In this paper we examine two approaches to the formal treatment of the notion of problem in the paradigm of algorithmic semantics. Namely, we will explore an approach based on Martin-Löf’s Constructive Type Theory, which can be seen as a direct continuation of Kolmogorov’s original calculus of problems, and an approach utilizing Tichý’s Transparent Intensional Logic, which can be viewed as a non-constructive attempt of interpreting Kolmogorov’s logic of problems. In the last section we propose Kolmogorov and CTT-inspired modifications to (...) TIL-based approach. The focus will be on non-empirical problems only. (shrink)

Many prominent writers on the philosophy of logic, including Michael Dummett, Dag Prawitz, Neil Tennant, have held that the introduction and elimination rules of a logical connective must be ‘in harmony ’ if the connective is to possess a sense. This Harmony Thesis has been used to justify the choice of logic: in particular, supposed violations of it by the classical rules for negation have been the basis for arguments for switching from classical to intuitionistic logic. The Thesis has also (...) had an influence on the philosophy of language: some prominent writers in that area, notably Dummett and Robert Brandom, have taken it to be a special case of a more general requirement that the grounds for asserting a statement must cohere with its consequences. This essay considers various ways of making the Harmony Thesis precise and scrutinizes the most influential arguments for it. The verdict is negative: all the extant arguments for the Thesis are weak, and no version of it is remotely plausible. (shrink)

We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the linearity rule, which (bottom-up) operates by linearising branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking (...) if data exists along certain paths within a nested sequent. Such rules require us to generalise our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalisation of the usual nested sequent formalism. Our calculi exhibit favourable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems. (shrink)

The aim of this article is to show that, just as in recent years Cobreros, Egré, Ripley and van Rooij have provided a non-transitive counterpart of classical logic (i.e. one in which all classically acceptable inferences are valid but Cut and other metainferences are not), the same can be done for every Tarskian logic, with full generality. To establish this fact, a semantic approach is taken by showing that appropriate structures can be devised to characterize a non-transitive counterpart of every (...) Tarskian logic, starting from the logical matrices that are usually taken to render them. (shrink)

The present endeavour aims at the clarification of the concept of the logical consequence. Initially we investigate the question: How was the concept of logical consequence discovered by the medieval philosophers? Which ancient philosophical foundations were necessary for the discovery of the logical relation of consequence and which explicit medieval contributions, such as the notion of the formality (formal validity), led to its discovery. Secondly we discuss which developments of modern philosophy effected the turn from the medieval concept of logical (...) consequence to its most recent conceptions, such as the semantic, syntactic, axiomatic and natural deductive approaches? Thirdly we examine which are the similarities and the differences between the logical concepts of consequence, inference, implication and entailment? Furthermore, we ask what kind of relation signifies the concept of the logical consequence? That is to say, which is the analytic definition of the consequence relation R between the premises p1, p2...pn and the conclusion c of a formally valid argument? Finally, we focus on the respective answers given through the developments in proof theory by David Hilbert and Gerhard Gentzen. (shrink)

This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate (...) or graduate level courses.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic. (shrink)

Además de los indemostrables, la lógica estoica incluye varias reglas de reducción. En este trabajo, analizo la primera de ellas con el fin de comprobar si fue derivada formalmente a partir de los indemostrables o los estoicos pudieron plantearla a partir del uso de sus capacidades naturales de razonamiento. De esta manera, trato de mostrar que tenemos razones para apoyar ambas posibilidades y, basándome en un enfoque semántico como el de la teoría de los modelos mentales, de ofrecer argumentos al (...) respecto. (shrink)

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used (...) for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics. (shrink)

What is a minimal proof-theoretical foundation of logic? Two different ways to answer this question may appear to offer themselves: reduce the whole of logic either to the relation of inference, or else to the property of incompatibility. The first way would involve defining logical operators in terms of the algebraic properties of the relation of inference—with conjunction $$\hbox {A}\wedge \hbox {B}$$ A ∧ B as the infimum of A and B, negation $$\lnot \hbox {A}$$ ¬ A as the minimal (...) incompatible of A, etc. The second way involves introducing logical operators in terms of the relation of incompatibility, such that X is incompatible with $$\{\lnot \hbox {A}\}$$ { ¬ A } iff every Y incompatible with X is incompatible with {A}; and X is incompatible with $$\{\hbox {A}\!\wedge \!\hbox {B}\}$$ { A ∧ B } iff X is incompatible with {A,B}; etc. Whereas the first route leads us naturally to intuitionistic logic, the second leads us to classical logic. The aim of this paper is threefold: to investigate the relationship of the two approaches within a very general framework, to discuss the viability of erecting logic on such austere foundations, and to find out whether choosing one of the ways we are inevitably led to a specific logical system. (shrink)

We examine the proof-theoretic verificationist justification procedure proposed by Dummett. After some scrutiny, two distinct interpretations with respect to bases are advanced: the independent and the dependent interpretation. We argue that both are unacceptable as a semantics for propositional intuitionistic logic.

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic. However, there are good reasons to avoid using full second-order logic for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by (...) augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory. Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof. (shrink)

ABSTRACT In this paper we state and prove Herbrand's properties for two modal systems, namely T and S4, thus adapting a previous result obtained for the system D [CIA 86a] to such theories. These properties allow the first order extension?along the lines of [CIA 91]?of the resolution method defined in [ENJ 86] for the corresponding propositional modal systems. In fact, the Herbrand-style procedures proposed here treat quantifiers in a uniform way, that suggests the definition of a restricted notion of unification (...) for quantifier free modal formulae obtained by means of a Skolem-like procedure. The differences among the three systems considered, D, T and S4, are taken into account by preliminary propositional transformations of the formulae, in such a way that the problem of quantifier elimination can be treated within the same pattern in the different systems. The methods employed to prove the fundamental theorems are purely proof-theoretical and make use of the formalization of the considered modal theories in terms of sequent calculi, where the cut elimination theorem holds. (shrink)

I interpret Mill?s view on logic as the instrumentalist view that logical inferences, complex statements, and logical operators are not necessary for reasoning itself, but are useful only for our remembering and communicating the results of the reasoning. To defend this view, I first show that we can transform all the complex statements in the language of classical first-order logic into what I call material inference rules and reduce logical inferences to inferences which involve only atomic statements and the material (...) inference rules. Then I explain why we introduce logical operators and logical inference rules into a system of the latter kind. In the end I determine what kind of negation is justified from this point of view. (shrink)

The entire development of modern logic is characterized by various forms of confrontation of what has come to be called proof theory with what has earned the label of model theory. For a long time the widely accepted view was that while model theory captures directly what logical formalisms are about, proof theory is merely our technical means of getting some incomplete grip on this; but in recent decades the situation has altered. Not only did proof theory expand into new (...) realms, generalizing the concept of proof in various directions; many philosophers also realized that meaning may be seen as primarily consisting in certain rules rather than in language-world links. However, the possibility of construing meaning as an inferential role is often seen as essentially compromised by the limits of prooftheoretical means. The aim of this paper is to sort out the cluster of problems besetting logical inferentialism by disentangling and clarifying one of them, namely determining the power of various inferential frameworks as measured by that of explicitly semantic ones. (shrink)

Axiomatic systems can be understood as subsets of syntactic systems. By a process of increasing abstraction, the notion of a syntactic system can become useful for understanding Meillassoux’s concept of hyper-Chaos.

We present an inferentialist account of the epistemic modal operator might. Our starting point is the bilateralist programme. A bilateralist explains the operator not in terms of the speech act of rejection ; we explain the operator might in terms of weak assertion, a speech act whose existence we argue for on the basis of linguistic evidence. We show that our account of might provides a solution to certain well-known puzzles about the semantics of modal vocabulary whilst retaining classical logic. (...) This demonstrates that an inferentialist approach to meaning can be successfully extended beyond the core logical constants. (shrink)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

The current formal theories trying to account for reasoning and language often need to resort to the thesis of the pragmatic or implicit premises to explain the real role pragmatics plays in the human inferential activity. The main aim of this paper is to show that, although they did not propose theories in this way, the Stoics, by means of their concept of ‘sign’, already offered the essential arguments and foundations necessary to describe the real action of both pragmatics and (...) the premises coming from general knowledge when we make inferences. (shrink)

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in proofs (...) and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics. (shrink)

This paper studies the relationship between Argumentation Logic, a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic. In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of sentences in AL and Natural Deduction proofs of the complement of these sentences. The proof of this equivalence uses a restricted form (...) of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies. (shrink)

Stoic logic assumes five inference schemata attributed to Chrysippus of Soli. Those schemata are the well-known indemonstrables. A problem related to them can be that, according to standard propositional calculus, only one of them, modus ponens, is clearly indemonstrable. Nevertheless, I try to show in this paper that the mental logic theory enables to understand why the Stoics considered such schemata to be basic kinds of arguments. Following that theory, four of them can be linked to ‘Core Schemata’ of mental (...) logic and the only one that is more controversial is modus tollens. However, as I also comment, some assumptions of Stoic philosophy, which can be interpreted from the mental logic theory, can explain why this last argument was included into the set of the indemonstrablesas well. (shrink)

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...) In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views. (shrink)

Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. For example, intersection types distinguish between call-by-name and call-by-value functions, because the subtyping law ∩≤A→ is unsound for the latter in (...) the presence of effects.In this paper we develop a proof-theoretic framework for analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered through linear logic, but we propose a fresh origin in Dummett’s program of justifying the logical laws through alternative verificationist or pragmatist “meaning-theories”, which include a bias towards either introduction or elimination rules. We revisit Dummett’s analysis using the tools of Martin-Löf’s judgmental method, and then show how to extend it to a unified polarized logic, with Girard’s “shift” connectives acting as intermediaries. This logic safely combines intuitionistic and dual intuitionistic reasoning principles, while simultaneously admitting a focusing interpretation for the classical sequent calculus.Then, by applying the Curry–Howard isomorphism to polarized logic, we obtain a single programming language in which evaluation order is reflected at the level of types. Different logical notions correspond directly to natural programming constructs, such as pattern-matching, explicit substitutions, values and call-by-value continuations. We give examples demonstrating the expressiveness of the language and type system, and prove a basic but modular type safety result. We conclude with a brief discussion of extensions to the language with additional effects and types, and sketch the sort of explanation this can provide for operationally sensitive typing phenomena. (shrink)

The purpose of this paper is to develop a class of semantic tableau systems for some counterfactual logics. All in all I will discuss 1024 systems. Possible world semantics is used to interpret our formal languages. Soundness results are obtained for every tableau system and completeness results for a large subclass of these.

The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)

RésuméOn défend ici l'idée que la définition des notions sémantiques à l'aide des fonctions de probabilité devrait être vue non pas comme une généralisation de la sémantique standard en termes d'assignations de valeurs de vérité, mais plutôt comme une généralisation aux degrés de conséquence logique, de la caractérisation de la relation de conséquence que l'on retrouve dans le calcul des séquents de Gentzen.

Legal terms, such as “ownership,”“contract,”“validity,”“negligence,” are used as middle terms in legal deduction. The paper distinguishes two problems regarding this use. One is the logical function of terms for deduction within a normative system. Specific problems dealt with in this connection are meaning, definition, and economy of expression. The other problem connected with middle terms is the “moulding” and possible manipulation of the meaning of legal terms, for arriving at desired conclusions in a given scheme of inference. It is indicated (...) how the moulding of contested legal concepts, if not restricted, will obscure the ratio of legal rules. This problem is relevant, inter alia, to arguments ex analogia in the law. (shrink)

This paper deals with Popper's little-known work on deductive logic, published between 1947 and 1949. According to his theory of deductive inference, the meaning of logical signs is determined by certain rules derived from ?inferential definitions? of those signs. Although strong arguments have been presented against Popper's claims (e.g. by Curry, Kleene, Lejewski and McKinsey), his theory can be reconstructed when it is viewed primarily as an attempt to demarcate logical from non-logical constants rather than as a semantic foundation for (...) logic. A criterion of logicality is obtained which is based on conjunction, implication and universal quantification as fundamental logical operations. (shrink)

Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs, either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule, thus in effect making them an auxiliary notion subservient (...) to proper propositions. The Curry–Howard correspondence is typically viewed as a formal counterpart of this conception. I will argue against this position and show that even though the Curry–Howard correspondence typically accommodates the placeholder view of assumptions, it is rather a matter of choice, not a necessity, and that another more assumption-friendly view can be adopted. (shrink)

It is usually said that there is a clear difference between pre-philosophical texts such as Homer’s Iliad and what is provided in the fragments corresponding to first philosophers such as Thales of Miletus. This paper tries to show that this is not undoubtedly so, and it does that by means of the analysis of a fragment of the Iliad in which Hypnos is speaking. In this way, the main argument is that, while the fragment can be interpreted both in a (...) literal and in a metaphorical or poetic sense, in both of those cases it can be thought that it is related to philosophy or rational thinking. If the interpretation is literal, the fragment expresses an idea very akin to an important thesis by Thales of Miletus. On the other hand, if the interpretation is metaphorical or poetic, we have to assume, following several contemporary reasoning theories, that the fragment is essentially based on an evident rational inference. (shrink)

The double negation has always been considered by the logical systems from ancient times to the present. In fact, that is an issue that the current syntactic theories studying human reasoning, for example, the mental logic theory, address today. However, in this paper, I claim that, in the case of some languages such as Spanish, the double negation causes problems for the cognitive theories mainly based on formal schemata and supporting the idea of a universal syntax of thought in the (...) human mind. Thus, I propose that, given those problems, semantic frameworks such as that of the mental models theory seem to be more appropriate for explaining the human inferential activity. (shrink)

Resumen Atendiendo a la lógica estándar, solo uno de los cinco indemostrables propuestos por Crisipo de Solos es realmente indemostrable. Sus otros cuatro esquemas son demostrables en tal lógica. La pregunta, por tanto, es: si cuatro de ellos no son verdaderamente indemostrables, por qué Crisipo consideró que sí lo eran. López-Astorga mostró que si ignoramos el cálculo proposicional estándar y asumimos que una teoría cognitiva contemporánea, la teoría de la lógica mental, describe correctamente el razonamiento humano, se puede entender por (...) qué Crisipo pensó que todos sus indemostrables eran tan básicos. No obstante, en este trabajo trato de argumentar que la teoría de la lógica mental no es el único marco que puede explicar esto. En concreto, sostengo que otra importante teoría sobre el razonamiento en el presente, la teoría de los modelos mentales, también puede ofrecer una explicación al respecto.According to standard logic, only one of the five indemonstrables proposed by Chrysippus of Soli is actually indemonstrable. The other four schemata are demonstra-ble in that logic. The question hence is, if four of them are not really indemonstrable, why Chrysippus considered them to be so. López-Astorga showed that, if we ignore standard propositional calculus and assume that a current cognitive theory, the mental logic theory, truly describes human reasoning, it can be explained why Chrysippus thought that all of his indemonstrables were so basic. However, in this paper, I try to argue that the mental logic theory is not the only framework that can account for that. In particular, I hold that another important reasoning theory at present, the mental models theory, can offer an explanation in that regard as well. (shrink)

For any extension $T$ of $I\Sigma_{1}$ having a cut-elimination property extending that of $I\Sigma_{1}$ , the number of different proofs that can be obtained by cut elimination from a single $T$ -proof cannot be bound by a function which is provably total in $T$.

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.

This book explains the first published consistency proof of PA. It contains the original Gentzen's proof, but it uses modern terminology and examples to illustrate the essential notions. The author comments on Gentzen's steps which are supplemented with exact calculations and parts of formal derivations. A notable aspect of the proof is the representation of ordinal numbers that was developed by Gentzen. This representation is analysed and connection to set-theoretical representation is found, namely an algorithm for translating Gentzen's notation into (...) Cantor normal form. The topic should interest researchers and students who work on proof theory, history of proof theory or Hilbert's program and who do not mind reading mathematical texts.​. (shrink)

Gentzen’s singular sequential system of first-order logic was an alternative notation for his system of natural deductions. His multiple sequential system was his symmetric generalization that was more appropriate to classical logic. Beth’s tableaus system was a system that was derived directly from the semantic analysis of connectives and quantifiers. It was soon realized that the Beth’s system and the Gentzen’s multiple system were only notational variants of each other. Kneale’s system of multiple natural deductions was a generalization of Gentzen’s (...) system of natural deductions. We prove that Kneale’s natural deductions are also a notational variant of Beth’s tableaus. (shrink)

We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill’s calculi, and focus on their fragments including multiplicative connectives, additive conjunction and disjunction, brackets and bracket modalities, and the! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut. We also prove algorithmic undecidability for both (...) calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages. (shrink)

Recientemente, se ha propuesto que la lógica no es el único factor que incide en el razonamiento condicional y que el valor y la utilidad son otros dos elementos importantes en este sentido. Al respecto, Bonnefon y Sloman añadieron en 2013 que la estructura de los condicionales es, del mismo modo, relevante en la actividad inferencial con condicionales de utilidad y trataron de demostrarlo por medio de varios experimentos. Sus resultados parecieron mostrar que, efectivamente, sus tesis eran correctas. No obstante, (...) plantearon igualmente que su propuesta y, por tanto, sus resultados eran compatibles con una teoría contemporánea de amplio alcance, la teoría de los modelos mentales, y en este artículo se trata de desarrollar esta idea con el propósito de comprobar si es cierta. (shrink)

The variety \ of semi-Heyting algebras was introduced by H. P. Sankappanavar [13] as an abstraction of the variety of Heyting algebras. Semi-Heyting algebras are the algebraic models for a logic HsH, known as semi-intuitionistic logic, which is equivalent to the one defined by a Hilbert style calculus in Cornejo :9–25, 2011) [6]. In this article we introduce a Gentzen style sequent calculus GsH for the semi-intuitionistic logic whose associated logic GsH is the same as HsH. The advantage of this (...) presentation of the logic is that we can prove a cut-elimination theorem for GsH that allows us to prove the decidability of the logic. As a direct consequence, we also obtain the decidability of the equational theory of semi-Heyting algebras. (shrink)