Mathematical modal logic: A view of its evolution
نویسنده
چکیده
Modal logic was originally conceived as the logic of necessary and possible truths. It is now viewed more broadly as the study of many linguistic constructions that qualify the truth conditions of statements, including statements concerning knowledge, belief, temporal discourse, and ethics. Most recently, modal symbolism and model theory have been put to use in computer science, to formalise reasoning about the way programs behave and to express dynamical properties of transitions between states. Over a period of three decades or so from the early 1930’s there evolved two kinds of mathematical semantics for modal logic. Algebraic semantics interprets modal connectives as operators on Boolean algebras. Relational semantics uses relational structures, often called Kripke models, whose elements are thought of variously as being possible worlds, moments of time, evidential situations, or states of a computer. The two approaches are intimately related: the subsets of a relational structure form a modal algebra (Boolean algebra with operators), while conversely any modal algebra can be embedded into an algebra of subsets of a relational structure via extensions of Stone’s Boolean representation theory. Techniques from both kinds of semantics have been used to explore the nature of modal logic and to clarify its relationship to other formalisms, particularly first and second order monadic predicate logic. The aim of this article is to review these developments in a way that provides some insight into how the present came to be as it is. The pervading theme is the mathematics underlying modal logic, and this has at least three dimensions. To begin with there are the new mathematical ideas: when and why they were
منابع مشابه
epage MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
there is no one fundamental logical notion of necessity, nor consequently of possibility. If this conclusion is valid, the subject of modality ought to be banished from logic, since propositions are simply true or false. . .
متن کاملSuhrawardi's Modal Syllogisms
Suhrawardi’s logic of the Hikmat al-Ishraq is basically modal. So to understand his modal logic one first has to know the non-modal part upon which his modal logic is built. In my previous paper ‘Suhrawardi on Syllogisms’(3) I discussed the former in detail. The present paper is an exposition of his treatment of modal syllogisms. On the basis of some reasonable existential presuppositi...
متن کاملModal Provability Foundations for Negation by Failure
This paper is a contribution to the foundation of negation by failure. It presents a view of negation by failure as a modal provability notion. Negation by failure is a central notion in Logic Programming and is used extensively in practice. There are various attempts at its foundations each with its own difficulties and limitations. We would like to present the modal point of view which, as fa...
متن کاملStrong Completeness of Coalgebraic Modal Logics
Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties – up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a...
متن کاملModal Foundations for Predicate Logic
The complexity of any logical modeling reeects both the intrinsic structure of a topic described and the weight of the formal tools. Some of this weight seems inherent in even the most basic logical systems. Notably, standard predicate logic is undecidable. In this paper, we investigatèlighter' versions of this general purpose tool, by modally`deconstructing' the usual semantics, and locating i...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Applied Logic
دوره 1 شماره
صفحات -
تاریخ انتشار 2003