Our investigation is concerned with the finite model property with respect to admissible rules. We establish general sufficient conditions for absence of fmp w. r. t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic λ containing K4 with the co-cover property and of width > 2 has fmp w. r. t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope (...) of this theorem–K4 itself, S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Thus the situation is completely opposite to the case of the ordinary fmp–the absolute majority of important logics have fmp, but not with respect to admissibility. As regards logics of width ≤ 2, there exists a zone for fmp w. r. t. admissibility. It is shown that all modal logics A of width ≤ 2 extending S4 which are not sub-logics of three special tabular logics have fmp w.r.t. admissibility. (shrink)

Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic (MALL) and for affine logic (LLW), i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL (which we call IMALL), and intuitionistic LLW (which we call ILLW). In addition, we shall show the finite model property for contractive linear logic (LLC), i.e., linear (...) logic with contraction, and for its intuitionistic version (ILLC). The finite model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL - -systems except FL c and GL - c of Ono [11], that will settle the open problems stated in Ono [12]. (shrink)

Tense logics in the bimodal propositional language are investigated with respect to the Finite Model Property. In order to prove positive results techniques from investigations of modal logics above K4 are extended to tense logic. General negative results show the limits of the transfer.

We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].

Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic and for affine logic, i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL, and intuitionistic LLW. In addition, we shall show the finite model property for contractive linear logic, i.e., linear logic with contraction, and for its intuitionistic version. The finite (...) model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL$^-$-systems except FL$_c$ and GL$^-_c$ of Ono [11], that will settle the open problems stated in Ono [12]. (shrink)

The tangle modality is a propositional connective that extends basic modal logic to a language that is expressively equivalent over certain classes of finite frames to the bisimulation-invariant fragments of both first-order and monadic second-order logic. This paper axiomatises several logics with tangle, including some that have the universal modality, and shows that they have the finite model property for Kripke frame semantics. The logics are specified by a variety of conditions on their validating frames, including (...) local and global connectedness properties. Some of the results have been used to obtain completeness theorems for interpretations of tangled modal logics in topological spaces. (shrink)

It is proved that the variety of relevant disjunction lattices has the finite embeddability property. It follows that Avron's relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron's result that RMImin is decidable.

The logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: γ, xn → y / γ, xm → y. It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has (...) the finite model property with respect to its algebraic semantics and hence that the logic is decidable. (shrink)

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.

The finite model property (FMP) in weakly transitive tense logics is explored. Let \(\mathbb {S}=[\textsf{wK}_t\textsf{4}, \textsf{K}_t\textsf{4}]\) be the interval of tense logics between \(\textsf{wK}_t\textsf{4}\) and \(\textsf{K}_t\textsf{4}\). We introduce the modal formula \(\textrm{t}_0^n\) for each \(n\ge 1\). Within the class of all weakly transitive frames, \(\textrm{t}_0^n\) defines the class of all frames in which every cluster has at most _n_ irreflexive points. For each \(n\ge 1\), we define the interval \(\mathbb {S}_n=[\textsf{wK}_t\textsf{4T}_0^{n+1}, \textsf{wK}_t\textsf{4T}_0^{n}]\) which is a subset of \(\mathbb (...) {S}\). There are \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) lacking the FMP, and there are \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) having the FMP. Then we explore the FMP in finitely alternative tense logics \(L_{n,m}=L\oplus \{\textrm{Alt}_n^F, \textrm{Alt}_m^P\}\) with \(n,m\ge 0\) and \(L\in \mathbb {S}\). For all \(k\ge 0\) and \(n,m\ge 1\), we define intervals \(\mathbb {F}^k_{n,m}\), \(\mathbb {P}^k_{n,m}\) and \(\mathbb {S}^k_{n,m}\) of tense logics. The number of logics lacking the FMP in them is either 0 or \(2^{\aleph _0}\), and the number of logics having the FMP in them is either finite or \(2^{\aleph _0}\). (shrink)

The aim of this paper is to show that the implicational fragment BKof the intuitionistic propositional calculus (IPC) without the rules of exchange and contraction has the finite model property with respect to the quasivariety of left residuation algebras (its equivalent algebraic semantics). It follows that the variety generated by all left residuation algebras is generated by the finite left residuation algebras. We also establish that BKhas the finite model property with respect to (...) a class of structures that constitute a Kripke-style relational semantics for it. The results settle a question of Ono and Komori [OK85]. (shrink)

MIPC is a well-known intuitionistic modal logic of Prior and Bull . It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property whenever L is Kripke-complete and universal.

This paper proposes a semantic description of the linear step-like temporal multi-agent logic with the universal modality \(\mathcal{LTK}.sl_U\) based on the idea of non-reflexive non-transitive nature of time. We proved a finite model property and projective unification for this logic.

An old conjecture of modal logics states that every splitting of the major systems K4, S4, G and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these (...) splitting frames namely that they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ/f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive. (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.

In this paper I shall look at the application of the ltration technique to omnitemporal logic . The principal result of the paper will be that the system BSeg of [3] has the nite model property; but I shall also make a few remarks about the system B+ of [2].

In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further (...) research. (shrink)

There are two known general results on the finite model property of commutators [L0,L1]. If L is finitely axiomatizable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so they have the fmp. On the negative side, if both L0 and L1 are determined by transitive frames and have frames of arbitrarily large depth, then [L0,L1] does not have the fmp. In this paper (...) we show that commutators with a “weakly connected” component often lack the fmp. Our results imply that the above positive result does not generalize to universally axiomatizable component logics, and even commutators without “transitive” components such as [K3,K] can lack the fmp. We also generalize the above negative result to cases where one of the component logics has frames of depth one only, such as [S4.3,S5] and the decidable product logic S4.3×S5. We also show cases when already half of commutativity is enough to force infinite frames. (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.

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.

We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus ( DFL ) whose algebraic semantics is the class of distributive residuated lattices ( DRL ). The problem was left open in [8, 5]. We use the method of nuclei and quasi–embedding in the style of [10, 1].

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of ♦-formulae, i.e. each ♦-formula has a ℒ-model iff it has a finite ℒ-model. Roughly speaking, a ♦-formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

Related Works: Part I: Michael Zakharyaschev. Canonical Formulas for $K4$. Part I: Basic Results. J. Symbolic Logic, Volume 57, Issue 4 , 1377--1402. Project Euclid: euclid.jsl/1183744119 Part II: Michael Zakharyaschev. Canonical Formulas for K4. Part II: Cofinal Subframe Logics. J. Symbolic Logic, Volume 61, Issue 2 , 421--449. Project Euclid: euclid.jsl/1183745008.

Related Works: Part I: Michael Zakharyaschev. Canonical Formulas for $K4$. Part I: Basic Results. J. Symbolic Logic, Volume 57, Issue 4, 1377--1402. Project Euclid: euclid.jsl/1183744119 Part II: Michael Zakharyaschev. Canonical Formulas for K4. Part II: Cofinal Subframe Logics. J. Symbolic Logic, Volume 61, Issue 2, 421--449. Project Euclid: euclid.jsl/1183745008.

Djordjević [Dj 1] proved that under natural technical assumptions, if a complete L n -theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of L n -theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces.

Reviews of the papers mentioned in the title.