Automated theorem proving amounts to solving search problems in usually tremendous search spaces. A lot of research therefore focuses on search space reductions. Our approach reduces the search space which arises when using so-called connection tableau calculi for first-order automated theorem proving. It uses disjunctive constraints over first-order equations to compress certain parts of this search space. We present the basics of our constrained-connection-tableau calculi, a constraint extension of connection tableau calculi, and deal with (...) the efficient handling of constraints during the search process. The new techniques are integrated into the automated connection tableau prover Setheo. (shrink)

We get cut-free sequent calculi for the interpolable propositional intermediate logics by translating suitable duplication-free tableau calculi developed within a semantical framework. From this point of view, the paper also provides semantical proofs of the admissibility of the cut-rule for appropriate cut-free sequent calculi.

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G 0(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G 0are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of (...) the cut-rule, the so-called analytical cut-rule.In addition we show that G 0is not compact (and therefore not canonical), and we proof with the tableau-method that G 0is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G 0is decidable and also characterised by the class of all frames for G 0. (shrink)

In this paper we provide cut-free tableau calculi for the intuitionistic modal logics IK, ID, IT, i.e. the intuitionistic analogues of the classical modal systems K, D and T. Further, we analyse the necessity of duplicating formulas to which rules are applied. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Specifically, we enlarge the language with the new signs Fc and CR near (...) to the usual signs T and F. In this work we establish the soundness and completeness theorems for these calculi with respect to the Kripke semantics proposed by Fischer Servi. (shrink)

In this paper, we provide non-clausal tableau calculi for the maximum satisfiability problem and its variants. We discuss both basic calculi to characterize the problem and their modifications to reduce the proof size.

Grammar logics were introduced by Fariñas del Cerro and Penttonen in 1988 and have been widely studied. In this paper we consider regular grammar logics with converse ( REG c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG c logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REG c logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned (...) problems, to which various optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in REG c logics is coNP-complete. (shrink)

Since 1993, when Hudelmaier developed an O(n log n)-space decision procedure for propositional Intuitionistic Logic, a lot of work has been done to improve the efficiency of the related proof-search algorithms. In this paper a tableau calculus using the signs T, F and Fc with a new set of rules to treat signed formulas of the kind T((A → B) → C) is provided. The main feature of the calculus is the reduction of both the non-determinism in proof-search and (...) the width of proofs with respect to Hudelmaier's one. These improvements have a significant influence on the performances of the implementation. (shrink)

We introduce a family of tableau calculi for deontic action logics based on finite boolean algebras, these logics provide deontic operators which are applied to a finite number of actions ; furthermore, in these formalisms, actions can be combined by means of boolean operators, this provides an expressive algebra of actions. We define a tableau calculus for the basic logic and then we extend this calculus to cope with extant variations of this formalism; we prove the soundness (...) and completeness of these proof systems. In addition, we investigate the computational complexity of the satisfiability problem for DAL and its extensions; we show this problem is NP-complete when the number of actions considered is fixed, and it is \-Hard when the number of actions is taken as an extra parameter. The tableau systems introduced here can be implemented in PSPACE, this seems reasonable taking into consideration the computational complexity of the logics. (shrink)

We define a tableau calculus for the logic of only knowing and knowing at most ON, which is an extension of Levesque's logic of only knowing O. The method is based on the possible-world semantics of the logic ON, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an (...) additional branch closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ON and O. With respect to [22], in which the tableau calculus was originally presented, in this paper we both provide a formal proof of soundness and completeness of the calculus, and prove the complexity results for the logic ON. (shrink)

The paper settles an open question concerning Negri-style labeled sequent calculi for modal logics and also, indirectly, other proof systems which make (more or less) explicit use of semantic parameters in the syntax and are thus subsumed by labeled calculi, like Brünnler’s deep sequent calculi, Poggiolesi’s tree-hypersequent calculi and Fitting’s prefixed tableau systems. Specifically, the main result we prove (through a semantic argument) is that labeled calculi for the modal logics K and D remain (...) complete w.r.t. valid sequents whose relational part encodes a tree-like structure, when the unique rule which contains an harmful implicit contraction—by which the condition that the premises be less complex than the conclusion is violated—is modified into a contraction-free one respecting the latter condition, thus making the proof-search space finite. (shrink)

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this (...) problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. (shrink)

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus T H(@,E,D, ♦ −) for hybrid logics with the satisfaction operators, the universal modality, (...) the difference modality and the inverse modality as well as the corresponding sequent calculus S H(@,E,D, ♦ −) . They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule. (shrink)

In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t, ⊧ f, ⊧ i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first (...) three entailment relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six. (shrink)

In this paper, first some propositional conditional logics based on Belnap and Dunn’s useful four-valued logic of first-degree entailment are introduced semantically, which are then turned into systems of weakly and unrestrictedly connexive conditional logic. The general frame semantics for these logics makes use of a set of allowable (or admissible) extension/antiextension pairs. Next, sound and complete tableau calculi for these logics are presented. Moreover, an expansion of the basic conditional connexive logics by a constructive implication is considered, (...) which gives an opportunity to discuss recent related work, motivated by the combination of indicative and counterfactual conditionals. Tableau calculi for the basic constructive connexive conditional logics are defined and shown to be sound and complete with respect to their semantics. This semantics has to ensure a persistence property with respect to the preorder that is used to interpret the constructive implication. (shrink)

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to (...) twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation. (shrink)

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. Hybrid logics are here divided into two groups: standard hybrid logics involving nominals as expressions of a separate sort, and non-standard hybrid logics, which do not involve nominals but whose expressive power matches the expressive power of binder-free standard hybrid logics.The original results of this book are split into two parts. This division reflects the division of (...) the book itself. The first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, the efforts focused on hybrid logics referred to as non-standard in this book. Non-standard hybrid logics are understood as modal logics with global counting operators ) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise: 1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators ), which can be easily extended onto other frame classes, 2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames ), MT), MD), MB), MK4) with numerical subscripts coded in binary. Establishing the exponential-size model property for this logic defined over the classes of Euclidean and equivalential frames ), MS5).Results of the second type consist of designing concrete deductive systems for standard and non-standard hybrid logics. More precisely, they include: 1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics. An interesting feature of indicated calculi is the nonbranching character of the rule, 2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics. The internalization technique applied to a tableau calculus for the modal logic with global counting operators is novel in the literature, 3. Devising the first hybrid algorithm involving an inequality solver for modal logics with global counting operators. Transferring the arithmetical part of reasoning to an inequality solver turned out to be sufficient in ensuring termination.The book is directed to philosophers and logicians working with modal and hybrid logics, as well as to computer scientists interested in deductive systems and decision procedures for logics. Extensive fragments of the first part of the book can also serve as an introduction to hybrid logics for wider audience interested in logic.The content of the book is situated in the areas of formal logic and theoretical computer science with some elements of the theory of computational complexity. (shrink)

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we (...) identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503. (shrink)

Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, cKN4, scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and cKN4 are connexive variants of the FDE-based modal logics BK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cBK}^-$$\end{document} and cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} are shown to be algebraizable. The algebraizability of cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} is derived from proving cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} to be definitionally equivalent to MBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MBL}$$\end{document}. All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics. (shrink)

Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. Completeness of these tableaux, however, relies on the cut rule.

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti and Reichenbach on the one hand, and by Cooper and Cantwell on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: (...) DF/TT allows for algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects. (shrink)

The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality (...) into account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)

We study a propositional bimodal logic consisting of two S4 modalities £ and [a], together with the interaction axiom scheme a £ϕ → £ aϕ. In the intended semantics, the plain £ is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation a. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The (...) class of topological Kripke frames characterised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although tableau completeness and decidability are still open questions. (shrink)

The Tableaux Work Bench (TWB) is a meta tableau system designed for logicians with limited programming or automatic reasoning knowledge to experiment with new tableau calculi and new decision procedures. It has a simple interface, a history mechanism for controlling loops or pruning the search space, and modal simplification.

David Nelson’s constructive logics with strong negation may beviewed as alternative paraconsistent logic. These logics have been developedbefore da Costa’s works. We address some philosophical aspects of Nelson’slogics and give technical results concerning Kripke models and tableau calculi. We also suggest possible applications of paraconsistent constructivelogics.

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and Cila. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa's logic C_1) but the whole hierarchy of da Costa's (...) class='Hi'>calculi C_n. This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each C_n, which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa's hierarchy of C-systems as a family of (non deterministically) (n+2)-valued logics, where n is the number of "inconsistently true" truth-values and 2 is the number of "classical" or "consistent" truth-values, for every C_n. (shrink)

ABSTRACT A computational logic, PLPR (Predicate Logic using Polymorphism and Recursion) is presented. Actually this logic is the object language of an automated deduction system designed as a tool for proving mathematical theorems as well as specify and verify properties of functional programs. A useful denotationl semantics and two general deduction methods for PLPR are defined. The first one is a tableau algorithm proved to be complete and also used as a guideline for building complete calculi. The second (...) is a sound and complete natural deduction system. Moreover a fixed point induction rule is introduced for formulas called continuous. The strategies for mechanizing the proofs of the final automated system are based on the previous deduction methods. As the examples of the use of the system show, the implemented theorem prover outperforms humans to a certain extent, retaining logic and calculi generality. (shrink)

The tableaux-constructions have a number of properties which advantageously distinguish them from equivalent axiomatic systems . The proofs in the form of tableaux-constructions have a full accordance with semantic interpretation and subformula property in the sense of Gentzen’s Hauptsatz. Method of tatleaux-construction gives a good substitute of Gentzen’s methods and thus opens a good perspective for the investigations of theoretical as well as applied aspects of logical calculi. It should be noted that application of tableau method in modal, (...) tense, relevant and other non-classical logics is connected with serious difficulties. Tableaux variants are constructed only for a few normal modal systems. As to relevant and paraconsistent logic, the absence of its tableau variants may be considered as a question of special interest. We shall formulate the tableaux for propositional modal system S4.1, S4.2, S4.3, S4.4 and relevant R∗ and E ∗ using Beth’s tableaux construction with indexed formulas. (shrink)

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational (...) variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators. (shrink)

The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are al- ways two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one’s attention since in the classical (two-valued) case the two systems coincide. (In two-valued logic the assignment of a (...) truth value and the exclusion of the opposite truth value describe the same situation.). (shrink)

Tautology elimination rule was successfully applied in automated deduction and recently considered in the framework of sequent calculi where it is provably equivalent to cut rule. In this paper we focus on the advantages of proving admissibility of tautology elimination rule instead of cut for sequent calculi. It seems that one may find simpler proofs of admissibility for tautology elimination than for cut admissibility. Moreover, one may prove its admissibility for some calculi where constructive proofs of cut (...) admissibility fail. As an illustration we present a cut-free sequent calculus for S5 based on tableau system of Fitting and prove admissibility of tautology elimination rule for it. (shrink)

We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite (...) distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers. (shrink)

We present a detailed proof of the admissibility of cut in sequent calculus for some congruent modal logics. The result was announced much earlier during the Trends in Logic Conference, Toruń 2006 and the proof for monotonic modal logics was provided already in Indrzejczak [5]. Also some tableau and natural deduction formalizations presented in Indrzejczak [6] and Indrzejczak [7] were based on this result but the proof itself was not published so far. In this paper we are going to (...) fill this gap. The delay was partly due to the fact that the author from time to time was trying to improve the result and extend it to some additional logics by testing other methods of proving cut elimination. Unfortunately all these attempts failed and cut elimination holds only for these logics which were proved to satisfy this property already in 2005. (shrink)

S5 is one of the most important modal logic with nice syntactic, semantic and algebraic properties. In spite of that, a successful (i.e. cut-free) formalization of S5 on the ground of standard sequent calculus (SC) was problematic and led to the invention of numerous nonstandard, generalized forms of SC. One of the most interesting framework which was very often used for this aim is that of hypersequent calculi (HC). The paper is a survey of HC for S5 proposed by (...) Pottinger, Avron, Restall, Poggiolesi, Lahav and Kurokawa. We are particularly interested in examining different methods which were used for proving the eliminability/admissibility of cut in these systems and present our own variant of a system which admits relatively simple proof of cut elimination. (shrink)

In [6] proof tableau formulations were given of the implication/negation fragments of the important zero-order relevant logics E and R and the semirelevant logic RM . The main purpose of this paper then, is to extend results by giving proof tableau formulations of the distribution-free fragments of E, R and RM and of their first order extensions EQ, RQ and RMQ. Where X is one of these logics, we shall follow [13] in calling its distribution-free fragment OX – (...) the ‘O’ standing for ‘ortho’ which is meant to signify the kinship of these logics to quantum logics or ortho-logics as they now tend to be known. Hence in what follows we shall refer to the logics OX simply as relevant ortho-logics. (shrink)

The paper introduces Hilbert– and Gentzen-style calculi which correspond to systems ${\mathsf{C}_{n}}$ from Gupta and Belnap [3]. Systems ${\mathsf{C}_{n}}$ were shown to be sound and complete with respect to the semantics of finite revision. Here, it is shown that Gentzen-style systems ${\mathsf{GC}_{n}}$ admit a syntactic proof of cut elimination. As a consequence, it follows that they are consistent.

Le Tableau naturel des rapports qui existent entre Dieu, l'homme et l'univers est un ouvrage fondamental pour ceux qui cherchent à comprendre les mystères de la Création. Pour Louis-Claude de Saint-Martin, l'homme porte en lui-même la clé de tous les mystères. Se connaître soi-même est donc la condition primordiale pour accéder à la Connaissance. Les textes sacrés nous enseignent que l'histoire de l'humanité est liée à un drame en trois actes. Le premier est celui de l'âge d'or ; le (...) deuxième celui d'une chute douloureuse dans la matière ; le troisième, qui reste à venir, sera marqué par une rédemption, une réintégration dans le Divin. L'auteur de ce livre s'efforce de percer le sens ésotérique de ces trois étapes. Il montre aussi que l'impasse dans laquelle se trouve l'homme résulte de la confusion qui règne dans ses pensées. S'il nourrissait sa volonté à la chaleur de l'Amour divin, il accéderait à la régénération d'une manière naturelle, et l'humanité trouverait la Connaissance menant à la Paix Profonde. (shrink)

Labelled tableau systems are developed for subintuitionistic logics \, \ and \. These subintuitionistic logics are embedded into corresponding normal modal logics. Hintikka’s model systems are applied to prove the completeness of labelled tableau systems. The finite model property, decidability and disjunction property are obtained by labelled tableau method.

In Symbolic Logic (1932), C. I. Lewis developed five modal systems S1 − S5. S4 and S5 are so-called normal modal systems. Since Lewis and Langford’s pioneering work many other systems of this kind have been investigated, among them the 32 systems that can be generated by the five axioms T, D, B, 4 and 5. Lewis also discusses how his systems can be augmented by propositional quantifiers and how these augmented logics allow us to express some interesting ideas that (...) cannot be expressed in the corresponding quantifier-free logics. In this paper, I will develop 64 normal modal semantic tableau systems that can be extended by propositional quantifiers yielding 64 extended systems. All in all, we will investigate 128 different systems. I will show how these systems can be used to prove some interesting theorems and I will discuss Lewis’s so-called existence postulate and some of its consequences. Finally, I will prove that all normal modal systems are sound and complete and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete. (shrink)

Epsilon Calculi are extended forms of the predicate calculus that incorporate epsilon terms. Epsilon terms are individual terms of the form ‘εxFx’, being defined for all predicates in the language. The epsilon term ‘εxFx’ denotes a chosen F, if there are any F’s, and has an arbitrary reference otherwise. Epsilon calculi were originally developed to study certain forms of Arithmetic, and Set Theory; also to prove some important meta-theorems about the predicate calculus. Later formal developments have included a (...) variety of intensional epsilon calculi, of use in the study of necessity, and more general intensional notions, like belief. An epsilon term such as ‘ εxFx’ was originally read ‘the first F’, and in arithmetical contexts ‘the least F’. More generally it can be read as the demonstrative description ‘that F’, when arising either deictically, i.e. in a pragmatic context where some F is being pointed at, or in linguistic cross-reference situations, as with, for example, ‘There is a red haired man in the room. That red haired man is Caucasian’. The application of epsilon terms to natural language shares some features with the use of iota terms within the theory of descriptions given by Bertrand Russell, but differs in formalising aspects of a slightly different theory of reference, first given by Keith Donnellan. More recently epsilon terms have been used by a number of writers to formalise cross-sentential anaphora, which would arise if ‘that red haired man’ in the linguistic case above was replaced with a pronoun such as ‘he’. There is then also the similar application in intensional cases, like ‘There is a red haired man in the room. Celia believed he was a woman.’. (shrink)

In this study, new sequent calculi for a minimal quantum logic ) are discussed that involve an implication. The sequent calculus \ for \ was established by Nishimura, and it is complete with respect to ortho-models. As \ does not contain implications, this study adopts the strict implication and constructs two new sequent calculi \ and \ as the expansions of \. Both \ and \ are complete with respect to the O-models. In this study, the completeness and (...) decidability theorems for these new systems are proven. Furthermore, some details pertaining to new rules and the strict implication are discussed. (shrink)

This paper contributes to the principled construction of tableau-based decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For converse-free formulas we present a terminating control that does not rely on the usual chain-based blocking scheme. Our tableau systems are based on a new model existence theorem.

The lambda-calculus lies at the very foundation of computer science. Besides its historical role in computability theory it has had significant influence on programming language design and implementation, denotational semantics and domain theory. The book emphasizes the proof theory for the type-free lambda-calculus. The first six chapters concern this calculus and cover the basic theory, reduction, models, computability, and the relationship between the lambda-calculus and combinatory logic. Chapter 7 presents a variety of typed calculi; first the simply typed lambda-calculus, (...) then Milner-style polymorphism and, finally the polymporphic lambda-calculus. Chapter 8 concerns three variants of the type-free lambda-calculus that have recently appeared in the research literature: the lazy lambda-calculus, the concurrent y-calculus and the lamdba omega-calculus. The final chapter contains references and a guide to further reading. There are exercises throughout. In contrast to earlier books on these topics, which were written by logicians, the book is written from a computer science perspective and emphasizes the practical relevance of many of the key theoretical ideas. The book is intended as a course text for final year undergraduates or first year graduate students in computer science. Research students should find it a useful introduction to more specialist literature. (shrink)

We discuss a propositional logic which combines classical reasoning with constructive reasoning, i.e., intuitionistic logic augmented with a class of propositional variables for which we postulate the decidability property. We call it intuitionistic logic with classical atoms. We introduce two hypersequent calculi for this logic. Our main results presented here are cut-elimination with the subformula property for the calculi. As corollaries, we show decidability, an extended form of the disjunction property, the existence of embedding into an intuitionistic modal (...) logic and a partial form of interpolation. (shrink)

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this paper, we consider logic systems called L'T without this kind of constants but limited to the case where T is a finite poset. We study the tableau method for this system and we prove its completeness for a class of formulas with respect to an (...) algebraic semantics. (shrink)

Product logic Π is an important t-norm based fuzzy logic with conjunction interpreted as multiplication on the real unit interval [0,1], while Cancellative hoop logic CHL is a related logic with connectives interpreted as for Π but on the real unit interval with 0 removed (0,1]. Here we present several analytic proof systems for Π and CHL, including hypersequent calculi, co-NP labelled calculi and sequent calculi.

The book aims to formalise tableau methods for the logics of propositions and names. The methods described are based on Set Theory. The tableau rule was reduced to an ordered n-tuple of sets of expressions where the first element is a set of premises, and the following elements are its supersets.