Modal Logics and Philosophy

Rod Girle

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781317492160
• Edition: 2
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

The first edition, published by Acumen in 2000, became a prescribed textbook on modal logic courses. The second edition has been fully revised in response to readers' suggestions, including two new chapters on conditional logic, which was not covered in the first edition. "Modal Logics and Philosophy" is a fully comprehensive introduction to modal logics and their application suitable for course use. Unlike most modal logic textbooks, which are both forbidding mathematically and short on philosophical discussion, "Modal Logics and Philosophy" places its emphasis firmly on showing how useful modal logic can be as a tool for formal philosophical analysis. In part 1 of the book, the reader is introduced to some standard systems of modal logic and encouraged through a series of exercises to become proficient in manipulating these logics. The emphasis is on possible world semantics for modal logics and the semantic emphasis is carried into the formal method, Jeffrey-style truth-trees. Standard truth-trees are extended in a simple and transparent way to take possible worlds into account. Part 2 systematically explores the applications of modal logic to philosophical issues such as truth, time, processes, knowledge and belief, obligation and permission.

CHAPTER 1

Argument and modality

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1.1 Introduction

One of the main tasks of logic is to give an analysis of argumentation. Traditionally this analysis is of premise–conclusion argumentation. If formal methods are used, one begins with propositional logic.

The most common form of propositional logic is truth-table logic. Truth-table logic extends very easily into truth-tree logic. We assume that the reader is familiar with propositional logic in both truth-table and truth-tree forms. From this point we shall use the term propositional logic to refer to truth-table and truth-tree propositional logic unless we make it clear otherwise. Most of this text can be read with a knowledge of propositional logic only.

Propositional logic is usually extended to predicate or first-order logic. To read the whole of this volume the reader should be familiar with predicate logic and the system of truth-trees for predicate logic. Several texts introduce the reader to propositional and predicate logic and the truth-tree system. They are listed in the further reading at the end of the chapter.

1.2 Argument analysis

There are arguments that are clearly valid, but that cannot be shown to be so by propositional logic. Arguments such as:

“All Athenians are Greeks.

Socrates is Athenian.

So Socrates is Greek.”

require a more detailed analysis of their logical form than can be given by propositional logic. We need a logic that can deal not only with the negation, disjunction and conjunction of propositions, but also with quantifiers (all, some), predicates (is Φ) and relations (loves). The result is predicate logic.

Similarly, there are yet further valid arguments that cannot be shown to be valid by propositional or predicate logic. Such an argument is:

[A] If a new course is to be offered next year, then submissions must be made to the Faculty Board before April. If submissions are to be made to the Faculty Board before April, then a Departmental meeting must be called. A week’s notice must be given if a Departmental meeting is to be called. Since it is not possible to give such notice, it follows that it is not possible to offer a new course next year.

To give a proper account of how this argument is valid we have to display what is expressed by must and what is expressed by possible. These English words express modal notions. Other such notions are expressed by the terms necessarily, might and can.

In the 1950s a group of logicians, chief among them Saul Kripke, developed the idea that the notions of possibility and necessity could be captured in terms of possible worlds. The idea is really quite simple. We live in one possible world. It is the actual world. A work of fiction can be seen as a description of a possible world other than the actual world. Some works of fiction are descriptions of possible worlds very like the actual one. Some, such as works of fantasy, describe worlds quite remote from the actual world.

The explanatory idea in the possible worlds logics is the idea that if someone says “It is possible that giant squids live in the sea”, then this is true just in case there is a possible world in which giant squids live in the sea. One such possible world could be the actual world. In the possible worlds logics a statement is true-in-a-world rather than just true. Furthermore, statements are said to be necessarily true just in case they are true in all possible worlds. If we say that statements such as “If it is raining then it is raining” are true in all possible worlds, then possible worlds logic holds us to be assuming that it is necessarily true that if it is raining then it is raining.

The possible worlds approach to modal terms provides a semantics for modal notions. That is, possible worlds provide an account of modal notions in terms of truth-values. Before the Kripke semantics, the main approach was by means of axiomatic and proof systems. In the following chapters we shall deal with both semantic and proof system approaches to modal logics, but, apart from Chapter 5 and sections of Chapter 6, we shall focus on the semantic approach.

The possible worlds approach to modal terms is quite persuasive. It is, in a sense, intuitively simple. But, does possible worlds logic really match the meaning that modal terms have in ordinary English? If the answer is “no”, then how great is the difference? This question is important if we are going to use modal logic to assess the validity of arguments couched in English. This question is...

Table of contents

• Cover Page
• Half Title page
• Title Page
• Copyright Page
• Contents
• Preface
• Acknowledgements
• Chapter 1 Argument and modality
• Part One Formal systems
• Chapter 2 A simple modal logic
• Chapter 3 The normal modal logics
• Chapter 4 The non-normal modal logics
• Chapter 5 Natural deduction and axiomatics
• Chapter 6 Conditional logic
• Chapter 7 Modal predicate logics
• Chapter 8 Quantifiers and existence
• Part Two Applications
• Chapter 9 Alethic modality
• Chapter 10 Temporal logic
• Chapter 11 Dynamic logic
• Chapter 12 Epistemic logic
• Chapter 13 Deontic logic
• Chapter 14 Conditionals and reliability
• Chapter 15 Synthesis and worlds
• Answers
• Index