Classical logic is often characterized through certain laws such as bi-valence and sharpness of concepts, among others. My view is that its most fundamental feature is a commitment to an objective conception of truth, which goes together with a realistic metaphysical view. Truth is objective in that it derives from the nature of reality, and is not dependent on beliefs, theories, practices, and the like. Classical logic is a theory of logical properties, logical truths, and logical (...) states of affairs. (shrink)

Connexive logics are based on two ideas: that no statement entails or is entailed by its own negation (this is Aristotle’s thesis) and that no statement entails both something and the negation of this very thing (this is Boethius' thesis). Usually, connexive logics are contra-classical. In this note, I introduce a reading of the connexive theses that makes them compatible with classical logic. According to this reading, the theses in question do not talk about validity alone; rather, (...) they talk in part about (a property related to) the soundness of arguments. (shrink)

Reasoning with conditionals is often thought to be non-monotonic, but there is no incompatibility with classical logic, and no need to formalise inference itself as probabilistic. When the addition of a new premise leads to abandonment of a previously compelling conclusion reached by modus ponens, for example, this is generally because it is hard to think of a model in which the conditional and the new premise are true.

In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc present novel and striking results about meta-inferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of non-classical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n, there (...) is a logic which agrees with classical logic on tautologies, inferences, meta-inferences, meta-meta-inferences, meta-meta-...-meta-inferences, but that disagrees with classical logic on n + 1-meta-inferences. They suggest that this shows that classical logic can only be characterized by defining its valid inferences at all orders. In this article, I invoke some simple symmetric generalizations of BPS’s results to show that the problem is worse than they suggest, since in fact there are logics that agree with classical logic on inferential validity to all orders but still intuitively differ from it. I then discuss the relevance of these results for truth theory and the classification problem. (shrink)

In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, intuitionistic, (...) paraconsistent and quantum logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages. (shrink)

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett (...) gives a mistaken elaboration of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen's. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic. (shrink)

Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will (...) encounter, originally as an example of what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (" demi-negation" ) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contra- intuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki's to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the "faithfully" means here is that not only is the translation of anything provable in the 'source' logic.. (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)

In the present paper we give the first proof-theoretical example of an embedding of classical logic into a quantum-like logic. This is performed in the framework of basic logic, where a proof-theoretical approach to quantum logic is convenient. We consider basic orthologic, that corresponds to a sequential formulation of paraconsistent quantum logic, and which is given by basic orthologic added with weakening and contraction, in a language with Girard's negation. In the paper we first (...) consider a convenient cut-free calculus for classical logic, in the same language; then, in a language enriched with a new kind of literals, we introduce basic orthologic with a primitive modality, where classical logic is embeddable. Similarly to Girard's negation, our modality is not a connective but conceivable as an operator defined inductively on the set of formulae. This allows us to obtain a calculus which enjoys cut-elimination. (shrink)

We describe here a simple method in order to obtain programs from proofs in second-order classical logic. Then we extend to classical logic the results about storage operators proved by Krivine for intuitionistic logic. This work generalizes previous results of Parigot.

The liar and kindred paradoxes show that we can derive contradictions when we reason in accordance with classical logic from the schema (T) about truth: S is true iff p, where ‘p’ is to be replaced with a sentence and ‘S’ with a name of that sentence. The paper presents two arguments to the effect that the blame lies not with (T) but with classical logic. The arguments derive contradictions using classical logic, but instead (...) of appealing to (T), they invoke semantic claims that seem even harder to reject. The first argument relies on two standard semantic principles that are not disquotational and on the claim that if there is such a thing as the property of being true, then ‘true’ expresses that property. The second argument relies on a schema about meaning: S means that p, where ‘S’ and ‘p’ are to be replaced as before. (shrink)

I show that it is not possible to uniquely characterize classical logic when working within classical set theory. By building on recent work by Eduardo Barrio, Federico Pailos, and Damian Szmuc, I show that for every inferential level (finite and transfinite), either classical logic is not unique at that level or there exist intuitively valid inferences of that level that are not definable in modern classical set theory. The classical logician is thereby faced (...) with a three-horned dilemma: Give up uniqueness but preserve characterizability, give up characterizability and preserve uniqueness, or (potentially) preserve both but give up modern classical set theory. After proving the main result, I briefly explore this third option by developing an account of classical logic within a paraconsistent set theory. This account of classical logic ensures unique characterizability in some sense, but the non-classical set theory also produces highly non-classical meta-results about classical logic. (shrink)

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of "increasingly classical" logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by (...) taking a somehow different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician. (shrink)

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, (...) and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

Semantic justifications of the classical rules of logical inference typically make use of a notion of bivalent truth, understood as a property guaranteed to attach to a sentence or its negation regardless of the prospects for speakers to determine it as so doing. For want of a convincing alternative account of classical logic, some philosophers suspicious of such recognition-transcending bivalence have seen no choice but to declare classical deduction unwarranted and settle for a weaker system; intuitionistic (...) logic in particular, buttressed by assertion-conditional semantics, is often considered to enjoy a degree of meaning-theoretical respectability unattainable by classical logic.The decision to forgo the classical inference rules is not always made lightly. Thus, Dummett: " In the resolution of the conflict between [the view that generally accepted classical modes of inference ought to be theoretically accommodated, and the demand that any such accommodation be achieved without recourse to bivalence] lies, as I see it, one of the most fundamental and intractable problems in the theory of meaning; indeed, in all philosophy. "The present article ventures to suggest that the conflict need not be irresoluble. Helping ourselves only to such conceptual resources as are standardly invoked by proponents of intuitionistic logic, we shall formulate a rudimentary meaning theory for a selection of logical constants, define a relation of valid inferability, and prove that the latter includes all of classical first-order logic. The result is essentially that reported in Sandqvist as Theorem 2.20.2. TheoryWe will be considering a standard-syntax first-order language with ⊃, ⊥ and ∀ as its only primitive logical constants. We assume countable supplies of individual constants, individual functors , and predicates , as well as a denumerably infinite set of individual variables. By a basic 1 formula we will mean one that …[Full Text of this Article]. (shrink)

This volume brings together a group of logic-minded philosophers and philosophically oriented logicians to address a diversity of topics on the structural analysis of non-classical logics. It mainly focuses on the construction of different types of models for various non-classical logics of current interest, including modal logics, epistemic logics, dynamic logics, and observational predicate logic. The book presents a wide range of applications of two well-known approaches in current research: structural modeling of certain philosophical issues in (...) the framework of non-classic logics, such as admissible models for modal logic, structural models for modal epistemology and for counterfactuals, and epistemological models for common knowledge and for public announcements; conceptual analysis of logical properties of, and formal semantics for, non-classical logics, such as sub-formula property, truthmaking, epistemic modality, behavioral strategies, speech acts and assertions. The structural analysis provided in this volume will appeal not only to graduate students and experts in non-classic logics, but also to readers from a wide range of disciplines, including computer science, cognitive science, linguistics, game theory and theory of action, to mention a few. (shrink)

One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot (...) of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference. (shrink)

This paper shows how to conservatively extend theories formulated in non-classical logics such as the Logic of Paradox, the Strong Kleene Logic and relevant logics with Skolem functions. Translations to and from the language extended by Skolem functions into the original one are presented and shown to preserve derivability. It is also shown that one may not always substitute s=f(t) and A(t, s) even though A determines the extension of a function and f is a Skolem function (...) for A. (shrink)

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)

In this paper, we make distinctions between Classical Logic (where the propositions are 100% true, or 100 false) and the Neutrosophic Logic (where one deals with partially true, partially indeterminate and partially false propositions) in order to respond to K. Georgiev’s criticism [1]. We recall that if an axiom is true in a classical logic system, it is not necessarily that the axiom be valid in a modern (fuzzy, intuitionistic fuzzy, neutrosophic etc.) logic system.

We offer a critical overview of two sorts of proof systems that may be said to characterize classical propositional logic indirectly (and non-standardly): refutation systems, which prove sound and complete with respect to classical contradictions, and rejection systems, which prove sound and complete with respect to the larger set of all classical non-tautologies. Systems of the latter sort are especially interesting, as they show that classical propositional logic can be given a paraconsistent characterization. In (...) both cases, we consider Hilbert-style systems as well as Gentzen-style sequent calculi and natural-deduction formalisms. (shrink)

Western (deductive) logic originated in Greek antiquity. It found its first expression in those works of the great philosopher Aristotle (384–322 BC) which have come to be known as the Organon, i.e., ‘instrument’. Aristotle’s logic, also known as syllogistics, was unsystematically concerned with patterns of reasoning and argumentation. It remained in this rudimentary state relatively unchanged and unchallenged until the second half of the nineteenth century. At that time, logic underwent a period of unprecedented reform and modernization, (...) due in large part to the German mathematician Gottlob Frege (1848– 1925) It thus became more and more a mathematical endeavor of studying the structure and peculiarities of artificial, formal languages. In this new form, logic gave rise in the twentieth century to disciplines such as theoretical informatics and programming languages, and transformed our lives through computation, information processing, and the Internet. A formal language consists, in effect, of a particular alphabet and some precise rules of forming, and transforming, strings over this alphabet. There exist several types of formal languages analyzed in logic. Depending on their structure, they are called first-order languages, second-order languages, and so on. Today a logic is considered a theory of such a language and is, correspondingly, referred to as a first-order logic, a second-order logic, and so on. In this chapter, we shall outline a first-order logic as a paragon of deductive logic. Its full name is: “Classical, first-order predicate logic with identity”. What all these expressions mean exactly, will become clear below. The logic we shall study first is termed classical logic because the idea to create such an instrument, or ‘organon’, is rooted in Greek antiquity. Owing to its origin, it is based on three time-honored Aristotelian doctrines. For these and several other reasons that we shall discuss later in this chapter, it is also called a logic of the Aristotelian style, or an Aristotelian logic for short. (shrink)

This paper shows how to conservatively extend classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truth—involving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system allows (...) for Cut—elimination, but the other does not.). (shrink)

We extend Barbanera and Berardi's symmetric lambda calculus [2] to second-order classical propositional logic and prove its strong normalization.

A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐order variable ranging over qualities. Similarly, (...) the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third‐order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third‐order item. Fourth‐order variables, and beyond, are characterized similarly. The phrase ‘higher‐order variable’ refers to the variables beyond first‐order. (shrink)

A simple method is provided for translating proofs in Grentzen's LK into proofs in Gentzen's LJ with the Peirce rule adjoined. A consequence is a simpler cut elimination operator for LJ + Peirce that is primitive recursive.

Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language.

Let me begin with a reminder of the crude but intuitive distinction from which the relativistic impulse springs. Any of the following claims would be likely to find both supporters and dissenters: That snails are delicious That cockroaches are disgusting That marital infidelity is alright provided nobody gets hurt That a Pacific sunset trumps any Impressionist canvas and perhaps That Philosophy is pointless if it is not widely intelligible That the belief that there is life elsewhere in the universe is (...) justified That death is nothing to fear Disputes about such claims may or may not involve quite strongly held convictions and attitudes. Sometimes they may be tractable disputes: there may be some other matter about which one of the disputing parties is mistaken or ignorant, where such a mistake or ignorance can perhaps be easily remedied, with the result of a change of heart about the original claim; or there may be a type of experience of which one of the disputing parties is innocent, and such that the effect of initiation into that experience is, once again, a change of view. (shrink)

. The effort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [4]. In this paper we show how to combine unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in constructively given models. These models are sheaf models over a $\sigma$ -complete Boolean algebra, whose topologies are generated by finite or countable covering relations. By a judicious choice of (...) the Boolean algebra we can directly extract effective content from $\Pi_2^0$ -statements true in the model. All the arguments of the present paper can be formalised in Martin-Löf's constructive type theory with generalised inductive definitions. (shrink)

The AGM theory of belief revision provides a formal framework to represent the dynamics of epistemic states. In this framework, the beliefs of the agent are usually represented as logical formulas while the change operations are constrained by rationality postulates. In the original proposal, the logic underlying the reasoning was supposed to be supraclassical, among other properties. In this paper, we present some of the existing work in adapting the AGM theory for non-classical logics and discuss their interconnections (...) and what is still missing for each approach. (shrink)

I argue that there are two ways of construing Wittgenstein’s slogan that meaning is use. One accepts the view that the notion of meaning must be explained in terms of truth-theoretic notions and is committed to the epistemic conception of truth. The other keeps the notion of meaning and the truth-theoretic notions apart and is not committed to the epistemic conception of truth. I argue that Dummett endorses the first way of construing Wittgenstein’s slogan. I address the issue by discussing (...) two of Dummett’s arguments against the realist truth-theoretic conception of meaning: the manifestation argument and the argument for the unintelligibility of classical logic. I examine the dialectic of those arguments and show that they rest on the assumption that meaning needs to be explained in terms of truth-theoretic notions.Tvrdim da postoje dva načina shvaćanja Wittgensteinova slogana da je značenje upotreba. Jedno prihvaća gledište da se pojam značenja mora objasniti na osnovi istinitosnoteoretskih pojmova te je obvezano na epistemičku koncepciju istine. Drugo pojam značenja i istinitosnoteoretske pojmove drži odvojenima te nije obvezano na epistemičku koncepciju istine. Tvrdim da Dummett prihvaća prvo shvaćanje Wittgensteinova slogana. Problemu pristupam tako što raspravljam o dvama Dummettovim argumentima protiv realističke istinitosnoteoretske koncepcije značenja: o argumentu iz manifestacije i argumentu za neshvatljivost klasične logike. Ispitujem dijalektiku tih argumenata i pokazujem da počivaju na pretpostavci da se značenje mora objasniti na osnovi istinitosnoteoretskih pojmova. (shrink)

Weatherson argues that whoever accepts classical logic, standard mereology and the difference between vague objects and any others, should conclude that there are no vague objects. Barnes and Williams claim that a supporter of vague objects who accepts classical logic and standard mereology should recognize that the existence of vague objects implies indeterminate identity. Even though it is not clearly stated, they all seem to be committed to the assumption that reality is ultimately constituted by mereological (...) atoms. This assumption is not granted by standard mereology which instead remains silent on whether reality is atomic or gunky; therefore, I contend that whoever maintains classical logic, standard mereology and the difference between vague objects and any others, is not forced to conclude with Weatherson that there are no vague objects; nor is she compelled to revise her point of view according to Barnes and Williams’s proposal and to accept that the existence of vague objects implies indeterminate identity. (shrink)

This is the first introductory textbook on non-classical propositional logics.

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)

The purpose of this paper is to provide a new system of logic for existence and essence, in which the traditional distinctions between essential and accidental properties, abstract and concrete objects, and actually existent and possibly existent objects are described and related in a suitable way. In order to accomplish this task, a primitive relation of essential identity between different objects is introduced and connected to a first order existence property and a first order abstractness property. The basic idea (...) is that possibly existent objects are completely determinate and that essentially identical objects are just different individuations of the same individual essence. Accordingly, essential properties are defined as properties that are invariant with respect to this kind of identity, while abstract objects are determined by being characterized by essential properties only. Once such ideas are implemented, a number of classical intuitions about objects, their essence, and their way of existence can be consistently interpreted. (shrink)

In distinction from the well-known double-negation embeddings of the classical logic we consider some variants of single-negation embeddings and describe some classes of superintuitionistic first-order predicate logics in which the classical first-order calculus is interpretable in such a way. Also we find the minimal extensions of Heyting's logic in which the classical predicate logic can be embedded by means of these translations.

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

An account of the logic of bivalent languages with truth-value gaps is given. This account is keyed to the use of tables introduced by S. C. Kleene. The account has two guiding ideas. First, that the bivalence property insures that the language satisfies classical logic. Second, that the general concepts of a valid sentence and an inconsistent sentence are, respectively, as sentences which are not false in any model and sentences which are not true in any model. (...) What recommends this approach is (1) its relative simplicity, and (2) the fact that it leaves the fundamental features of classical logic intact. (shrink)

The subject of Labelled Non-Classical Logics is the development and investigation of a framework for the modular and uniform presentation and implementation of non-classical logics, in particular modal and relevance logics. Logics are presented as labelled deduction systems, which are proved to be sound and complete with respect to the corresponding Kripke-style semantics. We investigate the proof theory of our systems, and show them to possess structural properties such as normalization and the subformula property, which we exploit not (...) only to establish advantages and limitations of our approach with respect to related ones, but also to give, by means of a substructural analysis, a new proof-theoretic method for investigating decidability and complexity of (some of) the logics we consider. All of our deduction systems have been implemented in the generic theorem prover Isabelle, thus providing a simple and natural environment for interactive proof development. Labelled Non-Classical Logics is essential reading for researchers and practitioners interested in the theory and applications of non-classical logics. (shrink)

In its first meaning, a logic is a collection of closely related artificial languages. There are certain languages called first‐order languages, and together they form first‐order logic. In the same spirit, there are several closely related languages called modal languages, and together they form modal logic. Likewise second‐order logic, deontic logic and so forth.

Bilateralists hold that the meanings of the connectives are determined by rules of inference for their use in deductive reasoning with asserted and denied formulas. This paper presents two bilateral connectives comparable to Prior's tonk, for which, unlike for tonk, there are reduction steps for the removal of maximal formulas arising from introducing and eliminating formulas with those connectives as main operators. Adding either of them to bilateral classical logic results in an incoherent system. One way around this (...) problem is to count formulas as maximal that are the conclusion of reductio and major premise of an elimination rule and to require their removability from deductions. The main part of the paper consists in a proof of a normalisation theorem for bilateral logic. The closing sections address philosophical concerns whether the proof provides a satisfactory solution to the problem at hand and confronts bilateralists with the dilemma that a bilateral notion of stability sits uneasily with the core bilateral thesis. (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 (...) class='Hi'>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)