We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also (...) briefly clarify the relationships of our falsification logic to some other logical systems. (shrink)

The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To (...) obtain a more precise mapping, both LJ and LDJ are extended by adding a "pseudo-difference" operator which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems and , and a simply consistent but -inconsistent Set Theory with Unrestricted Comprehension Schema based on LDJ is sketched. (shrink)

Dual-intuitionistic logics are logics proposed by Czermak , Goodman and Urbas . It is shown in this paper that there is a correspondence between Goodman's dual-intuitionistic logic and Nelson's constructive logic N−.

In this paper, we study the relationship among classical logic, intuitionistic logic, and quantum logic . These logics are related in an interesting way and are not far apart from each other, as is widely believed. The results in this paper show how they are related with each other through a dual intuitionistic logic . Our study is completely syntactical.

Mortensen studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complement-topos logic, which throws some light on the problem of giving a dualization for LJ.

We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.

The logic DHb is the result of extending Sylvan and Plumwood’s minimal De Morgan logic BM with a dual intuitionistic negation of the type Sylvan defined for the extension CCω of da Costa’s paraconsistent logic Cω. We provide Routley–Meyer ternary relational semantics with a set of designated points for DHb and a wealth of its extensions included in G3DH, the expansion of G3+ with a dual intuitionistic negation of the kind considered by Sylvan (...) (G3+ is the positive fragment of Gödelian 3-valued logic G3). All logics in the paper are paraconsistent. (shrink)

Da Costa's C systems are surveyed and motivated, and significant failings of the systems are indicated. Variations are then made on these systems in an attempt to surmount their defects and limitations. The main system to emerge from this effort, system CC , is investigated in some detail, and dual-intuitionistic semantical analyses are developed for it and surrounding systems. These semantics are then adapted for the original C systems, first in a rather unilluminating relational fashion, subsequently in a (...) more illuminating way through the introduction of impossible situations where and and or change roles. Finally other attempts to break out of impasses for the original and expanded C systems, by going inside them, are looked at, and further research directions suggested. (shrink)

Da Costa's C systems are surveyed and motivated, and significant failings of the systems are indicated. Variations are then made on these systems in an attempt to surmount their defects and limitations. The main system to emerge from this effort, system $CC_{\omega}$ , is investigated in some detail, and "dual-intuitionistic" semantical analyses are developed for it and surrounding systems. These semantics are then adapted for the original C systems, first in a rather unilluminating relational fashion, subsequently in a (...) more illuminating way through the introduction of impossible situations where and and or change roles. Finally other attempts to break out of impasses for the original and expanded C systems, by going inside them, are looked at, and further research directions suggested. (shrink)

Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not (...) use the machinery of neither saturated models nor elementary chains. (shrink)

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such (...) is equivalent to one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object . Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such (...) is equivalent to one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object. Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)

Two new propositional non-classical logics, referred to as symmetric intuitionistic logic (SIL) and conflated intuitionistic logic (CIL), are introduced as indexed and non-indexed Gentzen-style sequent calculi. SIL is regarded as a natural hybrid logic combining intuitionistic and dual-intuitionistic logics, whereas CIL is regarded as a variant of intuitionistic paraconsistent logic with conflation and without paraconsistent negation. The cut-elimination theorems for SIL and CIL are proved. CIL is shown to be conservative (...) over SIL. (shrink)

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

We present a multiple-assumption multiple-conclusion system for bi-intuitionistic logic. Derivations in the systems are graphs whose edges are labelled by formulas and whose nodes are labelled by rules. We show how to embed both the standard intuitionistic and dual-intuitionistic natural deduction systems into the proposed system. Soundness and completeness are established using translations with more traditional sequent calculi for bi-intuitionistic logic.

Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cut-free’ sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and (...) class='Hi'>dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. (shrink)

We combine intuitionistic logic and classical logic into a new, first-order logic called polarized intuitionistic logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzenʼs intuitionistic sequent calculus LJ. (...) In addition, this system also bears essential similarities to Girardʼs LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalinʼs multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. (shrink)

We explore a relation we call 'anticipation' between formulas, where A anticipates B (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective →) of the formula A → B. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as "a", governed (...) by rules which guarantee that for any formula B, aB is the (logically) strongest formula anticipating B. The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective. Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C. Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context. (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus (...) seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We (...) give the bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

I prove that the three basic propositions of Vasiliev's paraconsistent logic have a semantic interpretation by means of the intuitionist logic. The interpèretation is confirmed by amens of the da Costa's model of Vasiliev's paraconsistent logic.

We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich (...) Theorem, which characterizes the sentences preserved under sandwiches of Kripke models. We also define several intuitionistic formula hierarchies analogous to the classical formula hierarchies $\forall_n (= \Pi^0_n)$ and $\exists_n (=\Sigma^0_n)$ , and we show how our generalized syntactic preservation theorems specialize to these hierarchies. Each of these theorems implies the corresponding classical theorem in the case where the Kripke models force classical logic. (shrink)

In this contribution, we will present some results concerning the connectives of bi-intuitionistic logic in the setting of Arnold Koslow’s implication structures. Furthermore, we will present soundness and completeness results of Koslow’s implication structures with respect to bi-intuitionistic logic.

We introduce an extended intuitionistic linear logic with strong negation and modality. The logic presented is a modal extension of Wansing's extended linear logic with strong negation. First, we propose three types of cut-free sequent calculi for this new logic. The first one is named a subformula calculus, which yields the subformula property. The second one is termed a dual calculus, which has positive and negative sequents. The third one is called a triple-context calculus, (...) which is regarded as a natural extension or generalization of Hodas and Miller's dual-context calculus appearing in a linear logic programming language. Second, we present a concurrent process calculus based on the logic. This calculus is an extension of Okada's process calculus. Third, we introduce a Kripke type semantics for a fragment of the logic, and show the completeness theorems with respect to the semantics. Finally, we mention a logic programming language based on the triple-context calculus. (shrink)

This paper studies an economy whose agents perceive their consumption possibilities subjectively, and whose preferences are defined on what they subjectively experience, rather than on those alternatives that are objectively present. The model of agents' perceptions is based on intuitionistic logic. Roughly, this means that agents reason constructively: a solution to a problem exists only if there is a construction by which the problem can be solved. The theorems that can be proved determine how an agent perceives a (...) set of alternatives. A dual model relates perceived alternatives to a shared language, which the agents use in trading. So perceptions relate objective alternatives to an agent's subjective view of them, and reporting dually relates an agent's subjective world to a shared language. It turns out that an appropriately modified notion of competitive equilibrium always exists. However, in contrast with standard results in economic theory, competitive equilibrium need not be efficient. (shrink)

Girard introduced phase semantics as a complete set-theoretic semantics of linear logic, and Okada modified phase-semantic completeness proofs to obtain normal-form theorems. On the basis of these works, Okada and Takemura reformulated Girard’s phase semantics so that it became phase semantics for proof-terms, i.e., lambda-terms. They formulated phase semantics for proof-terms of Laird’s dual affine/intuitionistic lambda-calculus and proved the normal-form theorem for Laird’s calculus via a completeness theorem. Their semantics was obtained by an application of computability predicates. (...) In this paper, we first formulate phase semantics for proof-terms of second-order intuitionistic propositional logic by modifying Tait-Girard’s saturated sets method. Next, we prove the completeness theorem with respect to this semantics, which implies a strong normalization theorem. (shrink)

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional (...) logic . Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA. (shrink)

In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to (...) the announced proposition. We dually characterize the associated submodel-injection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs . The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras . Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL. (shrink)

The book presents logical foundations of dual tableaux together with a number of their applications both to logics traditionally dealt with in mathematics and philosophy (such as modal, intuitionistic, relevant, and many-valued logics) and to various applied theories of computational logic (such as temporal reasoning, spatial reasoning, fuzzy-set-based reasoning, rough-set-based reasoning, order-of magnitude reasoning, reasoning about programs, threshold logics, logics of conditional decisions). The distinguishing feature of most of these applications is that the corresponding dual tableaux (...) are built in a relational language which provides useful means of presentation of the theories. In this way modularity of dual tableaux is ensured. We do not need to develop and implement each dual tableau from scratch, we should only extend the relational core common to many theories with the rules specific for a particular theory. (shrink)

This volume examines the concept of falsification as a central notion of semantic theories and its effects on logical laws. The point of departure is the general constructivist line of argument that Michael Dummett has offered over the last decades. From there, the author examines the ways in which falsifications can enter into a constructivist semantics, displays the full spectrum of options, and discusses the logical systems most suitable to each one of them. While the idea of introducing falsifications into (...) the semantic account is Dummett's own, the many ways in which falsificationism departs quite radically from verificationism are here spelled out in detail for the first time. The volume is divided into three large parts. The first part provides important background information about Dummett’s program, intuitionism and logics with gaps and gluts. The second part is devoted to the introduction of falsifications into the constructive account and shows that there is more than one way in which one can do this. The third part details the logical effects of these various moves. In the end, the book shows that the constructive path may branch in different directions: towards intuitionistic logic, dual intuitionistic logic and several variations of Nelson logics. The author argues that, on balance, the latter are the more promising routes to take. "Kapsner’s book is the first detailed investigation of how to incorporate the notion of falsification into formal logic. This is a fascinating logico-philosophical investigation, which will interest non-classical logicians of all stripes." Graham Priest, Graduate Center, City University of New York and University of Melbourne. (shrink)

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we (...) present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. (shrink)

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, (...) and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional (...) logic . Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA. (shrink)

This paper investigates (modal) extensions of Heyting-Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gö;del-embedding of intuitionistic logic into S4, it is shown that all (modal) extensions of Heyting-Brouwer logic can be embedded into tense logics (with additional modal operators). An extension (...) of the Blok-Esakia-Theorem is proved for this embedding. (shrink)

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation (...) by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, (...) we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal (...) logic, intuitionist logic and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal (...) logic, intuitionist logic and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T 1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)

It is widely accepted that there is a clear sense in which the first-order paraconsistent constructive logic with strong negation of Almukdad and Nelson, QN4, is more constructive than intuitionistic first-order logic, QInt. While QInt and QN4 both possess the disjunction property and the existence property as characteristics of constructiveness (or constructivity), QInt lacks certain features of constructiveness enjoyed by QN4, namely the constructible falsity property and the dual of the existence property. This paper deals with (...) the constructiveness of the contra-classical, connexive, paraconsistent, and contradictory non-trivial first-order logic QC, which is a connexive variant of QN4. It is shown that there is a sense in which QC is even more constructive than QN4. The argument focuses on a problem that is mirror-inverted to Raymond Smullyan’s drinker paradox, namely the invalidity of what will be called the drinker truism and its dual in QN4 (and QInt), and on a version of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations that treats proofs and disproofs on a par. The validity of the drinker truism and its dual together with the greater constructiveness of QC in comparison to QN4 may serve as further motivation for the study of connexive logics and suggests that constructive logic is connexive and contradictory (the latter understood as being negation inconsistent). (shrink)

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order (...) logic. One device for making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999. (shrink)

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of 'true but unprovable' sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk's Logic, where modality can be interpreted as 'true and provable'. As we show, GS and Grzegorczyk's Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of (...) 'Essence and Accident' proposed by Marcos (Bull Sect Log 34(1):43-56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic. (shrink)

In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the (...) bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped. (shrink)

The basic quasi-Boolean negation expansions of relevance logics included in Anderson and Belnap’s relevance logic R are defined. We consider two types of QB-negation: H-negation and D-negation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with H-negation are paracomplete; logics with D-negation are paraconsistent. All logics defined in the paper are given a Routley-Meyer ternary relational semantics.

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧, →) homomorphisms, (∧, →, 0) homomorphisms, and (∧, →, ∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of (...) Zakharyaschev’s theorem that each intermediate logic can be axiomatized by canonical formulas. (shrink)

The purpose of this paper is to define a new logic $${\mathcal {SI}}$$ called semi-intuitionistic logic such that the semi-Heyting algebras introduced in [ 4 ] by Sankappanavar are the semantics for $${\mathcal {SI}}$$ . Besides, the intuitionistic logic will be an axiomatic extension of $${\mathcal {SI}}$$.