A theoretical investigation into quantitative modal logic

Decidability of order-based modal logics

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.

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

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.

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.

Some Modal Calculi Based On IC

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

Full classical S5 in natural deduction with weak normalization

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

1 Modal logic: a semantic perspective

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.

This paper characterizes the admissible rules for six interesting intuitionistic modal logics: iCK4, iCS4≡IPC, strong Löb logic iSL, modalized Heyting calculus mHC, Kuznetsov-Muravitsky logic KM, and propositional lax logic PLL. Admissible rules are rules that can be added to a logic without changing the set of theorems of the logic. We provide a Gentzen-style proof theory for admissibility that combines methods known for intuitionistic propositional logic and classical modal logic. From this proof theory, we extract bases for the admissible rules, i.e., sets of admissible rules that derive all other admissible rules. In addition, we show that admissibility is decidable for these logics.

We introduce a paraconsistent modal logic$$\mathbf {K}\mathsf {G}^2$$KG2, based on Gödel logic with coimplication (bi-Gödel logic) expanded with a De Morgan negation$$\lnot $$¬. We use the logic to formalise reasoning with graded, incomplete and inconsistent information. Semantics of$$\mathbf {K}\mathsf {G}^2$$KG2is two-dimensional: we interpret$$\mathbf {K}\mathsf {G}^2$$KG2on crisp frames with two valuations$$v_1$$v1and$$v_2$$v2, connected via$$\lnot $$¬, that assign to each formula two values from the real-valued interval [0, 1]. The first (resp., second) valuation encodes the positive (resp., negative) information the state gives to a statement. We obtain that$$\mathbf {K}\mathsf {G}^2$$KG2is strictly more expressive than the classical modal logic$$\mathbf {K}$$Kby proving that finitely branching frames are definable and by establishing a faithful embedding of$$\mathbf {K}$$Kinto$$\mathbf {K}\mathsf {G}^2$$KG2. We also construct a constraint tableau calculus for$$\mathbf {K}\mathsf {G}^2$$KG2over finitely branching frames, establish its decidability and provide a complexity evaluation.