In one of their papers, Michael De and Hitoshi Omori observed that the notion of classical negation is not uniquely determined in the context of so-called Belnap-Dunn logic, and in fact there are 16 unary operations that qualify to be called classical negation. These varieties are due to different falsity conditions one may assume for classical negation. The aim of this paper is to observe that there is an interesting way to make sense of (...) classical negation independent of falsity conditions. We discuss two equivalent semantics, and offer a Hilbert-style system that is sound and complete with respect to the semantics. (shrink)

*These notes were folded into the published paper "Probability and nonclassical logic*. Revising semantics and logic has consequences for the theory of mind. Standard formal treatments of rational belief and desire make classical assumptions. If we are to challenge the presuppositions, we indicate what is kind of theory is going to take their place. Consider probability theory interpreted as an account of ideal partial belief. But if some propositions are neither true nor false, or are half true, or whatever—then (...) it’s far from clear that our degrees of belief in it and its negation should sum to 1, as classical probability theory requires (?, cf.). There are extant proposals in the literature for generalizing (categorical) probability theory to a non-classical setting, and we will use these below. But subjective probabilities themselves stand in functional relations to other mental states, and we need to trace the knock-on consequences of revisionism for this interrelationship (arguably, degrees of belief only count as kinds of belief in virtue of standing in these functional relationships). (shrink)

Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic.

We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities (...) between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan. (shrink)

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.

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those (...) models. I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument. (shrink)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is (...) defined in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

Double negations are easily recognised in both the so-called “negative literature” and the original texts of some important scientific theories. Often they are not equivalent to the corresponding affirmative propositions. In the case the law of double negation fails they belong to non-classical logic, as first, intuitionist logic. Through a comparative analysis of the theories including them the main features of a new kind of theoretical organization governed by intuitionist logic are obtained. Its arguing proceeds through doubly negated (...) propositions and ad absurdum arguments. Then, the final doubly negated predicate is translated in the corresponding affirmative one, from which all the consequences are deduced in order to test them against reality. According to this model the true formalization of intuitionist logic is Kolmogorov’s 1932 paper. The above analysis also shows that modal words play an auxiliary or substitutive role to intuitionist propositions. In addition, it is shown that Vasiliev’s paraconsistent logic is represented through doubly negated propositions. By means of these three kinds of logic one can attribute a specific kind of logic to each of the three distinct steps of the process of building a scientific theory from the experimental data to its final theoretical system. It is also shown that the definition of negation is of a structural kind, rather than of an objective kind or a subjective kind: it has to be referred to the organization of the theory to which it belongs. (shrink)

Frege-Geach worries about embedding and composition have plagued metaethical theories like emotivism, prescriptivism and expressivism. The sharpened point of such criticism has come to focus on whether negation and inconsistency have to be understood in descriptivist terms. Because they reject descriptivism, these theories must offer a non-standard account of the meanings of ethical and normative sentences as well as related semantic facts, such as why certain sentences are inconsistent with each other. This paper fills out such a solution to (...) the negation problems, following some of the original emotivist ideas about the interplay of interests in conversation. We communicate both to share information and coordinate our actions, and we use distinctively normative language like deontic ‘must’ and ‘may’ to negotiate what people are to do. The kinds of disagreement involved in such negotiation can illuminate the issues with negation and inconsistency. This paper outlines a dynamic semantic system in which these ideas can bear fruit, developing the scorekeeping model of conversation. The result is clarification about what Frege-Geach worries can mean for nondescriptive semantics. (shrink)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of (...) this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, (...) and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

The focus of this paper are Dummett's meaning-theoretical arguments against classical logic based on consideration about the meaning of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett's outlook on the theory of meaning. (...) In particular, I shall grant Dummett his general bias towards verificationism, encapsulated in the slogan 'meaning is use'. The second general assumption I see no need to question is Dummett's particular breed of molecularism. Some of Dummett's assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inference in the Dummettian framework. (shrink)

I show that standard dynamic approaches to the semantics of epistemic modals invalidate the classical laws of excluded middle and non-contradiction, as well as the law of epistemic non-contradiction. I argue that these facts pose a serious challenge.

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory (...) validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to--whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities. (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)

This work contributes to the theory of judgement aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgement aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgement aggregation in non-classical logics is that they offer a number of modelling choices to represent agents’ reasoning in aggregation problems. By studying judgement aggregation (...) in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems. (shrink)

Formal ontologies are nowadays widely considered a standard tool for knowledge representation and reasoning in the Semantic Web. In this context, they are expected to play an important role in helping automated processes to access information. Namely: they are expected to provide a formal structure able to explicate the relationships between different concepts/terms, thus allowing intelligent agents to interpret, correctly, the semantics of the web resources improving the performances of the search technologies. Here we take into account a problem regarding (...) Knowledge Representation in general, and ontology based representations in particular; namely: the fact that knowledge modeling seems to be constrained between conflicting requirements, such as compositionality, on the one hand and the need to represent prototypical information on the other. In particular, most common sense concepts seem not to be captured by the stringent semantics expressed by such formalisms as, for example, Description Logics (which are the formalisms on which the ontology languages have been built). The aim of this work is to analyse this problem, suggesting a possible solution suitable for formal ontologies and semantic web representations. The questions guiding this research, in fact, have been: is it possible to provide a formal representational framework which, for the same concept, combines both the classical modelling view (accounting for compositional information) and defeasible, prototypical knowledge ? Is it possible to propose a modelling architecture able to provide different type of reasoning (e.g. classical deductive reasoning for the compositional component and a non monotonic reasoning for the prototypical one)? We suggest a possible answer to these questions proposing a modelling framework able to represent, within the semantic web languages, a multilevel representation of conceptual information, integrating both classical and non classical (typicality based) information. Within this framework we hypothesise, at least in principle, the coexistence of multiple reasoning processes involving the different levels of representation. (shrink)

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an (...) agent values accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings. (shrink)

In this paper, by suggesting a formal representation of science based on recent advances in logic-based Artificial Intelligence (AI), we show how three serious concerns around the realisation of traditional scientific realism (the theory/observation distinction, over-determination of theories by data, and theory revision) can be overcome such that traditional realism is given a new guise as ‘naturalised’. We contend that such issues can be dealt with (in the context of scientific realism) by developing a formal representation of science based on (...) the application of the following tools from Knowledge Representation: the family of Description Logics, an enrichment of classical logics via defeasible statements, and an application of the preferential interpretation of the approach to Belief Revision. (shrink)

At first sight the Theory of Computation i) relies on a kind of mathematics based on the notion of potential infinity; ii) its theoretical organization is irreducible to an axiomatic one; rather it is organized in order to solve a problem: “What is a computation?”; iii) it makes essential use of doubly negated propositions of non-classical logic, in particular in the word expressions of the Church-Turing’s thesis; iv) its arguments include ad absurdum proofs. Under such aspects, it is like (...) many other scientific theories, in particular the first theories of both mechanical machines and heat machines. A more accurate examination of Theory of Computation shows a difference from the above mentioned theories in its essentially including an odd notion, “thesis”, to which no theorem corresponds. On the other hand, arguments of each of the other theories conclude a doubly negative predicate which then, by applying the inverse translation of the ‘negative one’, is translated into the corresponding affirmative predicate. By also taking into account three criticisms to the current Theory of Computation a rational re-formulation of it is sketched out; to Turing-Church thesis of the usual theory corresponds a similar proposition, yet connecting physical total computation functions with constructive mathematical total computation functions. -/- . (shrink)

This paper develops co-ordinated multiple-domain supervenience relations to model determination and dependence relations between complex entities and their constituents by appealing to R-related pairs and by making use of associated isomorphisms. Supervenience relations are devised for order-sensitive and repetition-sensitive mereologies, for mereological systems that make room for many-many composition relations, as well as for hierarchical mereologies that incorporate compositional and hylomorphic structure. Finally, mappings are provided for theories that consider wholes to be prior to their parts.

Why must incompatibility be symmetric? An odd question, but recent work in the semantics of non-classical logic, which appeals to the notion of incompatibility as a primitive and defines negation in terms of it, has brought this question to the fore. Francesco Berto proposes such a semantics for negation argues that, since incompatibility must be symmetric, double negation introduction must be a law of negation. However, he offers no argument for the claim that incompatibility really (...) must be symmetric. Here, I provide such an argument, showing that, insofar as we think of incompatibility in normative pragmatic terms, it can play its basic pragmatic function only if it is symmetric. The upshot is that we can vindicate Berto’s claim about the symmetry of incompatibility but only if we, pace Berto, think about incompatibility, in the first instance, as a pragmatic relation between acts rather than a semantic relation between contents. (shrink)

Val Plumwood’s 1993 paper, “The politics of reason: towards a feminist logic” (hence- forth POR) attempted to set the stage for what she hoped would begin serious feminist exploration into formal logic – not merely its historical abuses, but, more importantly, its potential uses. This work offers us: (1) a case for there being feminist logic; and (2) a sketch of what it should resemble. The former goal of Plumwood’s paper encourages feminist theorists to reject anti-logic feminist views. The paper’s (...) latter aim is even more challenging. Plumwood’s critique of classical negation (and classical logic) as a logic of domination asks us to recognize that particular logical systems are weapons of oppression. Against anti-logic feminist theorists, Plumwood argues that there are other logics besides classical logic, such as relevant logics, which are suited for feminist theorizing. Some logics may oppress while others may liberate. We provide details about the sources and context for her rejection of classical logic and motivation for promoting relevant logics as feminist. (shrink)

This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered. First Part of this book presents some (...) theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes. Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification. Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well. (shrink)

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and (...) Klein bottle logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

In 1597 Francisco Suárez published a comprehensive treatise on beings of reason (entia rationis) as part of his Disputationes metaphysicae. Subsequent scholastic philosophers vigorously debated various aspects of Suárez’s theory. The aim of this paper is to identify some of the most controversial points of these debates, as they developed in the first half of the seventeenth century. In particular, I focus on the intension and the extension of ‘ens rationis’, its division (into negations, privations and relations of reason) and (...) its causes. Additionally, I will discuss how Suárez’s views sparked a number of debates within the classical view, debates which ultimately led to the emergence of various alternative theories, especially among the Jesuits. These non-classical views radically revise the previous classical conception of beings of reason. (shrink)

What were the true reasons of the second scientific revolution? – To answer the question, the epistemic model is applied, according to which radical breakthroughs in science were not due to the fanciful excogitation of new ideas ‘ex nihilo’, but rather to the tedious, long-term and troublesome processes of the mutual lapping, reconciliation, interpenetration and intertwinement of ‘old’ research traditions preceding such breaks. It is contended that Einstein's 'annus mirabilis' constituted an acme of the second scientific revolution. To fathom in (...) what felicitous ways Einstein's 1905 writings hang together one is bound to pay special tribute to his longed strive for unity evinced in incessant attempts to coordinate the profound research traditions of classical physics. Though Einstein's efforts sprung out of Max Planck's pioneering attempts to comprehend electromagnetic phenomena through the lenses of conceptual structures of statistical thermodynamics . It was Planck, who clearly realized the cross-contradiction between 'the physics of material bodies’ and ‘the physics of the ether' and outlined the sketch of its withdrawal: the paradigms 'must be modified to remain compatible'. And it was Planck who took the first step in modifying the physics of the ether and contending that 'not only matter itself but also the effects radiated from matter' possess discontinuous properties, which can be characterized by a new natural constant: the elementary quantum of action. Einstein's part consisted in that he took the second step in modifying the second component of the encounter – the physics of material bodies. Einstein’s foolhardy light quanta hypothesis and distinctive special theory of relativity turn out to be mere milestones of the unwinding of Maxwellian electrodynamics and statistical thermodynamics reconcilement research program. The notorious conception of luminiferous ether was a substantial obstacle for Einstein’s wayward statistical thermodynamics in which the pivotal lead was played by flagrant light quanta paper. Einstein was aware that his enticing light quanta hypothesis was too audacious to be taken literally and he laid out his version of 'electrodynamics of moving bodies' in a markedly Machian/ Duhemian, phenomenological way. As a result of the revision of the second scientific revolution key episode, the arguments are exhibited in favor of the necessity to modify the history of the genesis of general relativity; correspondingly, the process of unification of electromagnetic and weak interactions that took place in the second half of the XX-th century is scrutinized. Eventually, the encounter and interpenetration of the two dominating research traditions of modern physics – that of general relativity and quantum field theory – is elicited. (shrink)

This paper proposes a semantics for free choice permission that explains both the non-classical behavior of modals and disjunction in sentences used to grant permission, and their classical behavior under negation. It also explains why permissions can expire when new information comes in and why free choice arises even when modals scope under disjunction. On the proposed approach, deontic modals update preference orderings, and connectives operate on these updates rather than propositions. The success of this approach stems (...) from its capacity to capture the difference between expressing the preferences that give rise to permissions and conveying propositions about those preferences. (shrink)

Drawing on the puzzling behavior of ordinary knowledge ascriptions that embed an epistemic (im)possibility claim, we tentatively conclude that it is untenable to jointly endorse (i) an unfettered classical logic for epistemic language, (ii) the general veridicality of knowledge ascription, and (iii) an intuitive ‘negative transparency’ thesis that reduces knowledge of a simple negated ‘might’ claim to an epistemic claim without modal content. We motivate a strategic trade-off: preserve veridicality and (generalized) negative transparency, while abandoning the general validity of (...) contraposition. We criticize various approaches to incorporating veridicality into domain semantics, a paradigmatic ‘information-sensitive’ framework for capturing negative transparency and, more generally, the non-classical behavior of sentences with epistemic modals. We then present a novel information-sensitive semantics that successfully executes our favored strategy: stable acceptance semantics, extending a vanilla bilateral state-based semantics for epistemic modals with a knowledge operator loosely inspired by the defeasibility theory of knowledge. (shrink)

The present paper wants to promote epistemic pluralism as an alternative view of non-classical logics. For this purpose, a bilateralist logic of acceptance and rejection is developed in order to make an important di erence between several concepts of epistemology, including information and justi cation. Moreover, the notion of disagreement corresponds to a set of epistemic oppositions between agents. The result is a non-standard theory of opposition for many-valued logics, rendering total and partial disagreement in terms of epistemic (...) class='Hi'>negation and semi-negations. (shrink)

The review gives a short description of the content of the book and discusses the treatment of conditionals in it.

This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to (...) extend this constructive method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to (...) languages containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)

The main subject of Cusanus’ investigations was the name of God. He claimed to have achieved the best possible one, Not-Other. Since Cusanus stressed that these two words do not mean the corresponding affirmative word, i.e. the same, they represent the failure of the double negation law and there￾fore belong to non-classical, and above all, intuitionist logic. Some of his books implicitly applied intuitionist reasoning and the corresponding organization of a theory which is governed by intuitionist logic. A (...) comparison of two of Cusanus’ short writings shows that throughout his life he substantially improved his use of this kind of logic and ultimately was able to reason consistently within such a logic and recognize some of its basic laws. One important idea developed by him was that of a proposition composed of a triple repetition of “not-other” expressing “the Tri-unity of concordance” i.e. the “best name for the Trinity”. I complete his application of intuitionist logic to theological subjects by charac￾terizing the inner relationships within the Trinity in such a way that there are no longer contradictions in the notion. Generally speaking, the notion of the Trin￾ity implies a translation from intuitionist to classical logic, to which Cusanus closely approximated. Moreover, I show that the main aspects of Christian rev￾elation, including Christ’s teachings, are represented both by this translation and by some doubly negated propositions of intuitionist logic. Hence, intuition￾ist logic was introduced into the history of Western theological thinking with Christian revelation, as only Cusanus partly recognized. Appendix 1 summa￾rizes a detailed analysis of Cusanus’ second short writing. Appendix 2 shows that the Athanasian creed regarding the Christian Trinity is a consistent se￾quence of intuitionist propositions provided that some verbal emendations are added, showing that ancient trinitarian thinking was also close to intuitionist reasoning. (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)

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds (...) of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s (...) paraconsistent Set Theories〖NF〗n^C. (shrink)

I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of specific states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant implication. I then (...) investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics. (shrink)

For the classical Geometry, in a geometrical space, all items (concepts, axioms, theorems, etc.) are totally (100%) true. But, in the real world, many items are not totally true. The NeutroGeometry is a geometrical space that has some items that are only partially true (and partially indeterminate, and partially false), and no item that is totally false. The AntiGeometry is a geometrical space that has some item that are totally (100%) false. While the Non-Euclidean Geometries [hyperbolic and elliptic geometries] (...) resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom [and in general: theorem, concept, idea etc.] and even of more axioms [theorem, concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates, Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of any axiom (or concept, theorem, idea, etc.). Clearly, the AntiGeometry is a generalization of Non-Euclidean Geometries. (shrink)

A formal theory of oppositions and opposites is proposed on the basis of a non- Fregean semantics, where opposites are negation-forming operators that shed some new light on the connection between opposition and negation. The paper proceeds as follows. After recalling the historical background, oppositions and opposites are compared from a mathematical perspective: the first occurs as a relation, the second as a function. Then the main point of the paper appears with a calculus of oppositions, by means (...) of a non-Fregean semantics that redefines the logical values of various sorts of sentences. A num- ber of topics are then addressed in the light of this algebraic semantics, namely: how to construct value-functional operators for any logical opposition, beyond the classical case of contradiction; Blanché's "closure problem", i.e., how to find a complete structure connecting the sixteen binary sentences with one another. All of this is meant to devise an abstract theory of opposition: it encompasses the relation of consequence as subalternation, while relying upon the use of a primary "proto- negation" that turns any relatum into an opposite. This results in sentential negations that proceed as intensional operators, while negation is broadly viewed as a difference-forming operator without special constraints on it. (shrink)

The paper starts with some textual distinctions concerning the concept of God in the metaphysical framework of two classical schools of Hindu philosophy, Sāṃkhya and Yoga. Then the author focuses on the functional and pedagogical aspects of prayer as well as practical justification of “religious meditation” in both philosophical schools. A special attention is put on the practice called īśvarapraṇidhāna, recommended in Yoga school, which is interpreted by the author as a form of non-theistic devotion. The meaning of the (...) central object of this concentration, that is puruṣa-viśeṣa, is reconsidered in detail. The subject matter is discussed in the wider context of yogic self-discipline that enables a practitioner to overcome ignorance ( avidyā) and the narrowness of egotic perspective (asmitā), recognized in the Hindu darśanas as the root-cause of all suffering or never-fulfilled-satisfaction ( duḥkha). The non-theistic devotion and spiritual pragmatism assumed by the adherents of Sāṃkhya-Yoga redefines the concept of “God” ( īśvara) as primarily an object of meditative practice and a special tool convenient for spiritual pedagogy. (shrink)

The aim of this paper is to explore the peculiar case of infectious logics, a group of systems obtained generalizing the semantic behavior characteristic of the -fragment of the logics of nonsense, such as the ones due to Bochvar and Halldén, among others. Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as logics of formal inconsistency and logics of formal undeterminedness.

Many think that expressivists have a special problem with negation. I disagree. For if there is a problem with negation, I argue, it is a problem shared by those who accept some plausible claims about the nature of intentionality. Whether there is any special problem for expressivists turns, I will argue, on whether facts about what truth-conditions beliefs have can explain facts about basic inferential relations among those beliefs. And I will suggest that the answer to this last (...) question is, on most plausible attempts at solving the problem of intentionality, ‘no’. (shrink)

In a series of articles, Kit Fine presents some highly compelling objections to monism, the doctrine that spatially coincident objects are identical. His objections rely on Leibniz’s Law and linguistic environments that appear to be immune to the standard charge of non-transparency and substitution failure. In this paper, I respond to Fine’s objections on behalf of the monist. Following Benjamin Schnieder, I observe that arguments from Leibniz’s Law are valid only if they involve descriptive, rather than metalinguistic, negation. Then (...) I show that the monist is justified in treating the negation in Fine’s objections as metalinguistic in nature. Along the way I make a few methodological remarks about the interaction between the study of natural language and metaphysics. I also present evidence that some of the linguistic environments which Fine relies on are, contrary to appearances, non-transparent. (shrink)

Currently there are at least four sizeable projects going on to establish the gravitational acceleration of massive antiparticles on earth. While general relativity and modern quantum theories strictly forbid any repulsive gravity, it has not yet been established experimentally that gravity is attraction only. With that in mind, the Elementary Process Theory (EPT) is a rather abstract theory that has been developed from the hypothesis that massive antiparticles are repulsed by the gravitational field of a body of ordinary matter: the (...) EPT essentially describes the elementary processes by which the smallest massive systems have to interact with their environments for repulsive gravity to exist. In this paper we model a nonrelativistic, one-component massive system that evolves in time by the processes as described by the EPT in an environment described by classical fields: the main result is a semi-classical model of a process at Planck scale by which a non-relativistic onecomponent system interacts with its environment, such that the interaction has both gravitational and electromagnetic aspects. Some worked-out examples are provided, among which the repulsion of an antineutron by the gravitational field of the earth. The general conclusion is that the semi-classical model of the EPT corresponds to non-relativistic classical mechanics. Further research is aimed at demonstrating that the EPT has a model that reproduces the successful predictions of general relativity. (shrink)

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic (...) as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. (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)

Classical acquaintance theory is any version of classical foundationalism that appeals to acquaintance in order to account for non-inferential justification. Such theories are well suited to account for a kind of infallible non-inferential justification. Why am I justified in believing that I’m in pain? An initially attractive (partial) answer is that I’m acquainted with my pain. But since I can’t be acquainted with what isn’t there, acquaintance with my pain guarantees that I’m in pain. What’s less clear is (...) whether, given classical acquaintance theory, it’s possible to have non-inferential justification to believe something false. Classical acquaintance theorists try to make room for such a possibility, but I argue that the attempts of Richard Fumerton, Ali Hasan, and Evan Fales are inadequate. I’ll focus on introspective justification, but similar issues arise for a priori justification as well. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture (...) the following results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation of the standard (...) meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)