Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.

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)

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.

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)

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)

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)

We use the apparatus of the canonical formulas introduced by Zakharyaschev [10] to prove that all finitely axiomatizable normal modal logics containing K4.3 are decidable, though possibly not characterized by classes of finite frames. Our method is purely frame-theoretic. Roughly, given a normal logic L above K4.3, we enumerate effectively a class of frames with respect to which L is complete, show how to check effectively whether a frame in the class validates a given formula, and then (...) apply a Harropstyle argument to establish the decidability of L, provided of course that it has finitely many axioms. (shrink)

We show that modal logics characterized by a class of frames satisfying the insertion property are suitable for Reiter's default logic. We refine the canonical fix point construction defined by Marek, Schwarz and Truszczyński for Reiter's default logic and thus we addrress a new paradigm for nonmonotonic logic. In fact, differently from the construction defined by these authors. we show that suitable modal logics for such a construction must indeed contain K D4. When reflexivity is added to the modal (...) logic used for the fix point construction then we come to the Marek Schwarz and Truszczyński framework for Reiter's default logic. Our framework, in fact, is appropriate also to the family of modal logics in between S4 and S4f. If, instead, reflexivity is dropped, then we show that a new family of modal logics is gained, namely the modal logics in between KD4 and KD4Z. The upper bound can be extended to the modal logic KD4LZ whenever the propositional language taken into account is finite. (shrink)

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

In this paper we investigate those extensions of the bimodal provability logic ${\vec CSM}_{0}$ (alias ${\vec PRL}_{1}$ or ${\vec F}^{-})$ which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing ${\vec CSM}_{0}$ are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are (...) not even complete with respect to Kripke semantics. (shrink)

A finitely alternative normal tense logic \ is a normal tense logic characterized by frames in which every point has at most n future alternatives and m past alternatives. The structure of the lattice \\) is described. There are \ logics in \\) without the finite model property, and only one pretabular logic in \\). There are \ logics in \\) which are not finitely axiomatizable. For \, there are \ logics in \\) without the FMP, and infinitely (...) many pretabular extensions of \. (shrink)

Based on the results of [11] this paper delivers uniform algorithms for deciding whether a finitely axiomatizable tense logic has the finite model property, is complete with respect to Kripke semantics, is strongly complete with respect to Kripke semantics, is d-persistent, is r-persistent.It is also proved that a tense logic is strongly complete iff the corresponding variety of bimodal algebras is complex, and that a tense logic is d-persistent iff it is complete and its Kripke frames form a (...) first order definable class. From this we obtain many natural non-d-persistent tense logics whose corresponding varieties of bimodal algebras are complex. (shrink)

We study the modal logic M L r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and (...) show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. (shrink)

A conjecture of Gabbay (1981) states that any class of flows of time having the property known as finite H-dimension admits a finite set of expressively complete one-dimensional temporal connectives. Here we show that the class of 'circular' structures refutes the generalisation of this conjecture to Kripke frames. We then construct from this class, by a general method, a new class of irreflexive transitive flows of time that refutes the original conjecture. Our paper includes full descriptions of (...) a method for establishing finite H-dimension for a class of structures and of the technique for extending finite H-dimension to other classes, and an introduction surveying the area of expressive completeness. (shrink)

We modify the semantics of topological modal logic, a language due to Moss and Parikh. This enables us to study the corresponding theory of further classes of subset spaces. In the paper we deal with spaces where every chain of opens fulfils a certain finiteness condition. We consider both a local finiteness condition relevant to points and a global one concerning the whole frame. Completeness of the appearing logical systems, which turn out to be generalizations of the well-known modal (...) system G, can be obtained in the same manner as in the case of the general subset space logic. It is our main purpose to show that the systems differ with regard to the finite model property. (shrink)

In this paper we investigate those extensions of the bimodal provability logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec CSM}_{0}$\end{document} (alias \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec PRL}_{1}$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec F}^{-})$\end{document} which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely (...) axiomatizable subframe logics containing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\vec CSM}_{0}$\end{document} are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics. (shrink)

This paper presents two general results of decidability concerning logics based on an indeterministic metric tense logic, which can be applied to, among others, logics combining knowledge, time and agency. We provide a general Kripke semantics based on a variation of the notion of synchronized Ockhamist frames. Our proof of the decidability is by way of the finite frame property, applying subframe transformations and a variant of the filtration technique.

This paper deals with two main topics: One is a semantical investigation for a bimodal language with a modal operator \blacksquare associated with the intersection of the accessibility relation R and the inequality ≠. The other is a generalization of some of the former results to general extended languages with modal operators. First, for our language L\sb{\square\blacksquare}, we prove that Segerberg's theorem (equivalence between finite frame property and finite model property) fails and establish both van (...) Benthem-style and Goldblatt-Thomason-style characterizations. We extract the notion of \blacksquare-realizer (a generalization of bulldozing) as an essence from the proofs of our results. Second, we generalize the notion of \blacksquare-realizer and prove quite general versions of these semantical characterization results. The known and previously unknown characterization results for almost all of the languages extended with modal operators already proposed will be immediate corollaries. (shrink)

The veiled recession frame has served several times in the literature to provide examples of modal logics failing to have certain desirable properties. Makinson [4] was the first to use it in his presentation of a modal logic without the finite model property. Thomason [5] constructed a (rather complicated) logic whose Kripke frames have an accessibility relation which is reflexive and transitive, but which is satisfied by the (non-transitive) veiled recession frame, and hence incomplete. In Van (...) Benthem [2] the frame was an essential tool to find simple examples of incomplete logics, axiomatized by a formula in two proposition letters of degree 2, or by a formula in one proposition letter of degree 4 (the degree of a modal formula is the maximal number of nested occurrences of the necessity operator in ). In [3] we showed that the modal logic determined by the veiled recession frame is incomplete, and besides that, is an immediate predecessor of classical logic (or, more precisely, the modal logic axiomatized by the formula pp), and hence is a logic, maximal among the incomplete ones. Considering the importance of the modal logic determined by the veiled recession frame, it seems worthwhile to ask for an axiomatization, and in particular, to answer the question if it is finitely axiomatizable. In the present paper we find a finite axiomatization of the logic, and in fact, a rather simple one consisting of formulas in at most two proposition letters and of degree at most three. (shrink)

We are concerned with modal logics in the class EM0 of extensions of M0 . G denotes re exive frames. MG the modal logic on G in the sense of Kripke. M is nite if M = MG for some nite G. Finite G's will be drawn as framed diagrams, e.g. G = ! ; G = ! ; the latter shorter denoted by . EM0 is a complete lattice with zero M0 and one M . If M M0 (...) M0 is a succ of M. An ip of M is an immediate predecessor of M. E.g. M ! and M are the only ip's of M in ES4. M[P] denotes the extension of M adding P as a \new axiom", e.g. S4 = M0 [p ! p]. M is nitely axiomatizable if M = M0 [P] for some formula P. One easily shows if M is f.a. then each predecessor is separated from M by an ip of M. It is known that each nite M is f.a. and has only nitely many succ's all of which are nite. (shrink)

For each natural number n we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than n and no strictly ascending chains. The case n=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} is the Gödel-Löb provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number (...) of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When n=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1$$\end{document}, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to intuitionistic propositional logic. A topological semantic analysis shows that the n-th member of the sequence of extensions of S4 is the logic of hereditarily n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}-irresolvable spaces when the modality ◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document} is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of ◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document} as the derived set operation. The variety of modal algebras validating the n-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by n. Moreover each algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them. (shrink)

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem’s theorem, which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame (...) classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes–for instance, rooted frames, finite rooted frames, finite transitive frames, well-founded transitive frames, finite equivalence frames–as these are not elementary. Instead we develop and extend the game-based analysis over such classes and provide bisimulation preserving model constructions within these classes. Over most of the classes considered, we obtain finite model theory analogues of the classically expected characterisations, with new proofs also for the classical setting. The class of transitive frames is a notable exception, with a marked difference between the classical and the finite model theory of bisimulation invariant first-order properties. Over the class of all finite transitive frames in particular, we find that monadic second-order logic is no more expressive than first-order as far as bisimulation invariant properties are concerned — though both are more expressive here than basic modal logic. We obtain ramifications of the de Jongh–Sambin theorem and a new and specific analogue of the Janin–Walukiewicz characterisation of bisimulation invariant monadic second-order for finite transitive frames. (shrink)

We generalize the incompleteness proof of the modal predicate logic Q-S4+ p p + BF described in Hughes-Cresswell [6]. As a corollary, we show that, for every subframe logic Lcontaining S4, Kripke completeness of Q-L+ BF implies the finite embedding property of L.

The typical rules for truth-trees for first-order logic without functions can fail to generate finite branches for formulas that have finite models–the rule set fails to have the finite tree property. In 1984 Boolos showed that a new rule set proposed by Burgess does have this property. In this paper we address a similar problem with the typical rule set for first-order logic with identity and functions, proposing a new rule set that does have the (...) finite tree property. (shrink)

We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.

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)

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections Δ of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of Δ. A section Δ is shown to have the (...) JHDP if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results. (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.

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 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.

We prove, by a probabilistic argument, that a class of ω-categorical structures, on which algebraic closure defines a pregeometry, has the finite submodel property. This class includes any expansion of a pure set or of a vector space, projective space or affine space over a finite field such that the new relations are sufficiently independent of each other and over the original structure. In particular, the random graph belongs to this class, since it is a sufficiently independent (...) expansion of an infinite set, with no structure. The class also contains structures for which the pregeometry given by algebraic closure is non-trivial. (shrink)

A cutset of H is a subset of ∪ H which meets every element of H.H has the finite cutset property if every cutset of H contains a finite one. We study this notion, and in particular how it is related to the compactness of H for the natural topology. MSC: 04A20, 54D30.

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.

We argue that the simplicity condition on a probability function on sentences of a predicate language L that it takes only finitely many values on the sentences of any finite sublanguage of L can be viewed as rational. We then go on to investigate consequences of this condition, linking it to the model theoretic notion of quantifier elimination.

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 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)