Some Modal Calculi Based On IC

Decidability of order-based modal logics

Recently, Baltag and van Benthem introduced a decidable logic of functional\ndependence (LFD) that extends the logic of Cylindrical Relativized Set Algebras\n(CRS) with atomic local dependence statements. Its semantics can be given in\nterms of generalised assignment models or their modal counterparts, hence the\nlogic is both a first-order and a modal logic. We show that LFD has the finite\nmodel property (FMP) using Herwig's theorem on extending partial isomorphisms,\nand prove a bisimulation invariance theorem characterizing LFD as a fragment of\nfirst-order logic.\n

A theoretical investigation into quantitative modal logic

The main result of this paper is a complete semantics description of all modal logics with finite model property that preserves all admissible for S4 inference rules: a modal logic λ with fmp above S4 preserves the admissibility iff λ has so-called co-cover property. As an application it is shown that there are continuously many logics of this kind. On base mentioned above semantical criterion, as corollary, we give some precise description of all tabular logics that preserves admissible for S4 rules.

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

This paper considers the bi-modal logic with both $$\Box $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>□</mml:mo> </mml:math> and $$\Diamond $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>◊</mml:mo> </mml:math> arising from Kripke models with crisp accessibility whose propositions are valued over the standard Gödel algebra [0, 1]. Since this logic lacks the finite model property, we study the logic $$\textbf{GW}^\textrm{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>GW</mml:mi> <mml:mtext>c</mml:mtext> </mml:msup> </mml:math> relying on witnessed Kripke models where, for each modal formula, there is an assignment where the formula without the modality takes the same value as the modal one. We provide a cut-free sequent calculus and we exploit it to prove that $$\textbf{GW}^\textrm{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>GW</mml:mi> <mml:mtext>c</mml:mtext> </mml:msup> </mml:math> is decidable and meets the finite model property. Finally, we explore a connection between the witnessed models and the well-known bi-relational Kripke semantics.

It appears that the behaviour of interpolation over the modal S4 logic is similar to interpolation in superintuitionistic logics. It is shown that all extensions of S4 with interpolation property for deducibility IPD are modal companions of superintuitionistic logics with CIP, but there is an intermediate logic with CIP that has no modal companions with IPD. On the other hand, all modal companions of intermediate logics with CIP have a weaker version of interpolation, which is CIP restricted to those implications, where all occurrences of variables are preceded by necessity symbol. It is proved that there are no more than forty-nine modal logics with IPD in the family of normal extensions of S4; in this list there are twelve logics that have IPD but do not have CIP. All the forty-nine logics are finitely axiomatizable and have the finite model property. It is proved that IPD and CIP problems are deducible over S4, and amalgamation and superamalgamation are base-decidable in varieties of closure algebras.

419 The effect of environmental factors on spontaneous abortion: A case control study in Iran

This work presents three increasingly expressive Dynamic Logics in which the programs are described in a language based on CCS. Our goal is to build dynamic logics that are suitable for the description and verification of properties of communicating concurrent systems, in a similar way as PDL is used for the sequential case. In order to accomplish that, CCS’s operators and constructions are added to a basic modal logic. Doing this, the semantics of CCS’s parallel operator allows us to build dynamic logics that support communicating and concurrent programs. We build a simple Kripke semantics for these logics, provide complete axiomatizations for them and show that they have the finite model property. This contrasts with other dynamic logics with parallel operators presented in the literature, such as Peleg’s Concurrent PDL with Channels, where either the parallel programs cannot communicate, or at least one of the properties mentioned above (simple Kripke semantics, complete axiomatization and finite model property) is missing.

LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.

We consider an extension of the unary negation fragment of first-order logic in which arbitrarily many binary symbols may be required to be interpreted as equivalence relations. We show that this extension has the finite model property. More specifically, we show that every satisfiable formula has a model of at most doubly exponential size. We argue that the satisfiability (= finite satisfiability) problem for this logic is 2-ExpTime-complete. We also transfer our results to a restricted variant of the guarded negation fragment with equivalence relations.

The fluted fragment with transitive relations

1 Modal logic: a semantic perspective

A binary modal logic for the intersection types of lambda-calculus

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally $\vee $ complemented Heyting algebra (c$\vee $cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension ${\textrm {ILM}}$-${\vee }$ for c$\vee $cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and ${\textrm {ILM}}$-${\vee }$ are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and ${\textrm {ILM}}$-${\vee }$ are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics $K_{im}$ and $K_{im-{\vee }}$ are discussed, where the language does not include implication. The $K_{im}$-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the $K_{im}$-algebras and $K_{im-{\vee }}$-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee }}$ is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and ${\textrm {ILM}}$-${\vee }$.