We show that the modal µ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [4]. Further, we introduce the modal µ~-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result (...) allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formula. (shrink)

We show that the modal µ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [4]. Further, we introduce the modal µ~-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result (...) allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formula. (shrink)

We study the strictness of the modal μ-calculus hierarchy over some restricted classes of transition systems. First, we prove that over transitive systems the hierarchy collapses to the alternationfree fragment. In order to do this the finite model theorem for transitive transition systems is proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment. Finally, we show that the hierarchy is strict over reflexive frames. By proving the finite (...) model theorem for reflexive systems the same results holds for finite models. (shrink)

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.

This paper treats a kind of a modal logic based on the intuitionistic propositional logic which arose from the provability predicate in the first order arithmetic. The semantics of this calculus is presented in both a relational and an algebraic way.Completeness theorems, existence of a characteristic model and of a characteristic frame, properties of FMP and FFP and decidability are proved.

The first example of an intuitively meaningful propositional logic without the finite model property, and still the simplest one in the literature. The question of its decidability appears still to be open.

The modal -calculus extends basic modal logic with second-order quantification in terms of arbitrarily nested fixpoint operators. Its satisfiability problem is EXPTIME-complete. Decision procedures for the modal -calculus are not easy to obtain though since the arbitrary nesting of fixpoint constructs requires some combinatorial arguments for showing the well-foundedness of least fixpoint unfoldings. The tableau-based decision procedures so far also make assumptions on the unfoldings of fixpoint formulas, e.g., explicitly require formulas to be in guarded (...) normal form. In this paper we present a tableau calculus for deciding satisfiability of arbitrary, i.e., not necessarily guarded -calculus formulas. It is based on a new unfolding rule for greatest fixpoint formulas which allows unguarded formulas to be handled without an explicit transformation into guarded form, thus avoiding a exponential blow-up in formula size. We prove soundness and completeness of the calculus, and compare it empirically to using guarded transformation instead. The new unfolding rule can also be used to handle nested star operators in PDL formulas correctly. (shrink)

The first example of an intuitively meaningful propositional logic without the finite model property, and still the simplest one in the literature. The question of its decidability appears still to be open.

The modal object calculus is the system of logic which houses the (proper) axiomatic theory of abstract objects. The calculus has some rather interesting features in and of itself, independent of the proper theory. The most sophisticated, type-theoretic incarnation of the calculus can be used to analyze the intensional contexts of natural language and so constitutes an intensional logic. However, the simpler second-order version of the calculus couches a theory of fine-grained properties, relations and propositions (...) and serves as a framework for defining situations, possible worlds, stories, and fictional characters, among other things. In the present paper, we focus on the second-order calculus. The second-order modal object calculus is so-called to distinguish it from the second-order modal predicate calculus. Though the differences are slight, the extra expressive power of the object calculus significantly enhances its ability to resolve logical and philosophical concepts and problems. (shrink)

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a (...) translation of mHC into K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

Global properties of canonical derivability predicates in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, (...) finite model property, interpolation and the fixed point theorem. (shrink)

The tetravalent modal logic is one of the two logics defined by Font and Rius :481–518, 2000) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the (...) algebras is not so good as in $${{\mathcal {TML}}}^N$$ TML N, but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus :481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al., we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic. (shrink)

We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

Masini, A., 2-Sequent calculus: a proof theory of modalities, Annals of Pure and Applied Logic 58 229–246. In this work we propose an extension of the Getzen sequent calculus in order to deal with modalities. We extend the notion of a sequent obtaining what we call a 2-sequent. For the obtained calculus we prove a cut elimination theorem.

Dieser Aufsatz zeigt die Möglichkeiten auf, aus den von Leibniz im April 1679 geschriebenen logischen Abhandlungen ein numerisches Modell für die durch Widerspruchsfreiheit bestimmten Modalbegriffe zu lesen. Dazu wird als erstes die Definition der Modalisatoren durch die Widerspruchsfreiheit betrachtet und die Probleme‚ die das Leibniz'sche ‘continere’ oder ‘implicare contradictionem’ auslöst. Danach werden die in den Schriften von 1679 entwickelten numerischen Modelle untersucht‚ besonders dasjenige‚ das sich auf den Mechanismus des charakteristischen Zahlenpaares stützt. Dieses Modell wird für den Aufbau formaler Definitionen (...) von Möglichkeit‚ Unmöglichkeit‚ Kompatibilität und Inkompatibilität benutzt. Schließlich werden kurz einige spezifische Probleme angeführt‚ die die vorgeschlagene Interpretation offenläßt: die Darstellung der einfachen Begriffe innerhalb des numerischen Kalküls und die Beziehung zwischen den Wortpaaren Kompatibilität-Inkompatibilität und Kompossibilität-Inkompossibilität. Was diese letzteren betrifft‚ wird die Bedeutung betont‚ die eine diachronische Betrachtung der Leibnizschen Lehre für ein besseres Verständnis der theoretischen Zusammenhänge hat‚ in denen diese Bezeichnungen auftreten. (shrink)

We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

A new proof style adequate for modal logics is defined from the polynomial ring calculus. The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra???Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S 5, and can be easily (...) extended to other modal logics. (shrink)

We characterize the expressive power of the modal mu-calculus on monotone neighborhood structures, in the style of the Janin-Walukiewicz theorem for the standard modal mu-calculus. For this purpose we consider a monadic second-order logic for monotone neighborhood structures. Our main result shows that the monotone modal mu-calculus corresponds exactly to the fragment of this second-order language that is invariant for neighborhood bisimulations.

Quasi-Boolean modal algebras are quasi-Boolean algebras with a modal operator satisfying the interaction axiom. Sequential quasi-Boolean modal logics and the relational semantics are introduced. Kripke-completeness for some quasi-Boolean modal logics is shown by the canonical model method. We show that every descriptive persistent quasi-Boolean modal logic is canonical. The finite model property of some quasi-Boolean modal logics is proved. A cut-free Gentzen sequent calculus for the minimal quasi-Boolean logic is developed and we show (...) that it has the Craig interpolation property. (shrink)

The paper contains an exposition of some non standard approach to gentzenization of modal logics. The first section is devoted to short discussion of desirable properties of Gentzen systems and the short review of various sequential systems for modal logics. Two non standard, cut-free sequent systems are then presented, both based on the idea of using special modal sequents, in addition to usual ones. First of them, GSC I is well suited for nonsymmetric modal logics The (...) second one, GSC II is devised specially for symmetric, i.e. Blogics. GSC I and GSC II are not different formalizations, from the theoretical point of view GSC I may be seen as a simplification of the more general approach present in GSC II. They are considered separately, mainly because Blogics demand different, and more complicated, strategy in completeness proof, whereas non symmetric logics are easily and uniformly characterised by means of Fitting's Consistency Properties. The weakest modal logic captured by this formalization is minimal regular logic C, but many stronger logics are obtainable by addition of suitable structural rules, which conforms to Do en's methodology. Both variants of GSC satisfy also other, besides cut-freedom, desirable properties. (shrink)

Motivated by semantic inferentialism and logical expressivism proposed by Robert Brandom, in this paper, I submit a nonmonotonic modal relevant sequent calculus equipped with special operators, □ and R. The base level of this calculus consists of two different types of atomic axioms: material and relevant. The material base contains, along with all the flat atomic sequents (e.g., Γ0, p |~0 p), some non-flat, defeasible atomic sequents (e.g., Γ0, p |~0 q); whereas the relevant base consists of (...) the local region of such a material base that is sensitive to relevance. The rules of the calculus uniquely and conservatively extend these two types of nonmonotonic bases into logically complex material/relevant consequence relations and incoherence properties, while preserving Containment in the material base and Reflexivity in the relevant base. The material extension is supra-intuitionistic, whereas the relevant extension is stronger than a logic slightly weaker than R. The relevant extension also avoids the fallacies of relevance. Although the extended material consequence relation is defeasible and insensitive to relevance, it has local regions of indefeasibility and relevance (the latter of which is marked by the relevant extension). The newly introduced operators, □ and R, codify these local regions within the same extended material consequence relation. (shrink)

We present a rooted hypersequent calculus for modal propositional logic S5. We show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well.

We propose a translation into Modal Logic of the ideas that formalise belief change in the Situation Calculus. This translation is extended to the case of revision. In the conclusion is presented a set of open issues.