S5

Abstract: Abstract For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators - `like it is necessary' in philosophy, `it is believed' in cognitive science, `it is provable' in mathematics and `it is true after executing a program' in computer science. It is an advanced text which starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantical and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist can use the book as a source of references. For the first time results and methods of many directions in propositional modal logic - from completeness and duality to algorithmic problems - are collected and systematically presented in one volume. Unlike other books, modal logic is treated here as a uniform theory rather than a collection of a few particular systems. It is the only book presenting the theory of superintuitionistic logics.

Abstract: Publisher Summary This chapter focuses on normal logics and first order properties of the accessibility relation. There are three main themes, a positive, a negative, and a comparative. The chapter outlines that the first order relational semantics is adequate for all tense logics and that there is an equivalent algebra for each first order relational frame. The same may be done for normal modal logics. And by defining a first order neighborhood semantics, one may—via an algebraic semantics—prove the adequacy for all classical modal logics. This theorem is strong enough because it does not require any connections between the modal and Boolean operations. For tense logic or normal modal logic, stronger representation theorems are needed.

Abstract: In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the (complete) extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment.