This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.
In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(Λ). This summarizes results that can be found already in [13, 14] and . Furthermore, we determine the (...) structure of the lattice of extensions of n(LC). (shrink)
This paper shows that non-normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic.
This papers gives a survey of recent results about simulations of one class of modal logics by another class and of the transfer of properties of modal logics under extensions of the underlying modal language. We discuss: the transfer from normal polymodal logics to their fusions, the transfer from normal modal logics to their extensions by adding the universal modality, and the transfer from normal monomodal logics to minimal tense extensions. Likewise, we discuss simulations of normal polymodal logics by normal (...) monomodal logics, of nominals and the difference operator by normal operators, of monotonic monomodal logics by normal bimodal logics, of polyadic normal modal logics by polymodal normal modal logics, and of intuitionistic modal logics by normal bimodal logics. (shrink)
Given a normal (multi-)modal logic a characterization is given of the finitely presentable algebras A whose logics L A split the lattice of normal extensions of . This is a substantial generalization of Rautenberg  and  in which is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok  that for all cycle-free and finite A L A splits the lattice of normal extensions of K. Although we (...) firmly believe it to be true, we have not been able to prove that if a logic splits the lattice of extensions of then is the logic of an algebra finitely presentable over ; in this respect our result remains partial. (shrink)
This book argues that languages are composed of sets of ‘signs’, rather than ‘strings’. This notion, first posited by de Saussure in the early 20th century, has for decades been neglected by linguists, particularly following Chomsky’s heavy critiques of the 1950s. Yet since the emergence of formal semantics in the 1970s, the issue of compositionality has gained traction in the theoretical debate, becoming a selling point for linguistic theories. Yet the concept of ‘compositionality’ itself remains ill-defined, an issue this book (...) addresses. Positioning compositionality as a cornerstone in linguistic theory, it argues that, contrary to widely held beliefs, there exist non-compositional languages, which shows that the concept of compositionality has empirical content. The author asserts that the existence of syntactic structure can flow from the fact that a compositional grammar cannot be delivered without prior agreement on the syntactic structure of the constituents. (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)
This paper shows that non-normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic.
The present paper is the result of a long struggle to understand how the notion of compositionality can be used to motivate the structure of a sentence. While everyone seems to have intuitions about which proposals are compositional and which ones are not, these intuitions generally have no formal basis. What is needed to make such arguments work is a proper understanding of what meanings are and how they can be manipulated. In particular, we need a definition of meaning that (...) bans all mentioning of syntactic structure; it is not the task of semantics to state in which way things are put together in syntax. The present paper presents such a theory of meaning. This, in tandem with some minimal assumptions on the syntactic process (that there can be no deletion) yield surprisingly deep insights into natural language. First, it rehabilitates a lot of linguistic work as necessary on semantic grounds and defends it against potential claims of redundancy. For example, θ-roles and linking are an integral part of semantics, and not syntax. To assume the latter is to put the cart before the horse. Second, as a particular example we shall show that Dutch is not strongly context free even if weakly context free. To our knowledge, this is the first formal proof of this fact. (shrink)
The present paper deals with the semantics of locative expressions. Our approach is essentially model-theoretic, using basic geometrical properties of the space-time continuum. We shall demonstrate that locatives consist of two layers: the first layer defines a location and the second a type of movement with respect to that location. The elements defining these layers, called localisersand modalisers, tend to form a unit, which is typically either an adposition or a case marker. It will be seen that this layering is (...) not only semantically but in many languages also morphologically manifest. There are numerous languages in which the morphology is sufficiently transparent with respect to the layering. The consequences of this theory are manifold. For example, we shall show that it explains the contrast between English and Finnish concerning directionals, which is discussed in Fong (1997). In addition, we shall be concerned with the question of orientation of locatives, as discussed in Nam (1995). We propose that nondirectional locatives are oriented to the event, while directional locatives are oriented to certain arguments, called movers. (shrink)
In this paper we show that a variety of modal algebras of finite type is semisimple iff it is discriminator iff it is both weakly transitive and cyclic. This fact has been claimed already in  (based on joint work by the two authors) but the proof was fatally flawed.
The present paper is based on , where a number of conjectures are made concerning the structure of the lattice of normal extensions of the tense logicKt. That paper was mainly dealing with splittings of and some sublattices, and this is what I will concentrate on here as well. The main tool in analysing the splittings of will be the splitting theorem of . In  it was conjectured that each finite subdirectly irreducible algebra splits the lattice of normal extensions (...) ofK4t andS4t. We will show that this is not the case and that on the contrary only very few and trivial splittings of the mentioned lattices exist. (shrink)
A logic Λ bounds a property P if all proper extensions of Λ have P while Λ itself does not. We construct logics bounding finite axiomatizability and logics bounding finite model property in the lattice of intermediate logics and in the lattice of normal extensions of K4.3. MSC: 03B45, 03B55.
In this paper I argue that a variety of consequence relations can be subsumed under a common core. The reduction proceeds by taking the unconditional consequence, or judgment, as basic and deriving the conditional consequence via a uniform abstraction scheme. A specific outcome is that it is better not to base such a scheme on the semantic notion of a matrix and valuation but rather on theories and substitutions. I will also briefly look at consequence relations that are not reducible (...) in this way. (shrink)
In this paper I argue that in contrast to natural languages, logical languages typically are not compositional. This does not mean that the meaning of expressions cannot be determined at all using some well-defined set of rules. It only means that the meaning of an expression cannot be determined without looking at its form. If one is serious about the compositionality of a logic, the only possibility I see is to define it via abstraction from a variable free language.
The transition from form to meaning is not neatly layered: there is no point where form ends and content sets in. Rather, there is an almost continuous process that converts form into meaning. That process cannot always take a straight line. Very often we hit barriers in our mind, due to the inability to understand the exact content of the sentence just heard. The standard division between formula and interpretation (or value) should therefore be given up when talking about the (...) process of understanding. Interestingly, when we do this it turns out that there are 'easy' formulae, those we can understand without further help, and 'difficult' ones, which we cannot. (shrink)
The present paper investigates the groups of automorphisms for some lattices of modal logics. The main results are the following. The lattice of normal extensions of S4.3, NExtS4.3, has exactly two automorphisms, NExtK.alt1 has continuously many automorphisms. Moreover, any automorphism of NExtS4 fixes all logics of finite codimension. We also obtain the following characterization of pretabular logics containing S4: a logic properly extends a pretabular logic of NExtS4 iff its lattice of extensions is finite and linear.
The idea that language is a homogeneous code is a massive simplification. In actual fact, we constantly use a wide array of codes, be they other languages, dialects, registers, or special purpose codes . In this paper we provide a formal analysis of code switching.
Modal logic originated in philosophy as the logic of necessity and possibility. Now it has reached a high level of mathematical sophistication and has many applications in a variety of disciplines, including theoretical and applied computer science, artificial intelligence, the foundations of mathematics, and natural language syntax and semantics. This volume represents the proceedings of the first international workshop on Advances in Modal Logic, held in Berlin, Germany, October 8-10, 1996. It offers an up-to-date perspective on the field, with contributions (...) covering its proof theory, its applications in knowledge representation, computing and mathematics, as well as its theoretical underpinnings. (shrink)
In a recent paper, Godblatt and Kowalski show that if we add to monomodal logic just a single propositional constant then instead of two coatoms, we suddenly have continuum many. In this note we shall provide an alternative proof of that fact by showing that the simulation results of Kracht and Wolter can be sharpened.
Modal logic is concerned with the analysis of sentential operators in the widest sense. Originally invented to analyse the notion of necessity applications have been found in many areas of philosophy, logic, linguistics and computer science. This in turn has led to an increased interest in the technical development of modal logic.
In transformational grammar the notion of a chain has been central ever since its introduction in the early 80's. However, an insightful theory of chains has hitherto been missing. This paper develops such a theory of chains. Though it is applicable to virtually all chains, we shall focus on movement-induced chains. It will become apparent that chains are far from innocuous. A proper formulation of the structures and algorithms involved is quite a demanding task. Furthermore, we shall show that it (...) is possible to define structures in which the notion of a chain coincides with that of a constituent, so that the notion of a chain becomes redundant, generally without making the theory more complicated. These structures avoid some of the problems that beset the standard structures (such as unbound traces created by remnant movement). (shrink)
A moda logic Λ is called invariant if for all automorphisms α of NExt K, α = Λ. An invariant ogic is therefore unique y determined by its surrounding in the attice. It wi be established among other that a extensions of K.alt1S4.3 and G.3 are invariant ogics. Apart from the results that are being obtained, this work contributes to the understanding of the combinatorics of finite frames in genera, something wich has not been done except for transitive frames. Certain (...) useful concepts will be established, such as the notion of a d-homogeneous frame. (shrink)
The Kuznetsov-Index of a modal logic is the least cardinal such that any consistent formula has a Kripke-model of size if it has a Kripke-model at all. The Kuznetsov-Spectrum is the set of all Kuznetsov-Indices of modal logics with countably many operators. It has been shown by Thomason that there are tense logics with Kuznetsov-Index . Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov-Index . We will show here that for each countable ordinal there are logics with Kuznetsov-Index (...) . Furthermore, we show that the Kuznetsov-Spectrum is identical to the spectrum of indices for -theories which is likewise defined. A particular consequence is the following. If inaccessible (weakly compact, measurable) cardinals exist, then the least inaccessible (weakly compact, measurable) cardinal is also a Kuznetsov-Index. (shrink)