Nominal tense logic and other sorted intensional frameworks

This thesis introduces of a system of tense logic called nominal tense logic (NTL), and several extensions. Its primary aim is to establish that these systems are logically interesting, and can provide useful models of natural language tense, temporal reference, and their interaction. Languages of nominal tense logic are a simple augmentation of Priorean tense logic. They add to the familiar Priorean languages a new sort of atomic symbol, nominals. Like propositional variables, nominals are atomic sentences and may be freely combined with other wffs using the usual connectives. When interpreting these languages we handle the Priorean components standardly, but insist that nominals must be true at one and only one time. We can think of nominals as naming this time. Logically, the change increases the expressive power of tensed languages. There are certain intuitions about the flow of time, such as irreflexivity, that cannot be expressed in Priorean languages; with nominals they can. The effects of this increase in expressive power on the usual model theoretic results for tensed languages discussed, and completeness and decidability results for several temporally interesting classes of frames are given. Various extensions of the basic system are also investigated and similar results are proved. In the final chapter a brief treatment of similarly referential interval based logics is presented. As far as natural language semantics is concerned, the change is an important one. A familiar criticism of Priorean tense logic is that as it lacks any mechanism for temporal reference, it cannot provide realistic models of natural language temporal usage. Natural language tense is at least partly about referring to times, and nowadays the deictic and anaphoric properties of tense are a focus of research. The thesis presents a uniform treatment of certain temporally referring expressions such as indexicals, and simple discourse phenomena

Hybrid type theory: a quartet in four movements

This paper sings a song -a song created by bringing together the work of four great names in the history of logic: Hans Reichenbach, Arthur Prior, Richard Montague, and Leon Henkin. Although the work of the first three of these authors have previously been combined, adding the ideas of Leon Henkin is the addition required to make the combination work at the logical level. But the present paper does not focus on the underlying technicalities (these can be found in Areces, Blackburn, Huertas, and Manzano [to appear]) rather it focusses on the underlying instruments, and the way they work together. We hope the reader will be tempted to sing along

Simple cut elimination proof for hybrid logic

In the paper we present a relatively simple proof of cut elimination theorem for variety of hybrid logics in the language with satisfaction operators and universal modality. The proof is based on the strategy introduced originally in the framework of hypersequent calculi but it works well also for standard sequent calculi. Sequent calculus examined in the paper works on so called satisfaction formulae and cover all logics adequate with respect to classes of frames defined by so called geometric conditions

Hybrid Languages and Temporal Logic

Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev) the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well-behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full first-order expressive strength, is demonstrated for a weaker local language capable of defining the Until operator, we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths

Modal Hybrid Logic

This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory

Temporal location of events in language and (non) persistence of the past

The article reviews some analyses of temporal language in logical approaches to natural language semantics. It considers some asymmetries between past and future, manifested in language, which motivate the “standard view” of the non-reversibility of time and the persistence of the past. It concludes with a puzzle about the changing past which challenges the standard view

Towards World Identification in Description Logics

Logical analysis of the applicability of nominals (which are introduced by hybrid logic) in the formal descriptions of the world (within modern knowledge representation and semantics-based systems) is very important because nominals, as second sorts of propositional symbols, can support logical identification of the described world at specific [temporal and/or spacial] states. This paper will focus on answering the philosophical-logical question of ‘how a fundamental world description in description logic (DL) and a nominal can be related to each other?’. Based on my assumption that nominals can support more adequate identification of the world in DL, this paper will deal with the concept of ‘world identification’. Accordingly, based on a logical-terminological analysis of nominals, the paper will analyse hybridised fundamental world descriptions. The research will finally reach the idea that we can have a hybrid description logic based on the analysed concepts