Modal Logic S4 Research Articles

Logical Extensions of Aristotle’s Square

We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is $$\square\neg\alpha$$ , I is $$\neg\square\neg\alpha$$ and O is $$\square\neg\alpha$$ , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether $$\neg$$ is involutive or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of $$\{\alpha,\square\alpha\}$$ by the logical operations $$\{\neg,\wedge,\vee\}$$ , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.

Full classical S5 in natural deduction with weak normalization

Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [D. Prawitz, Natural Deduction: A Proof-theoretical Study, in: Stockholm Studies in Philosophy, vol. 3, Almqvist and Wiksell, Stockholm, 1965. Reprinted at: Dover Publications, Dover Books on Mathematics, 2006] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic only for a language without ∨ , ∃ and with a restricted application of reduction ad absurdum. Reduction steps related to ∨ , ∃ and classical negation bring about many problems solved only rather recently. For classical S5 modal logic, Prawitz defined a normalizable system, but for a language without ∨ , ∃ , ◊ and, for a propositional language without ◊ , Medeiros [M.da P.N. Medeiros, A new S4 classical modal logic in natural deduction, Journal of Symbolic Logic 71 (3) (2006) 799–809] presented a normalizable system for classical S4. We can mention many cut-free Gentzen systems for S4, S5 and K45/K45D, some normalizable natural deduction systems for intuitionistic modal logics and one more for full classical S4, but not for full classical S5. Here our focus is on the definition of a classical and normalizable natural deduction system for S5, taking not only □ and ◊ as primitive symbols, but also all connectives and quantifiers, including classical negation, disjunction and the existential quantifier. The normalization procedure is based on the strategy proposed by Massi [C.D.B. Massi, Provas de normalizaçaõ para a lógica clássica, Ph.D. Thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990] and Pereira and Massi [L.C. Pereira, C.D.B. Massi, Normalização para a lógica clássica, in: O que nos faz pensar, Cadernos de Filosofia da PUC-RJ, vol. 2, 1990, pp. 49–53] for first-order classical logic to cope with the combined use of classical negation, disjunction and the existential quantifier. Here we extend such results to deal with □ and ◊ too. The elimination rule for ◊ uses the notions of connection and of essentially modal formulas already proposed by Prawitz for the introduction of □ . Beyond weak normalization, we also prove the subformula property for full S5.

Sufficient conditions for cut elimination with complexity analysis

Sufficient conditions for first-order-based sequent calculi to admit cut elimination by a Schütte–Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related to the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.

Classical Modal Display Logic in the Calculus of Structures and Minimal Cut-free Deep Inference Calculi for S5

We begin by showing how to faithfully encode the Classical Modal Display Logic (CMDL) of Wansing into the Calculus of Structures (CoS) of Guglielmi. Since every CMDL calculus enjoys cut-elimination, we obtain a cut-elimination theorem for all corresponding CoS calculi. We then show how our result leads to a minimal cut-free CoS calculus for modal logic S5. No other existing CoS calculi for S5 enjoy both these properties simultaneously.

A Deep Inference System for the Modal Logic S5

We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms.

The Embedding Theorem: Its Further Developments and Consequences. Part 1

We outline the Gödel-McKinsey-Tarski Theorem on embedding of Intuitionistic Propositional Logic Int into modal logic S4 and further developments which led to the Generalized Embedding Theorem. The latter in turn opened a full-scale comparative exploration of lattices of the (normal) extensions of modal propositional logic S4, provability logic GL, proof-intuitionistic logic KM, and others, including Int. The present paper is a contribution to this part of the research originated from the Gödel-McKinsey-Tarski Theorem. In particular, we show that the lattice ExtInt of intermediate logics is likely to be the only constructing block with which ExtS4, the lattice of the extensions of S4, can be formed. We, however, advise the reader that our exposition is different from the historical lines along which some of the results discussed below came to light. Part 1, presented here, deals mostly with structural issues of extensions of logics, where algebraic semantics, though underlying this approach, is used merely occasionally. Part 2 will be devoted to algebraic analysis of the Embedding Theorem.

A new S4 classical modal logic in natural deduction

AbstractWe show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.

Identical Twins, Deduction Theorems, and Pattern Functions: Exploring the Implicative BCSK Fragment of S5

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives.

’In yaqum vs ’in qāma: un'ipotesi modale

This paper considers all systems of conditional structures of the type ’ in sarṭ jawāb al-sarṭ as treated by main Arab scholars. It investigates the opposition between yaqum vs qāma forms as a reason for including/excluding some verbal structures in/from the conditional system of each considered grammarian. An assumption is made that the frontier between yaqum vs qāma verbal forms in the conditional context can be interpreted in terms of the opposition between the two Aristotelian modal categories of “possibility” and “necessity”. The hypothesis is tested on the occurrences of ’ in sarṭ jawāb al-sarṭ structures in the Koran. A formalization of conditional systems by means of standard modal logic S4 is then given and discussed.

Natural Deduction for Full S5 Modal Logic with Weak Normalization

Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [Prawitz, D., “Natural Deduction: A Proof-theoretical Study”, Stockholm Studies in Philosophy 3, Almqvist and Wiksell, Stockholm, 1965] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic just for a language without ∨, ∃ and with a restricted application of reduction ad absurdum. Reduction steps related to ∨, ∃ and classical negation brings about a lot of problems solved only rather recently [Seldin, J., Normalization and Excluded Middle I, Studia Logica 48 (1989), 193–217; Stalmark, G., Normalizations Theorems for Full First Order Classical Natural Deduction, The Journal of Symbolic Logic 56 (1991); Massi, C.D.B., “Provas de Normalização para a Lógica Clássica”, Ph.D. thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990; Pereira, L.C., and C.D.B. Massi, Normalização para a Lógica Clássica, O que nos faz pensar, Cadernos de Filosofia da PUC-RJ 2 (1990), 49–53]. For classical S4/S5 modal logics, Prawitz defined normalizable systems, but for a language without ∨, ∃ and ◊. We can mention cut-free Gentzen systems for S4/S5 [Meré, M.C., and L.C. Pereira. “A New Cut-Free System for Intuitionistic S4”, Encontro Brasileiro de Lógica, Salvador, 1996; Mints, G.E., “Lewis' Systems and System T (1965–1973)”, Selected Papers in Proof Theory, Studies in Proof Threory, North-Holland, 1992, 221–293; Mints, G.E., Cut-Free Calculi of the S5 Type Studies in Constructive Mathematics and Mathematical Logic, Part II, Seminars in Mathematics 8 (1970), 79–82; Bragança, R.C.M., E.H. Haeusler, and L.C. Pereira, A New Cut-Free Sequent calculus for S5, The Bulletin of Symbolic Logic 5 (1999), 497–498; Braüner, T., A Cut-Free Gentzen Formulation of the Modal Logic S5, Logic Journal of the IGPL 8 (2000), 629–643; H. Wansing. Sequent Calculi for Normal Modal Propositional Logics, Journal of Logic and Computation 4 (1994), 125–142], normalizable natural deduction systems for intuitionistic modal logics [Simpson, A.K., “The Proof Theory and Semantics of Intuitionistic Modal Logic”, Ph.D. thesis, University of Edinburgh, 1994; Masini, A., 2-Sequent Calculus: Intuitionism and Natural Deduction, Journal of Logic and Computation 3 (1993), 533–562] and for full classical S4 [Martins, L.R., and A.T. Martins, “Normalizable Natural Deduction Rules for S4 Modal Operators”, World Congress on Universal Logic, Montreux, 2005, 79–80], but not for full classical S5. Here our focus is in the definition of a classical and normalizable natural deduction system for S5, taken not only □ and ◊ as primitive symbols, but also all connectives and quantifiers, including classical negation, disjunction and the existential quantifier. The normalization procedure will be based on the strategy proposed by [Massi, C.D.B., “Provas de Normalização para a Lógica Clássica”, Ph.D. thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990; Pereira, L.C., and C.D.B. Massi, Normalização para a Lógica Clássica, O que nos faz pensar, Cadernos de Filosofia da PUC-RJ 2 (1990), 49–53] to cope with the combined use of classical negation, ∨ and ∃. We will extend such results to deal with ◊ too. The elimination rule for ◊ will use the notions of connection and of essentially modal formulas already proposed by Prawitz for the introduction of □. Weak normalization and subformula property is proved for full S5.

Normal derivability in modal logic

The standard rule of necessitation in systems of natural deduction for the modal logic S4 concludes □A from A whenever all assumptions A depends on are modal formulas. This condition prevents the composability and normalization of derivations, and therefore modifications of the rule have been suggested. It is shown that both properties hold if, instead of changing the rule of necessitation, all elimination rules are formulated in the manner of disjunction elimination, i.e. with an arbitrary consequence. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Ground Nonmonotonic Modal Logic S5: New Results

We study logic programs under Gelfond's translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T and S5 are equivalent. Furthermore, we also show that these logics are equivalent to a nonmonotonic logic that we construct using the well known F O U R bilattice. We will call this semantic GNM-S5 as a reminder of its origin in the logic S5. Finally we show that, for normal programs, our approach is closely related to theWell-Founded-by-Cases Semantics introduced by Schlipf and the WFS+ proposed by Dix. We prove that GNM-S5 has the properties of classicality and extended cut. While WFS+ also supports classicality it fails to satisfy the extended cut principle, an important property available in other semantics such as stable models. Hence, we claim that GNM-S5 is a good candidate for defining a nonmonotonic semantics closer to the direction of classical logic.

Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLScw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLScw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.

On the complexity of the disjunction property in intuitionistic and modal logics

In this article we study the complexity of disjunction property for intuitionistic logic, the modal logics S4 , S4.1 , Grzegorczyk logic, Gödel-Löb logic, and the intuitionistic counterpart of the modal logic K . For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs. The techniques we use do not require proving structural properties of the calculi in hand, such as the cut-elimination theorem or the normalization theorem. This is a key point of our approach, since it allows us to treat logics for which only Hilbert-style characterizations are known.

A spatial modal logic with a location interpretation

A spatial modal logic (SML) is introduced as an extension of the modal logic S4 with the addition of certain spatial operators. A sound and complete Kripke semantics with a natural space (or location) interpretation is obtained for SML. The finite model property with respect to the semantics for SML and the cut-elimination theorem for a modified subsystem of SML are also presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Knowledge updates: Semantics and complexity issues

We consider the problem of updating of an agent's knowledge. We propose a formal method of knowledge update on the basis of the semantics of modal logic S5. In our method, an update is specified according to the minimal change on both the agent's actual world and knowledge. We discuss general minimal change properties of knowledge update and show that our knowledge update operator satisfies all the update postulates of Katsuno and Mendelzon. We characterize several specific forms of knowledge update which have important applications in reasoning about change of agents' knowledge. We also examine the persistence property of knowledge and ignorance associated with knowledge update. We then investigate the computational complexity of model checking for knowledge update. We first show that in general the model checking for knowledge update is Σ 2 P -complete. We then identify a subclass of knowledge update problems that has polynomial time complexity for model checking. We point out that some important knowledge update problems belong to this subclass. We further address another interesting subclass of knowledge update problems for which the complexity of model checking is NP-complete.

Towards OntoClean 2.0: A framework for rigidity

The OntoClean methodology was based on a set of formal meta-properties whose semantics were specified in S5 modal logic. One of these metaproperties, Rigidity, has come under more focused scrutiny ...

Path calculus in the modal logic S4

We describe the tableau and inverse calculi for the propositional modal logic S4. The formulas are treated as sets of paths. We obtain the upper bound for the number of applications of rules in the deduction tree.

Resolution method for some class of formulas of modal logic S4

There is not abstract.

The Resolution Method for One Reducible Class of Formulas of the First-Order Modal Logic S4

In the paper, we describe a resolution method for the family of the formulas of the form ∀*□i∀*∃*(L1 ∨ ⋯ ∀ Ls), where i = 0 1 and Lj are modal literals. Negations here stand directly before classical atomic formulas, the formulas may contain constants. We also present the absorption tactics for a set of such formulas.