Modal Logic: An Introduction to Its Syntax and Semantics

Modal Logic: An Introduction to Its Syntax and Semantics

Modal Logic: An Introduction to Its Syntax and Semantics

Modal Logic: An Introduction to Its Syntax and Semantics


In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight. All of the S1-S5 modal logics of Lewis and Langford, among others, are constructed. A matrix, or many-valued semantics, for sentential modal logic is formalized, and an important result that no finite matrix can characterize any of the standard modal logics is proven. Exercises, some of which show independence results, help to develop logical skills.

A separate sentential modal logic of logical necessity in logical atomism is also constructed and shown to be complete and decidable. On the first-order level of the logic of logical necessity, the modal thesis of anti-essentialism is valid and everyde resentence is provably equivalent to adedictosentence.

An elegant extension of the standard sentential modal logics into several first-order modal logics is developed. Both a first-order modal logic for possibilism containing actualism as a proper part as well as a separate modal logic for actualism alone are constructed for a variety of modal systems. Exercises on this level show the connections between modal laws and quantifier logic regarding generalization into, or out of, modal contexts and the conditions required for the necessity of identity and non-identity.

Two types of second-order modal logics, one possibilist and the other actualist, are developed based on a distinction between existence-entailing concepts and concepts in general. The result is a deeper second-order analysis of possibilism and actualism as ontological frameworks. Exercises regarding second-order predicate quantifiers clarify the distinction between existence-entailing concepts and concepts in general.

Modal Logicis ideally suited as a core text for graduate and undergraduate courses in modal logic, and as supplementary reading in courses on mathematical logic, formal ontology, and artificial intelligence.


Modal logic is a theoretical field that is important not only in philosophy, where logic in general is commonly studied, but in mathematics, linguistics, and computer and information sciences as well. This book will be useful for students, researchers, and professionals in all of these and related disciplines. the only requirement is some familiarity with first-order logic and elementary set-theory. the main outline of this book is a development of the logical syntax and semantics of modal logic in three stages of increasing logical complexity. the first stage is a comprehensive development of sentential modal logic, which is followed by a similarly comprehensive development of first-order modal predicate logic. the final stage is a development of second-order modal predicate logic. These stages are introduced gradually, with increasing difficulty at each stage. Most of the important results in modal logic are described and proved in each of their respective stages.

This book is based on a series of lectures given over a number of years at Indiana University by the first author. a draft of the book has also been used by the second author in Costa Rica and Mexico. the book is organized as follows. We begin in chapter 1 with concatenation theory and the logistic method. By means of this theory and method we describe the construction of expressions, formal languages, and formal systems or calculi. Different modal calculi are then constructed in chapter 2. These cover all of the well-known systems, S1–S5, of Lewis and Langford’s 1932 classic Symbolic Logic. As already indicated, these systems are constructed first on the level of sentential (or propositional) logic and then later in chapter 7 on the level of first-order predicate logic, where we distinguish the quantified modal logic of actualism from that of possibilism. the systems are then extended yet again to the level of second-order modal predicate logic in chapter 9, where the notion of existence that is central to the actualism-possibilism distinction is given a deeper and finer-grained analysis in terms of existence-entailing concepts, as opposed to concepts that do not entail existence.

Our distinction between actualism and possibilism in quantified modal logic is similar to what is sometimes described as variable-domain versus fixed-domain semantics. But possibilism on our version goes beyond a fixed-domain semantics by including actualism (and hence a variable-domain semantics) as a proper subsystem. in addition, our development of second-order actualist and possibilist modal logic are subjects that are not covered in other textbooks on modal . . .

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