Nonclassical Logic, Misc

In [Is Classical Mathematics Appropriate for Theory of Computation?] we show there is a contradiction which in [“Fuzzy Time”, a solution of Unexpected Hanging Paradox (A Fuzzy interpretation of Quantum Mechanics), Philpapers 2019-04-13] we give a solution for that. This is the starting point for new Theories, Theory of Fuzzy Time Computation and Fuzzy Time –Particle interpretation of quantum Mechanics. A question is remained which was mentioned in [Two points and two questions, F.Didehvar, Philpapers, Researchgate, 2025]. Is this contradiction a (...) contradiction in Mathematics too? Here, we try to answer this question. The article belongs to a group of articles which presents the logical back bone of this line of research. (shrink)

This series is an attempt to discover foundational truths, to increase investment in spreading truths, to improve marginal logic, and to summarize from a different perspective. Extracting some value from previous content would be a good start, although there is no logical necessity. I hope to replicate the same input-output ratio as my previous conversations with AI. While there are no plans, I believe logic will bring good luck. Since last time it started with AI and soul liberation, this time (...) I might as well start with Homo sapiens and science. After a few days, I discovered that I could form a somewhat interactive parallel relationship with the previous AI dialogue. If the logical foundation is to strive to reason smaller and smaller, this time we can strive to expand outward and systematical. In order to improve marginal logic, as always, I only have two options for AI's comments: delete or keep. (shrink)

This paper explores an alternative non-relational semantics for modal logic, framing modal systems as "poly-logics"—intersections of simpler, foundational logics. Building on pioneering work by J. Kearns and subsequent developments, we demonstrate how established systems such as K series (K, K4, K5, K45), KD series (KD, KD4, KD5, KD45), KB series (KDB, KB, KB4, KB5, KB45) emerge as intersections of logics like KT, KTB, FN, TR, and their extensions. Utilizing Resolution Matrix Semantics (RMS), we establish soundness and completeness for key systems (...) with sketched proofs and introduce an innovative tableau-based proof procedure tailored to RMS. This approach not only unifies disparate modal frameworks but also yields efficient algorithmic tools for validity checking, offering fresh insights into the algebraic structure of modal deduction. Examples illustrate the method's efficacy, highlighting its advantages over traditional Kripke-style model checking. (shrink)

Human thinking does not proceed within a single logic. It stabilizes meaning at the intersection of multiple, partially incompatible logics while tolerating indeterminacy. This paper develops quantum-inspired polylogical systems - formal framework in which this cognitive fact becomes a principle of inference. Building on Resolution Matrix Semantics, indeterminate truth values are interpreted as semantic superpositions, and logical systems themselves form a space of interacting constraints. Inference is reconceived not as derivation within a fixed logic, but as the emergence of stable (...) semantic configurations through interference and suppression. Unlike conventional quantum AI, which accelerates classical reasoning, the proposed model alters the semantics of inference itself. Poly-Logic Quantum AI thus offers a structural model of human reasoning—one that explains coherence without certainty, context sensitivity without logic switching, and the emergence of new ideas without randomness. The result is a theory of artificial reasoning closer to human thinking than to calculation. (shrink)

A logic is said to be hyperintensional when it admits connectives or operators creating hyperintensional contexts, i.e. sentential contexts not respecting the intersubstitutivity salva veritate of co-intensionals. Odintsov and Wansing (2021) suggest a formal criterion for classifying a logic as hyperintensional based on whether its consequence relation is self-extensional. I argue that satisfaction of Odintsov and Wansing's criterion is neither necessary nor sufficient for characterising a logic as hyperintensional. Indeed, the criterion classifies as hyperintensional logics which are not meant to (...) capture hyperintensional behaviour, and fails to classify as hyperintensional logics which admit some forms of hyperintensional behaviour. I suggest several possible criteria for a logic to count as hyperintensional, arguing that each of them corresponds to a different understanding of hyperintensionality for logics. Then, I weigh these criteria against a family of logics, which display different "nuances'' of hyperintensional behaviour. Finally, I gesture towards a diagnosis of hyperintensionality for logics. (shrink)

In this paper, we investigate the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an 'actuality' operator among the connectives. This logic was initially considered by L. Humberstone, and, more recently, also by S. Niki and H. Omori to present a possible application of intuitionism to empirical discourse. Niki and Omori's idea to consider the notion of actuality based on intuitionistic logic was presented, among other things, using Gentzen sequents. Unfortunately, their proof system is not (...) cut-free. In this paper, we define another proof system for Niki and Omori’s logic, namely a hypersequent calculus, and present a proof of cut-elimination for it. (shrink)

This chapter gives some elements of the history of trivalent logics and presents the key technical notions. We stress that Boole and Frege were aware of reasons to go beyond bivalence, in ways that influenced Łukasiewicz in particular. Then we put particular emphasis on the 1930s as a pivotal moment in the application of trivalence to a range of interconnected phenomena, such as probability and hypothetical reasoning, quantum indeterminacy, computability theory, and the semantic paradoxes. The chapter goes on to present (...) basic elements regarding the connection of trivalent logics to other logics, including intuitionistic logic, modal logics, fuzzy and many-valued logics, and finally to classical and non-classical logics more broadly. Along the way, we present basic systems of mixed consequence, and provide glimpses into the theory of ternary computation. (shrink)

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

Modal Interpretations of quantum mechanics distinguish between the realm of possibility, where the dynamical state determines what may be the case, and the realm of actuality, where the value state represents what actually is the case. In particular, the Modal-Hamiltonian Interpretation proposes an ontology in which there are no individual objects: quantum systems are characterized as non-individual bundles of properties. The aim of the present paper is to propose some rst steps towards the development of a new modal logic adapted (...) to describe a possibilist modality in an ontology of properties. The main idea consists in conceiving possibility as fundamental and introducing actuality as an operator acting on possible facts. On this basis, propositions referring to possible facts and propositions referring to actual facts are distinguished, and some relations between the realm of possibility and the realm of actuality are introduced. (shrink)

Following a suggestion of Birkhoff & von Neumann [Ann. Math. 1936;37:23–32], we pursue a joint study of quantum logic and intuitionistic logic. We exhibit a linear-time translation which for each quantum logic Q and each superintuitionistic logic I yields an axiomatization of the intersection of Q and I from axiomatizations of Q and of I⁠. The translation is centered around a certain axiom (Ex) which (together with introduction and elimination rules for connectives) is shown to axiomatize the intersection of orthologic (...) and intuitionistic logic, solving a problem of Holliday [Logics 2023;1:36–79]. We prove that the lattice of all super-Ex logics is isomorphic to the product of the lattices of quantum logics and superintuitionistic logics. We prove that there are infinitely many sub-Ex logics extending Holliday’s fundamental logic. (shrink)

This paper examines a range of logical systems within the family of variable inclusion logics—also known as containment logics. We focus on those logics that restrict classically valid inferences to ones meeting specific variable inclusion constraints, hence called variable inclusion companions of classical logic. These constraints can be seen as enforcing varying degrees of relevance between premises and conclusions, placing these systems within the broader tradition of relevance logics. We review established companions of Classical Logic, including Weak Kleene logics (Bochvar, (...) 1938, Halldén, 1949), pure variable inclusion logics (Paoli et al., 2021), uniform Weak Kleene logics (Borzi and Zirattu, forth.), and their intersection—pure uniform logics. We then introduce two new systems named analytic and synthetic companions of Classical Logic. We characterize their consequence relations and interpret them, as their names suggest, as validating only analytic and synthetic inferences. Lastly, we argue that these systems more effectively address the irrelevance cases that motivated earlier proposals, by correctly tracing them to the Monotonicity of the consequence relation. (shrink)

This paper presents a sequent calculus for Inquisitive Propositional Logic obtained by expanding the sequent calculus g3ip for intuitionistic propositional logic with suitable rules for double negation elimination for atoms and the Split Property. A suitable rule for the Split Property is obtained by taking advantage of the connection between the truth-conditional fragment in Inquisitive Logic and Harrop formulas. The paper proves admissibility of cut for the sequent calculus and uses the sequent calculus to prove interpolation for Inquisitive Propositional Logic. (...) Interpolation is obtained using Maehara’s lemma. (shrink)

There are essentially three ways to treat conditionals with impossible antecedents in a formal framework that employs classical truth values: one can hold that such conditionals are all true, that they are all false, or that some are true while others are false. These three options will be examined under the background hypothesis that a conditional is true when its antecedent is incompatible with the negation of its consequent. It will be argued that the third option can be coherently developed (...) by defining a suitable notion of compatibility. (shrink)

El análisis de los vínculos que pueden ser establecidos entre la lógica cuántica, las semánticas no deterministas de Nmatrices y las teorías de cuasiconjuntos es el núcleo central de este trabajo. La necesidad de tal análisis ha surgido de forma natural luego de que la relación entre Nmatrices y lógica cuántica quedó explicitada en un trabajo previo: los estados cuánticos, entendidos como medida de probabilidad, pueden ser interpretados como valuaciones de una cierta Nmatriz para el retículo cuántico de proyectores. El (...) Teorema de Kochen-Specker priva a la lógica cuántica de veritativo funcionalidad, pero no de admitir una semántica no determinista Nmatricial. Esto es, la semántica de Nmatrices es compatible con los requerimientos semánticos y algebraicos del retículo de proyectores cuánticos. Si tenemos en cuenta que las teorías de cuasiconjuntos (Quasi-sets y Quasets) han surgido motivadas por la ontología cuántica, que las teorías de conjuntos están fuertemente vinculadas con la lógica y que originalmente uno de los objetivos de la teoría de quasets QST era brindar una semántica para los entes cuánticos, entonces el puente establecido entre la lógica cuántica y las semánticas no deterministas puede ser visto como el detonador que inicia la búsqueda de vínculos entre estas teorías. Por un lado, la lógica cuántica de Birkhoff y von Neumann está comprometida con la representación de espacios de Hilbert de la Mecánica cuántica. Por otro lado, esta representación (como quedará claro más adelante) está comprometida con una ontología de individuos. Por lo tanto, la lógica cuántica y la ontología están vinculadas a través del formalismo de espacios de Hilbert. Como dijimos, las teorías de cuasiconjuntos han surgido motivadas por la ontología de los entes del micromundo. Esto implica que los cuasiconjuntos y la lógica cuántica mantendrían de forma indirecta cierta relación. (shrink)

Work in the field of feminist logic is still rather scarce and the field itself remains a contested area of study, but still, it is developing. One approach concentrates on analyzing logical systems with respect to structural features that may perpetuate sexism and oppression or, on the other hand, features that may be helpful for resisting and opposing these social phenomena. Upon this assumption, I want to investigate possible applications of queer feminist views on (philosophy of) logic with respect to (...) a very specific group, namely contradictory logics, i.e., logical systems containing contradictions in their set of theorems. I want to show that, on the one hand, the formal set-up of contradictory logics makes them well-suited from the perspectives of feminist logic and, on the other hand, that queer feminist theories provide a relevant, and so far undeveloped, conceptual motivation for contradictory logics. Thus, bringing together contradictory logics and queer feminist theories may prove fruitful both as a ‘real-life’ motivation for these peripheral logical systems and as a formal basis for a philosophical field that is still characterized by a distrust of formalism. (shrink)

This paper presents a comprehensive proof-theoretic analysis of Jaśkowski’s discussive (or discursive) logic, working with a set of connectives including classical negation and disjunction, as well as so-called (right-)discussive conjunction and discussive implication. By employing established techniques two labelled frameworks are introduced: sequent and natural deduction systems. The paper explores the ability of the proposed calculi to accurately represent Jaśkowski’s discussive logic, particularly in light of its paraconsistent nature, and establishes cut- admissibility and normalization theorems. Additionally, the introduced sequent calculus (...) – shown to allow terminating proof search – is employed to prove the embedding of discussive logic within modal logic S5. Finally, it is proved that the natural deduction calculus translates into the corresponding sequent system, with soundness and completeness established for both calculi. Concluding remarks highlight the potential for expanding this study and suggest directions for future research. (shrink)

The aim of this paper is to provide a proof-theoretic characterization of relevant logics including fusion and fission connectives, as well as Ackermann’s truth constant. We achieve this by employing the well-established methodology of labelled sequent calculi. After having introduced several systems, we will conduct a detailed proof-theoretic analysis, show a cut-admissibility theorem, and establish soundness and completeness. The paper ends with a discussion that contextualizes our current work within the broader landscape of the proof theory of relevant logics.

The key idea of this paper is that grounds are a special kind of reasons, so their logic is part of the logic of reasons. We outline a natural deduction calculus that provides a basic formal characterization of reasons and enables us to obtain some distinctive and relatively uncontentious principles about grounds. Then we show that the calculus outlined is consistent and decidable, which we take to be an interesting result in its own right.

Resolution Matrix Semantics (RMS) introduces the alternative truth-value-based framework for modal logic, providing a substantive alternative to Kripke’s relational semantics of possible worlds. Drawing inspiration from Y. Ivlev’s substantive semantics, RMS utilizes a 4-valued structure—necessary truth (tn), contingent truth (tc), contingent false (fc), and necessary false (fn)—augmented by indeterminate values (t, f, t/f) to define modal systems Km, KDm, KTm, S4m, and S5m, analogous to Kripke’s K, KD, T, S4, and S5. By directly assigning determined and indeterminate truth values via (...) an interpretation function, RMS validates formulas without relying on accessibility relations, as demonstrated through soundness and completeness proofs and a tailored tableau method. The framework’s versatility extends to applications in deontic logic, artificial intelligence, and quantum computing, enabling context-dependent reasoning with computational efficiency. RMS thus bridges philosophical logic and practical domains, offering a truth-centric perspective on modal reasoning’s complexities. (shrink)

This paper introduces Resolution Matrix Semantics (RMS), a novel framework for modal logic that prioritizes indeterminate truth values and sub-interpretations over traditional relational structures, offering a poly-logic model that mirrors human cognition. Drawing on Vladimir Bibler’s concept of poly-logic substantive control, RMS captures the pluralistic, concurrent nature of human reasoning by evaluating logical formulas across multiple interpretive threads, resolving ambiguities akin to quantum cognitive processes. By integrating insights from quantum cognition, neuroscience, and parallel computing, the paper argues that RMS reflects (...) the brain’s intrinsic ability to navigate uncertainty through superposition-like states and entangled judgments. Philosophically, RMS challenges classical mono-logic assumptions, aligning with non-classical logics and pluralistic epistemologies. Its implications extend to AI, NLP, and quantum computing, promising rapid, precise decision-making under uncertainty. RMS thus redefines rationality as a dynamic, dialogic process, bridging logic, cognition, and the fundamental indeterminacy of reality. (shrink)

In this chapter, we will provide some background on the volume and the topics of the papers. We begin by presenting the context for the workshop that gave rise to the volume. We then present a short historical overview of relevant logics. We then provide context to situate each of the papers in this volume, organized by the topics of the parts, namely Philosophical Foundations, Model Theory, Proof Theory, and Applications.

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties—requiring that in valid entailments $A\to B$, some atom must appear in both A and B with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse (...) trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf {BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf {B}$ ) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan’s notion of hyperformalism, and for the very definition of a relevant logic itself. (shrink)

We explore depth substitution invariance, or hyperformalism, and extend known results in this realm to justification logics extending weak relevant logics. We then examine the surprising invariance of justifications over formulas and restrict our attention to the substitution of proofs in the original relevant logic. The results of this paper indicate that depth invariance is a recalcitrant feature of the logic and that proof structures in hyperformal logics are quite inflexible.

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of partitions. Or, putting (...) it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (= ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives). (shrink)

The systematic account of deductive reasoning and the development of a formal logic to reveal the principles of such reasoning began with Aristotle's syllogistic. It was a term logic, a logic that dominated the field until the rise of modern predicate logic at the end of the Nineteenth century. That system quickly supplanted the old logic of terms. However, in the middle of the Twentieth century Fred Sommers took up the challenge to build a revised and strengthened term logic, one (...) fit to challenge the hegemony of predicate logic. The aim was to devise a formal logic that could better serve as the logic of natural language. -/- In recent years a new group of logicians have taken this version of term logic into many new directions. This book presents here for the first time some of the best examples of their work with new essays by philosophers, logicians, mathematicians, computational theorists, and historians of logic. The topics addressed include relative terms, logical copulation, Aristotelian diagrams, modality, truth, semantics, epistemic term logic, non-classical quantifiers, and more. (shrink)

Timothy Williamson has recently argued that the applicability of classical mathematics in the natural and social sciences raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We first reconstruct his argument and present its restriction to the case of quantum logic (QL). Then we show that there is no problematic tension between the applicability of classical mathematical models to quantum phenomena and the endorsement of QL in the reasoning about the latter. Once we (...) identify the premise in Williamson’s argument that turns out to be false when restricted to QL, we argue that the same premise fails for a wider variety of non-classical logics. In the end, we use our discussion to draw some general lessons concerning the relationship between applied logic and applied mathematics. (shrink)

Sengzhao (c.374–414) was a Chinese Neo-Daoist who converted to Mahāyāna Buddhism, and few people doubt his influence on Chinese Buddhist philosophy. In this article, provided his Neo-Daoism (xuanxue) and Madhyamaka Buddhism, I will present how Sengzhao featured a symbolic meaning of ‘void’ (śūnya) as rooted originally in Daoism. The Daoist contradictions, in particular between ‘being’ (you) and ‘nothing [non-being]’ (wu), are essential to the development of his doctrine of ‘no ultimate void’ (不真空論, Buzhenkonglun). To understand what Sengzhao meant by ‘void’, (...) which is in denial about the ultimate reality, I broach a notion of nihil (‘nothing’ but also ‘no value’) that bears on his discursive practice. In this light, I formulate a Daoist argument for contradictions and ECN (ex contradictione nihil—nothing follows from contradictions) from Laozi’s Daodejing. Furthermore, I elaborate on Sengzhao’s defence of ECN in his Buzhenkonglun. Reconstructing his negative approach to contradictions within the scope of the four-valued expressions (catuṣkoṭi) in the Madhyamaka tradition from Nāgārjuna, I consider a likely objection that a fifth value such as the ineffable may be inferred as void. Instead of subsuming the ineffable value under his discourse, I finally endorse Sengzhao’s purpose of linguistic and conventional approximation to the ultimate reality as silence. As such, I conclude the significance of void in Sengzhao’s denials via contradictions (ECN), i.e. an early philosophical peak of Chinese Buddhism from Daoism. (shrink)

A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and a characterizing (...) frame property. While the identification of negation as impossibility and negation as unnecessity does solve Onishi’s problems, in general it nevertheless seems desirable to keep the two notions separate. The present paper addresses the first problem by introducing a reformulation of Onishi’s display sequent calculus in a language of sequents that incorporates Boolean negation. Instead of identifying negation as impossibility and negation as unnecessity, a notion of weak frame correspondence is defined. It is observed that DNE weakly corresponds with a certain frame property, and two structural sequent rules are presented that together also weakly correspond with that frame property and allow one to derive DNE. Moreover, the reformulated display calculus has an independent motivation by considerations on proof-theoretic semantics. (shrink)

This chapter responds to Saul Kripke’s critique of the idea of adopting an alternative logic. It defends an anti-exceptionalist view of logic, on which coming to accept a new logic is a special case of coming to accept a new scientific theory. The approach is illustrated in detail by debates on quantified modal logic. A distinction between folk logic and scientific logic is modelled on the distinction between folk physics and scientific physics. The importance of not confusing logic with metalogic (...) in applying this distinction is emphasized. Defeasible inferential dispositions are shown to play a major role in theory acceptance in logic and mathematics as well as in natural and social science. Like beliefs, such dispositions are malleable in response to evidence, though not simply at will. The chapter considers the Quinean objection that accepting an alternative logic involves changing the subject rather than denying the doctrine. The objection is shown to depend on neglect of the social dimension of meaning determination, akin to descriptivism about proper names and natural kind terms criticized by Kripke and Putnam. Normal standards of interpretation indicate that disputes between classical and non-classical logicians are genuine disagreements. (shrink)

This comprehensive volume features the proceedings of the Third International Workshop on Logics for New-Generation Artificial Intelligence and the International Workshop on Logic, AI and Law, held in Hangzhou, China on September 8-9 and 11-12, 2023. The collection offers a diverse range of papers that explore the intersection of logic, artificial intelligence, and law. With contributions from some of the leading experts in the field, this volume provides insights into the latest research and developments in the applications of logic in (...) these areas. It is an essential resource for researchers, practitioners, and students interested in the latest advancements in logic and its applications to artificial intelligence and law. (shrink)

As one of the basic tasks in natural language processing (NLP), named entity recognition (NER) is an important basic tool for downstream tasks of NLP, such as information extraction, syntactic analysis, machine translation and so on. The internal operation logic of the current name entity recognition model is black-box to the user, so the user has no basis to determine which name entity makes more sense. Therefore, a user-friendly explainable recognition process would be very useful for many people. In this (...) paper, we propose a novel interpretable method, BTPK (Binary Talmudic Public Announcement Logic model), to help users understand the internal recognition logic of the name entity recognition tasks based on Talmudic Public Announcement Logic. BTPK model can also capture the semantic information in the input sentences, that is, the context dependency of the sentence. We observed the public announcement of BTPK presents the inner decision logic of Bidirectional Recurrent Neural Networks (BRNNs), and the explanations obtained from a BTPK model show us how BRNNs essentially handle NER tasks. (shrink)

In this paper we present an overview, with historical and critical remarks, of two articles by S. Jaśkowski ([20, 21] 1948 and [22, 23] 1949), which contain the oldest known formulation of a paraconsistent logic. Jaśkowski has built the logic – he termed discussive (D2) – by defining two new connectives and by introducing a modal translation map from D2 systems into Lewis’ modal logic S5. Discussive systems, for their formal details and their original philosophical justification, have attracted discrete attention (...) among experts. Indeed, in what follows, after having introduced Jaśkowski methodology of building D2 and his main philosophical motivations for providing such a system, we will explore some of the main contributions to the development of D2. (shrink)

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with (...) semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

In his paper ‘The possibility of vagueness’ (Fine in Synthese 194(10):3699–3725, 2017), Kit Fine proposes a new logic of vagueness, CL, that promises to provide both a solution to the sorites paradox and a way to avoid the impossibility result from Fine (Philos Perspect 22(1):111–136, 2008). The present paper presents a challenge to his new theory of vagueness. I argue that the possibility theorem stated in Fine (Synthese 194(10):3699–3725, 2017), as well as his solution to the sorites paradox, fail in (...) certain reasonable extensions of the language of CL. More specifically, I show that if we extend the language with any negation operator that obeys reductio ad absurdum, we can prove a new impossibility result that makes the kind of indeterminacy that Fine takes to be a hallmark of vagueness impossible. I show that such negation operators can be conservatively added to CL and examine some of the philosophical consequences of this result. Moreover, I demonstrate that we can define a particular negation operator that behaves exactly like intuitionistic negation in a natural and unobjectionable propositionally quantified extension of CL. Since intuitionistic negation obeys reductio, the new impossibility result holds in this propositionally quantified extension of CL. In addition, the sorites paradox resurfaces for the new negation. (shrink)

There are fundamentally two mathematical logics. One the Boolean logic of subsets, usually presented today in the special case of propositional logic, which has many sublogics and extensions, the most important being the intuitionistic logic usually modeled by the open subsets of a topological space. The other co-fundamental mathematical logic is the topic of this book, the logic of partitions. We are using ”logic” in a mathematical sense as being about basic mathematical objects, subsets of a universe set or partitions (...) on a universe set.1 Logic, in this mathematical sense, is not about propositions, although, as with any mathematical theory, it involves propositions about the basic objects, e.g., that an element is in a subset or that a distinction is made by a partition. Moreover, by taking the universe set to be the one element set 1, with two subsets ∅ and 1, there is a special case of subset logic, namely propositional logic, that can be interpreted as being about propositions with ∅ representing falsehood and 1 representing truth. (shrink)

Brock and Glasgow recently introduced a new definition of paradox and argue that this conception of paradox itself leads to paradox, the so-called Paradox Paradox. I show that they beg the questions during the course of their argument, but, more importantly, do so in a philosophically interesting way: it reveals a counterexample to the equivalence between being a logical truth and having a probability of one. This has consequences regarding norms of rationality, undermining the grounds for the Conjunction Fallacy.

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; (...) if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

Philosophe japonais polyglotte au savoir encyclopédique, Yamauchi Tokuryū est à n’en point douter l’un des auteurs les moins étudiés de l’école de Kyōto. La présente étude vient corriger ce qui ne constitue rien d’autre qu’un accident de l’histoire, tant l’ampleur du projet philosophique de Yamauchi est à même de susciter l’intérêt du philosophe, du savant et de l’amateur cultivé. La démarche de ce penseur japonais, disciple de Nishida Kitarō, est remarquable en ce qu’il chercha à proposer un dépassement englobant de (...) ce qu’il nomme, d’une part, la « logique du logos » – qui figurait déjà chez Aristote et se structure autour des principes d’identité, de contradiction et du tiers exclu – et, d’autre part, la « logique du lemme » – méthode d’appréhension intuitive des étants qui, née en Inde, serait d’inspiration essentiellement bouddhique. Cette étude, dans laquelle les conceptions bouddhiques qui traversent le propos de Yamauchi sont explicitées, est essentiellement consacrée à l’analyse de ses deux essais les plus importants Logos et lemme (1974) et Philosophie de la latence (1993). Le lecteur y rencontrera une discussion tant de la logique aristotélicienne que d’un régime de logicité tel que le tétralemme, tant du coeur des philosophies européennes que de celui des pensées bouddhiques. Il y rencontrera en outre une discussion de l’idée de néant, de la causalité et de la question du fondement de l’être, nourrie de conceptions bouddhiques telles que la « coproduction conditionnée » (pratītya-samutpāda) et la « latence » (anuśaya). (shrink)

In this article, we perform a detailed proof theoretic investigation of a wide number of relevant logics by employing the well-established methodology of labelled sequent calculi to build our intended systems. At the semantic level, we will characterise relevant logics by employing reduced Routley-Meyer models, namely, relational structures with a ternary relation between worlds along with a unique distinct element considered as the real (or actual) world. This paper realizes the idea of building a variety of modular labelled calculi by (...) reflecting, at the syntactic level, semantic informations taken from reduced Routley-Meyer models. Central results include proofs of soundness and completeness, as well as a proof of cut- admissibility. (shrink)

This dissertation concerns dynamic semantics and the broader normative and epistemic consequences of theorizing with dynamic contents. Dynamic semantics deviates significantly from canonical approaches to meaning in that it treats the meanings of sentences as well as the contents of attitudes as context-change-potentials rather than propositions. While some of the consequences of this deviation have been recognized, several crucial consequences remain, heretofore, unexplained. In particular, I argue that dynamic theories not only differ from more traditional static theories with respect to (...) their preferred representational objects, but also with respect to how they must treat the normative statuses of these representational objects, their idealizing assumptions, and how content interfaces with choice and decision. In each case, I demonstrate how and why dynamic accounts differ and explore the consequences. In certain cases, I argue why these consequences ought to be embraced. In others, I provide solutions to the problems they pose. (shrink)

There was a time when 'logic' just meant classical logic. The climate is slowly changing and non-classical logic cannot be dismissed off-hand. However, a metatheory used to study the properties of non-classical logic is often classical. In this paper, we will argue that this practice of relying on classical metatheories is problematic. In particular, we will show that it is a bad practice because the metatheory that is used to study a non-classical logic often rules out the very logic it (...) is designed to study. (shrink)

In order to define some interesting consequence relations, certain generalizations have been proposed in a many-valued semantic setting that have been useful for defining what have been called pure, mixed and ordertheoretic consequence relations. But these generalizations are insufficient to capture some other interesting relations, like other intersective mixed relations (a relation that cannot be defined as a mixed relation, but only as the intersection of two mixed relations) or relations with a conjunctive (or, better, “universal”) interpretation for multiple conclusions. (...) We propose a broader framework to define these cases, and many others, and to set a common background that allows for a direct compared analysis. At the end of the work, we illustrate some of these comparisons. (shrink)

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...) extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics. (shrink)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...) of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

There is a tension between the definition of empty logic as a logic with no valid arguments and no valid meta-arguments, on the one hand, and the way in which we have usually interpreted the validity of meta-arguments, on the other. Here we argue that one way to eliminate the tension is understanding the “If. . . then. . . ” in a meta-argument, at least in the case of an empty logic, as a transplication (aka the de Finetti conditional) (...) instead of an extensional or material conditional. (shrink)

In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of (...) the systems have the variable sharing property). I emphasize the intuitive content of the semantics and discuss how natural informational considerations underly each of the examined systems. (shrink)

We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true of (...) all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi. (shrink)

This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic paradigms: labelled and nested sequent calculi. The formalism of labelled sequents has been successful in that cut-free calculi in possession of desirable proof-theoretic properties can be automatically generated for large classes of logics. Despite these qualities, labelled systems make use of a complicated syntax that explicitly incorporates the (...) semantics of the associated logic, and such systems typically violate the subformula property to a high degree. By contrast, nested sequent calculi employ a simpler syntax and adhere to a strict reading of the subformula property, making such systems useful in the design of automated reasoning algorithms. However, the downside of the nested sequent paradigm is that a general theory concerning the automated construction of such calculi (as in the labelled setting) is essentially absent, meaning that the construction of nested systems and the confirmation of their properties is usually done on a case-by-case basis. The refinement method connects both paradigms in a fruitful way, by transforming labelled systems into nested (or, refined labelled) systems with the properties of the former preserved throughout the transformation process. To demonstrate the method of refinement and some of its applications, we consider grammar logics, first-order intuitionistic logics, and deontic STIT logics. The introduced refined labelled calculi will be used to provide the first proof-search algorithms for deontic STIT logics. Furthermore, we employ our refined labelled calculi for grammar logics to show that every logic in the class possesses the effective Lyndon interpolation property. (shrink)

Bypassing Lewis’ Triviality Results. A Kripke-Style Partial Semantics for Compounds of Adams’ Conditionals Alberto Mura University of Sassari Abstract According to Lewis’ Triviality Results (LTR), conditionals cannot satisfy the equa­tion (E) P(C if A) = P(C | A), except in trivial cases. Ernst Adams (1975), however, provided a probabilistic semantics for the so-called simple conditionals that also sat­isfies equation (E) and provides a probabilistic counterpart of logical consequence (called p-entailment). Adams’ probabilistic semantics is coextensive to Stalnaker­Thomason’s (1970) and Lewis’ (1973) (...) semantics as far as simple conditionals are concerned. A theorem, proved in McGee 1981, shows that no truth-functional many-valued logic allows a relation of logical consequence coextensive with Adams’ p-entailment. This paper presents a modified modal (Kripke-style) version of de Finetti’s se­mantics that escapes McGee’s result and provides a general truth-conditional se­mantics for indicative conditionals. It agrees with Adams’ logic and is not affected by LTR. The new framework encompasses and extends Adams’ probabilistic se­mantics (APS) to compounds of conditionals. A generalised set of axioms for prob­ability over the set of tri-events is provided, which coincide with the standard axioms over the set of the two-valued ordinary sentences. -/- Keywords: Conditionals, Probability logic, de Finetti, Tri-events, Adams’ logic, Stal­naker’s thesis, Partial logic, Lewis’ triviality results, Ramsey test. (shrink)

We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If ∼A, then A, should not hold, since the conditional’s antecedent ∼A contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event A|~A is p(A|~A)=0 . Moreover, connexive logics aim to capture the intuition that conditionals should express (...) some “connection” between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle’s Theses, Aristotle’s Second Thesis, Abelard’s First Principle and selected versions of Boethius’ Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles. (shrink)