Many-Valued Logic

Does it make sense to employ modern logical tools for ancient philosophy? This well-known debate2 has been re-launched by the indologist Piotr Balcerowicz, questioning those who want to look at the Eastern school of Jainism with Western glasses. While plainly acknowledging the legitimacy of Balcerowicz’s mistrust, the present paper wants to propose a formal reconstruction of one of the well-known parts of the Jaina philosophy, namely: the saptabhaṅgī, i.e. the theory of sevenfold predication. Before arguing for this formalist approach to philosophy, let us return to the reasons to be reluctant at it. For one thing, Balcerowicz (2008) made a clear-cut distinction between a constructivist and a reductionist method of interpretation. The former is defined by Balcerowicz (2008: 4), where the so-called “constructivist” method is not to be confused with the philosophy of mathematics and merely amounts to the exercise of formal reconstruction:

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.

This course expands on material learned in Logic I: classical and modal logic. Where the first course introduced classical logic along with its modal expansions, the logics learned in this course are more naturally thought of as rivals of classical logics: all of them deny some of the valid consequences of classical logic, and thus imply that classical logic gets it wrong in certain cases. The course aims to introduce the student to the philosophical issues involved in these logics, while instilling technical proficiency in each of them.

This paper reports results and some new problems in one of the domains to which automatic first-order theorem provers have been most successfully applied: axiomatics of nonclassical propositional logics. It is well known that one of the standard axioms of the denumerable-valued pure implication logic of Łukasiewicz becomes derivable from the remainder in the presence of negation. Here it is shown that the same axiom is similarly derivable using conjunction and disjunction instead of negation. This closes a problem left open by Harris and Fitelson (Journal of Automated Reasoning 27, 2001). Related problems are discussed, and five such open problems are proposed as challenges to the automated reasoning community.

Offering four truth states: truth, falsity, impossibility and uncertainty, U4 represents an extension of Boolean algebra and classical logic. While fundamental theorems of classical logic are found in the inference theorems of U4, its discrimination can be seen by the invalidation of the genuine paradoxes of implication and the anomolous theorems of propositional logic. Compared to nineteen rules given for classical logic there are sixteen rules of inference for U4. Not only is U4 consistent and sound, it is probably complete. As U4 implication is closest to the meaning of " if-then " in natural language, it closely models human reasoning.

A rational interpretation is proposed for two ancient Indian logics: the Jaina saptabhangi, and the Madhyamika catuskoti. It is argued that the irrationality currently imputed to these logics relies upon some philosophical preconceptions inherited from Aristotelian metaphysics. This misunderstanding can be corrected in two steps: by recalling their assumptions about truth; by reconstructing their ensuing theory of judgment within a common conceptual framework.

يناقش الكتاب مراحل وآليات تطور المنطق الرمزي المعاصر متعدد القيم بأنساقه المختلفة، مركزًا على مشكلة الغموض المعرفي للإنسان بأبعادها اللغوية والإبستمولوجية والأنطولوجية، والتي تتجلى – على سبيل المثال – فيما تحفل به الدراسات الفلسفية والمنطقية والعلمية من مفارقات تمثل تحديًا قويًا لثنائية الصدق والكذب الكلاسيكية، وكذلك في اكتشاف «هيزنبرج» لمبدأ اللايقين  ، وتأكيده وعلماء الكمّ على ضرورة التفسيرات الإحصائية في المجال دون الذري، الأمر الذي يؤكد عدم فعالية قانون الثالث المرفوع في التعامل مع معطيات الواقع الفعلي، واستحالة مشروع إقامة لغة مثالية أو اصطناعية أو كاملة منطقيًا  تتجاوز عيوب ونقائص اللغة العادية التي نفكر ونتعامل بها. وينتهي الكتاب إلى نتيجة مؤداها أن ما واجهه المنطق الكلاسيكي –  ثنائي القيم – من مشكلات أدت إلى تطويره، لاسيما مشكلة الغموض، لابد وأن يواجهه بالمثل المنطق متعدد القيم، ذلك أن الغموض ظاهرة إبستمولوجية في المحل الأول، مردودها إلى الذات العارفة وقصور إمكاناتها الإدراكية والقياسية، لا إلى الوجود ذاته. وأننا حتى لو سمحنا لأية قضية منطقية بقيمة صدق ثالثة، أو بأكثر من قيمة تتوسط بين الصدق التام والكذب التام، فسوف تظل القضية – كتمثيل لغوي لإحدى وقائع العالم – صادقة أو كاذبة، سواء أدركنا ذلك أو لم ندركه، وذلك تأكيد لنزعة أفلاطون الواقعية القائلة بوجود أزلي وثابت للحقائق في عالم المثل، تحول دون معرفتنا الظنية بظلالها في عالم الحس المتغير. ويعني ذلك بعبارة أخرى أن الشك في قانون الثالث المرفوع هو إسقاط من الذات على الموضوع، مبعثه عدم اكتمال العملية المعرفية ومحدوديتها، وأن ظهور الأنساق المنطقية ذات القيم المتعددة ما هو إلا حلقة من حلقات العلاقة الجدلية اللامنتهية بين الإنسان والطبيعة، أو بيم ما هو مُدرك وما هو موجود بالفعل. والكتاب بصفة عامة هو أول كتاب باللغة العربية يعرض لأنساق المنطق متعدد القيم بعد مرحلة البرينكيبيا ماثيماتيكا

Are modus ponens and its counterpart, modus tollens, universal laws of logic? A deconstruction of modus ponens in binary propositional logic finds that the propositional formula of binary logic has exceeded it's definition .

In a number of logic systems, including U4 , modus ponens is not a theorem. An examination of the truth table for modus ponens in U 4 will show that modus ponens as an inference rule can be preserved.

"This dissertation consists of a collection of papers on pure and applied modal logics preceded by an introduction that ties them together. The papers and the variety of topics discussed in them can be viewed as a contribution to a logic toolbox. More specifically, a logic toolbox streamlined to model information, knowledge, and beliefs. The logic tools have many applications in various fields such as computer science, philosophy, mathematics, linguistics, economics, and other social sciences, but in this thesis the focus will be on their application in social epistemology and multi-agent systems.

After the introduction, which outlines the logic toolbox and some of its applications, follow four technical chapters that expand the logic toolbox (chapters 2, 3, 4, and 5) and two chapters that show the logic tools at work (chapters 6 and 7). In Chapter 2 a many-valued hybrid logic is introduced and a sound, complete, and terminating tableau system is given. Chapter 3 is a supplement to Chapter 2 and discusses various alternative definitions of the semantics of the hybrid part of the many-valued hybrid logic. Chapter 4 combines public announcement logic with hybrid logic and gives a sound and complete axiomatisation of the logic as well as discussing extensions with other modalities such as distributed knowledge. In Chapter 5, terminating tableau systems using reduction axioms as rules are given for standard dynamic epistemic logic as well as the hybrid public announcement logic of Chapter 4. Chapter 6 uses a dynamic epistemic logic to model the phenomenon of pluralistic ignorance. Finally, Chapter 7 uses first-order logic and description logic to discuss foundational issues in knowledge representation of regulatory relations in biomedical pathways.

This dissertation makes numerous contributions to the field of which the following three are the most important:
- A detailed investigation of the combination of hybrid logic and public announcement logic, showing in particular that the proof theoretic advantages of hybrid logic (such as automatic completeness with pure formulas) extend to hybrid public announcement logic (chapters 4 and 5).
- An approach to constructing terminating tableau systems for hybrid logics is shown to be extremely general in the sense that it extends to both many-valued hybrid logic (Chapter 2), dynamic epistemic logic, and hybrid public announcement logic (Chapter 5).
- This dissertation touches upon the issue of how knowledge and beliefs of a group of agents relate to the knowledge and beliefs of the individuals of the group. How knowledge or beliefs of the individual agents can be aggregated to yield group knowledge or group beliefs is studied in Section 1.3.3 of the introduction, and how information and beliefs flow among a group of agents involved in the phenomenon of pluralistic ignorance is studied in Chapter 6."

A number of similarities between fuzzy logic and probability can be identified. Historically, fuzzy logic is founded upon a probability based many valued logic of Lukasiewicz. The fuzzy membership function and the probability density function closely resemble as they both have similar shapes and have values that range between zero and one. Vague linguistic terms can be viewed as being probabilistic. The fuzzy rules of inference can be identified with probabilistic inference of inductive reasoning. The centroid method of calculating the output of fuzzy control is identical to that used in probability density functions, is namely the mean. Fuzzy logic could be probability as there as are so many close resemblances.

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Supercomplex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item's essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super-and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with 'broken symmetry' might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic 'asymmetry' of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics-which is presently unfolding in physics, mathematics, life and cognitive sciences-may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics.

Equilibrium logic, introduced in [20], is a conservative extension of answer set semantics for logic programs to the full language of propositional logic. In this paper we initiate the study of first-order variants of equilibrium logic. In particular, we focus on a quantified version QN 5 of the propositional many-valued logic N 5 of here-and-there with strong negation, and define the condition of equilibrium via a minimal model construction. We verify Skolem forms and Herbrand theorems for QN 5 and show that, like its propositional counterpart, the ...

Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomatizability on the one hand, and algebraical/model theoretic investigations on the other. The proof theory of many-valued systems has not been investigated to any comparable extent. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. Several people have, since the 1950's, proposed ways to generalize such formalisms from the classical to the many-valued case. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. (Section 3.1 contains a short overview of work done in this area). The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. This report is actually a template, from which all results can be specialized to particular logics. This idea has also found its way into the notation: Throughout this report, we use V as denoting a set of m truth values, and P and Q as dummies representing n-ary connectives and quantifiers, respectively. Replace V by the set of values true and false, and P and Q by the usual logical connectives and quantifiers, and you have a treatise on proof theoretic systems for classical logic.

The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition.

This is the Spanish translation of Haack's first book, DEVIANT LOGIC, here in full text. It covers many issues raised by non-standard (and specifically deviant) logics, and studies of many different many-valued, intuitionist, free, etc. logical systems.

Guy Kahane (2011, 2012) asks an axiological question: what value would (or does) God's existence bestow on the world? Supposing God's existence is a matter of necessity, this axiological question faces a problem because answering it will require assessing the truth-value of counterpossibles. I argue that Kahane, Moser, and Davis and Franks fail in their attempts to render the axiological question substantive. I then offer my own solution by bringing work in cognitive psychology and philosophy of mind to bear on the possibility of assessing counterpossibles. I argue that humans can engage in counterpossible reasoning by 'accepting' or 'supposing' that the antecedent is true and 'screening out' those beliefs that would result in contradiction when combined in inferences with the acceptance or supposition. These screened out propositions are not treated as false, but are ignored. I offer a three-valued logic for counterpossible reasoning. I then conclude by outlining some implications for the axiological question.

I study here the implications of Suszko-like reductive results for the common wisdom on the internal logic of a topos. My aim here is to question certain slogans about topos logic, and especially on its many-valuedness. The originality of this paper comes not from the observation that the internal logic of a topos is bivalent because it is in the scope of Suszko’s or any other reduction; that is easy. Rather, the originality comes from (1) recognizing the import of applying Suszko’s and similar results in a topos-theoretical setting, (2) the suggestions to give categorial content to the reduction, and (3) the extrapolation of the debate about Suszko’s thesis to the topos-theoretical framework, which gives us some insight about the scope of another theorem, namely that stating the intuitionistic character of the internal logic of a topos.

In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar many-valued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them discusses our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction.

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This volume examines the entire logical and philosophical production of Nikolai A. Vasil’ev, studying his life and activities as a historian and man of letters. Readers will gain a comprehensive understanding of this influential Russian logician, philosopher, psychologist, and poet.
The author frames Vasil’ev’s work within its historical and cultural context. He takes into consideration both the situation of logic in Russia and the state of logic in Western Europe, from the end of the 19th century to the beginning of the 20th. Following this, the book considers the attempts to develop non-Aristotelian logics or ideas that present affinities with imaginary logic. It then looks at the contribution of traditional logic in elaborating non-classical ideas. This logic allows the author to deal with incomplete objects just as imaginary logic does with contradictory ones. Both logics are objects of interesting analysis by modern researchers.

This paper argues: first, that a presentist, powers based, diachronic account of modality can provide a satisfactory account of our intuitions about modality as well as a compelling rebuttal to alternative accounts; second that taking this account seriously requires a more radically attenuated modal logic than even partisans of such an account realize. Attempting to navigate this problem, I provide a formal, non-normal semantics for a tensed modal logic, along with a philosophical interpretation of it.

This paper deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames, over the very lattice. And we also give axiomatizations for the case of a finite MV chain but this time without canonical constants.

Lukasiewicz has often been criticized for his motive for inventing his three-valued logic, namely the avoidance of determinism. First of all, I want to show that almost all of the critcism along this line was wrong. Second I will indicate that he made mistakes, however, in constructing his system, because he had other motives at the same time. Finally I will propose some modification of his system and its interpretation which can attain his original purpose in some sense.

While participating in a symposium on Dave Ripley's forthcoming book Uncut, I had proposed that employing a strict-tolerant interpretation of the weak Kleene matrices provided a content-theoretical conception of the bounds of conversational norms that enjoyed advantages over Ripley's use of the strong Kleene matrices. During discussion, I used the case of sentences that are taken to be out-of-bounds for being secrets as an example of a case in which the setting of conversational bounds in practice diverged from the account championed by Ripley. In this paper, I consider an objection that my treatment of quantifiers was mistaken insofar as the confidentiality of a sentence ϕ(t) may not lift to the sentence ∃xϕ(x) and draw from this objection that neither the strong nor the weak Kleene interpretation of quantifiers suffices, but that a novel interpretation may do so.

The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxes. There is something very distinctive about ST: namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes. For example, the logic ST + (ST for a language with a transparent truth predicate and self-referential sentences) is a conservative extension of classical logic. That is, ST + is not only non-trivial, but it has exactly the same valid inferences as classical logic for the T-free fragment. How is this possible? Well, because ST + preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST +. In a recent paper, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST + metainferences are closely related to LP inferences. In this paper we review their result and put the connection in a broader context.

The contemporary usage of inductive logic can be modelled by the U4 implication truth table. In particular inductive logic can be transposed to the cases when the conclusion is uncertain. U4 implication is shown to offer a systemisation of the methods of both strong and weak arguments and cogent and uncogent arguments for inductive logic. The application of the implication truth table of U4 to inductive logic therefore validates the U4 implication truth table.

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 I1 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 I1 products.

This article argues that resolutions to the sorites paradox offered by epistemic and supervaluation theories fail to adequately account for vagueness. After explai​ning the paradox, I examine the epistemic theory defended by Timothy Williamson and discuss objections to his semantic argument for vague terms having precise boundaries. I then consider Rosanna Keefe's supervaluationist approach and explain why it fails to accommodate the problem of higher-order vagueness. I conclude by discussing how fuzzy logic may hold the key to resolving the sorites paradox without positing indefensible borders to the correct application of vague terms.

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.

This paper describes an application of fuzzy logic in analysis o f delays in construction projects using Fuzzy toolbox of MATLAB Program So ftware. Delays in construction projects are inevitable and may result in claims and disputes among different construction parties. Delays in construction projects can be due to a number of causes, which need to be classified and identified. For the success of a construction project, estimation of likelihood of delay resulting fro m different factors is a must. Fuzzy logic provides a simple way to arrive at a definite conclusion based u pon vague, imprecise or missing input informat ion. Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact.

Inspired by Tarski’s works, in the late 1970’s Priest proposed a new technique to build a hierarchy of logics, which he considered a re-foundation of normal modal logic. By a brief retrospective of the questions of both modal and many-valued logics, we argue that his proposal deeply diverges from what Lewis had in mind and yields an entirely different and non-modal logic. We treat normal modal logics as infinite-valued logics and suggest that their finite-valued extensions are simpler and more promising alternatives to the Priest’s re-foundation, like Béziau’s basic logics.

A special algebra called EQ-algebra has been recently introduced by Vilém Novák. Its original motivation comes from fuzzy type theory, in which the main connective is fuzzy equality. EQ-algebras have three binary operations – meet, multiplication, and fuzzy equality – and a unit element. They open the door to an alternative development of fuzzy (many-valued) logic with the basic connective

This dissertation describes the development of automated synthesis algorithms that construct reversible quantum circuits for reversible functions with large number of variables. Specifically, the research area is focused on reversible, permutative and fully specified binary and ternary specifications and the applicability of the resulting circuit to the physical limitations of existing quantum technologies. Automated synthesis of arbitrary reversible specifications is an NP hard, multiobjective optimization problem, where 1) the amount of time and computational resources required to synthesize the specification, 2) the number of primitive quantum gates in the resulting circuit (quantum cost), and 3) the number of ancillary qubits (variables added to hold intermediate calculations) are all minimized while 4) the number of variables is maximized. Some of the existing algorithms in the literature ignored tenth of my PhD. I close by thanking my friends Naveed and