e. with one designated value. In the preceding literature both approaches were developed independently. Moreover, we present the lattices of the functional expansions of DM4.

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching (...) theory to cognitive modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for (...) this system and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)

In the monograph [1] of Chang and Keisler, a considerable extent of model theory of the first order continuous logic is ingeniously developed without using any notion of provability.In this paper we shall define the notion of provability in continuous logic as well as the notion of matrix, which is a natural extension of one in finite-valued logic in [2], and develop the syntax and semantics of it mostly along the line in the preceding paper [2]. Fundamental theorems of (...) model theory in continuous logic, which have been proved with purely model-theoretic proofs in [1], will be proved with proofs which are natural extensions of those in two-valued logic. (shrink)

Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any (...) new equivalences. (shrink)