Nonclassical Logic, Misc

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.

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)

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)

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.

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)

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)

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)

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)

This dissertation offers a proof of the logical possibility of testing empirical/factual theories that are inconsistent, but non-trivial. In particular, I discuss whether or not such theories can satisfy Popper's principle of falsifiablility. An inconsistent theory Ƭ closed under a classical consequence relation implies every statement of its language because in classical logic the inconsistency and triviality are coextensive. A theory Ƭ is consistent iff there is not a α such that Ƭ ⊢ α ∧ ¬α, otherwise it is inconsistent. (...) We say, instead, that Ƭ is non-trivial iff there is at least one α such that Ƭ ⊢ α, otherwise we say that it trivial. This happens because classical logic satisfies the principle of explosion, according ex contradictione sequitur quodlibet (from a contradiction anything follows). Under these conditions inconsistent classical theories would be compatible with any well-formed formula, which makes them useless for science. There are, however, so-called paraconsistent logics in which the principle of explosion does not generally hold and in which a theory can be (simply) inconsistent, but also absolutely consistent. It is in this logical framework that we can prove that some inconsistent theories can be falsifiable. (shrink)

Could there be a single logical system that would allow us to work simultaneously with classical, paraconsistent, and paracomplete negations? These three negations were separately studied in logics whose negations bear their names. Initially we will restrict our analysis to propositional logics by analyzing classical negation, ¬c, as treated by Classical Propositional Logic (LPC); the paraconsistent negation, ¬p, as treated through the hierarchy of Paraconsistent Propositional Calculi Cn (0 ≤ n ≤ ω); and the paracomplete negation, ¬q, as treated by (...) the hierarchy of Paracomplete Propositional Calculi Pn (0 ≤ n ≤ ω). In “Logics that are both paraconsistent and paracomplete” (1989), Newton da Costa proposed a system with approximate characteristics to what we are looking for. In the hierarchy of Non-Alethical Propositional Calculi Nn (0 ≤ n ≤ ω), only one negation is introduced (as primitive), called a “non-alethic” (¬n), whose operation preserves the properties of classical, or paraconsistent or paracomplete negation -- depending on the well or ill behavior of the formula connected to it. However, as we shall see, in the hierarchy Nn we can not reiterate negations with different behaviors in a same formula (e.g., ¬p¬cα or ¬q¬c¬p α), or even analyze a formula like ¬cα → ¬pα. In view of these problems, can we really say that the hierarchy Nn allows us to understand the relationships and interactions of the three types of negations? In order to deal with this, given the initial problem, we will present four axiomatic systems (KG) in which, unlike Nn, the three negations are directly introduced -- offering a semantics and a method of proofs by analytic tableaux. Through the KG Systems we will show how the negations interact, obtaining non-demonstrable theorems in LPC, Cn, Pn, and Nn (0 ≤ n ≤ ω). Finally, we will also offer a first-order extension for the KG Systems. (shrink)

I argue against abductivism about logic, which is the view that rational theory choice in logic happens by abduction. Abduction cannot serve as a neutral arbiter in many foundational disputes in logic because, in order to use abduction, one must first identify the relevant data. Which data one deems relevant depends on what I call one's conception of logic. One's conception of logic is, however, not independent of one's views regarding many of the foundational disputes that one may hope to (...) solve by abduction. (shrink)

The first degree entailment (FDE) family is a group of logics, a many-valued semantics for each system of which is obtained from classical logic by adding to the classical truth-values true and false any subset of {both, neither, indeterminate}, where indeterminate is an infectious value (any formula containing a subformula with the value indeterminate itself has the value indeterminate). In this paper, we see how to extend a version of star semantics for the logics whose many-valued semantics lack indeterminate to (...) star semantics for logics whose many-valued semantics include indeterminate. The equivalence of the many-valued semantics and star semantics is established by way of a soundness and completeness proof. The upshot of the novel semantics in terms of the applied semantics of these logics, and specifically infectiousness, is explored, settling on the idea that infectiousness concerns ineffability. (shrink)

Logic systems that can handle contradictions were being used for some time without having a general technical name. One of the main proposers of these systems, Newton da Costa, asked Francisco Miró Quesada to suggest him a name for those systems. In the historical letter that here we translate into English for the first time, Miró Quesada suggests three names to da Costa for this purpose: ‘ultraconsistent’, ‘metaconsistent’, and ‘paraconsistent’; explaining their pros and cons. -/- Paper based on a letter (...) by Francisco Miró Quesada Cantuarias to Newton da Costa, edited, translated, and annotated by Luis Felipe Bartolo Alegre. (shrink)

En este trabajo presentaré una forma de evitar los problemas más recurrentes en cierta versión del pluralismo lógico, aquella que defiende que incluso considerando un lenguaje fijo existen múltiples sistemas lógicos legítimos. Para ello, será necesario considerar los puntos de partida del programa pluralista y explicitar los problemas que de ellos surgen, principalmente el Desafío de Quine y el Problema del Colapso. Luego, propondré una modificación respecto de lo que se entiende por consecuencia lógica, para poder considerar una familia de (...) sistemas lógicos, las lógicas mixtas, que abarcan tanto a las lógicas puras como a las impuras. Finalmente, mostraré que con una interpretación razonable del formalismo se puede eludir aquellos problemas a la vez que se respeta el espíritu del programa pluralista. (shrink)

Neste artigo discutimos alguns aspectos da lógica nos dias atuais. O propósito central é mostrar a evolução dessa disciplina. Começamos com uma breve introdução onde especificamos o que queremos dizer com o termo “lógica”. A seguir, exporemos e discutiremos o que consideramos ser algumas das principais áreas de investigação da lógica atual. Concluímos o artigo com algumas observações sobre lógicas não clássicas e seus impactos sobre a filosofia. Ao final do texto se encontram mais detalhes que apontam para um aprofundamento (...) e uma ampliação das questões aqui tratadas. (shrink)

The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic method is also defined. In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions.

This paper formalizes part of the cognitive architecture that Kant develops in the Critique of Pure Reason. The central Kantian notion that we formalize is the rule. As we interpret Kant, a rule is not a declarative conditional stating what would be true if such and such conditions hold. Rather, a Kantian rule is a general procedure, represented by a conditional imperative or permissive, indicating which acts must or may be performed, given certain acts that are already being performed. These (...) acts are not propositions; they do not have truth-values. Our formalization is related to the input/ output logics, a family of logics designed to capture relations between elements that need not have truth-values. In this paper, we introduce KL3 as a formalization of Kant’s conception of rules as conditional imperatives and permissives. We explain how it differs from standard input/output logics, geometric logic, and first-order logic, as well as how it translates natural language sentences not well captured by first-order logic. Finally, we show how the various distinctions in Kant’s much-maligned Table of Judgements emerge as the most natural way of dividing up the various types and sub-types of rule in KL3. Our analysis sheds new light on the way in which normative notions play a fundamental role in the conception of logic at the heart of Kant’s theoretical philosophy. (shrink)

In this work we answer a long standing request for temporal embeddings of deontic STIT logics by introducing the multi-agent STIT logic TDS . The logic is based upon atemporal utilitarian STIT logic. Yet, the logic presented here will be neutral: instead of committing ourselves to utilitarian theories, we prove the logic TDS sound and complete with respect to relational frames not employing any utilitarian function. We demonstrate how these neutral frames can be transformed into utilitarian temporal frames, while preserving (...) validity. Last, we discuss problems that arise from employing binary utility functions in a temporal setting. (shrink)

En el presente artículo, introducimos el conjunto plitogénico (como generalización de conjuntos nítidos, borrosos, intuicionistas, borrosos y neutrosóficos), que es un conjunto cuyos elementos se caracterizan por los valores de muchos atrib utos. Un valor de atributo v tiene un grado correspondiente (difuso, intuicionista difuso o neutrosófico) de pertenencia d (x, v) del elemento x, al conjunto P, con respecto a algunos criterios dados. Para obtener una mejor precisión para los operadores d e agregación plitogénica en el conjunto plitogénico, y (...) para una inclusión más exacta (orden parcial), se define un grado de contradicción (disimilitud difusa, intuicionista difusa o neutrosófica) entre cada atributo del valor y el valor del atributo domina nte (el más importante). La intersección y la unión plitogénicas son combinaciones lineales de los operadores difusos t norm y tconorm, mientras que el complemento plitogénico, la inclusión (desigualdad), la igualdad está influenciados por los grados de co ntradicción (disimilitud) de los valores de los atributos. Por tal motivo el objetivo del presente trabajo es ofrecer algunos ejemplos y aplicaciones de los nuevos conceptos que se proponen, para su aplicación en la vida cotidiana. (shrink)

The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic 2Int, which Wansing presents in [2016a]. There he also gives a natural deduction system for this logic, N2Int, to which SC2Int is equivalent in terms of what is derivable. What is important is that these calculi represent a kind of bilateralist reasoning, since they (...) do not only internalize processes of verifcation or provability but also the dual processes in terms of falsifcation or what is called dual provability. In [Wansing, 2017] a normal form theorem for N2Int is stated, here, I want to prove a cut-elimination theorem for SC2Int, i.e., if successful, this would extend the results existing so far. (shrink)