A New Introduction to Modal Logic

M.J. Cresswell, G.E. Hughes

• Publisher: Routledge
• Year: 2012
• Print ISBN: 9780415125994
• eBook ISBN: 9781134800278
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

This long-awaited book replaces Hughes and Cresswell's two classic studies of modal logic: An Introduction to Modal Logic and A Companion to Modal Logic.
A New Introduction to Modal Logic is an entirely new work, completely re-written by the authors. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of exposition and approachability that were essential features of their earlier works.
The book takes readers from the most basic systems of modal propositional logic right up to systems of modal predicate with identity. It covers both technical developments such as completeness and incompleteness, and finite and infinite models, and their philosophical applications, especially in the area of modal predicate logic.

Part I

BASIC MODAL
PROPOSITIONAL
LOGIC

1

THE BASIC NOTIONS

In this chapter we introduce the basic notions of modal propositional logic. Modal logic is based upon the ‘ordinary’ (two-valued) Propositional Calculus, and when we use the expression ‘Propositional Calculus’ (or the abbreviation ‘PC’) simpliciter, it is to this non-modal system of logic that we shall be referring.¹ The present chapter begins by outlining, in a very summary fashion, those elements of PC which we shall take for granted in what follows, and at the same time explains some of the terminology which we shall use throughout the book.

The language of PC

We take as primitive (or undefined) symbols of PC the following:

A set of letters: p, q, r, ... (with or without numerical subscripts). We suppose ourselves to have an unlimited number of these.

The following four symbols: ˜, ∨, (, ).

Any symbol in the above list, or any sequence of such symbols, we call an expression. An expression is either a formula - more exactly a well-formed formula (wff) - or else it is not. We are concerned only with expressions which are well-formed formulae (wff). The following formation rules of PC specify which expressions are to count as wff:

FR1	A letter standing alone is a wff.
FR2	If α is a wff, so is ˜ α.
FR3	If α and β are wff, so is (α ∨ β).

In these rules the symbols α and β are used to stand indifferently for any expressions. Thus the meaning of FR2 is: the result of prefixing ˜ to any wff is itself a wff. Symbols used as α and β are used here are known as metalogical variables. They are not among the symbols of the system (PC in this case), but are used in talking about the system.

Examples of wff are: p, ˜ q, ˜ ˜ ˜q, (p ∨ ˜ q), ((p ∨ r) v ˜ (q v ˜ (˜ r ∨ p))). For convenience, however, we allow ourselves to omit the outermost brackets round any complete wff (though not any subordinate part thereof). No ambiguity in interpretation or unclarity about what is permitted by the rules will result from this notational simplification.

Interpretation

We interpret the letters as variables whose values are propositions. We shall usually call them propositional variables. We assume that the reader is familiar with the notion of a proposition, and shall not enter into the philosophical issues which this notion raises. Rough synonyms of ‘proposition’ are ‘statement’ and ‘assertion’, where these words are used to refer to what is stated or asserted, not to the act of stating or asserting. Every proposition is either true or false, and no proposition is both true and false. (Hence if something is neither true nor false, or is capable of being both true and false, it is not to count as a proposition in the present context.) Truth and falsity are said to be the truth-values of propositions.

Now it is possible to form more complex propositions out of simpler ones. E.g., out of the proposition that Brutus killed Caesar we can form the proposition that it is not the case that Brutus killed Caesar. This is a proposition which is true if the original proposition is false, and false otherwise. In general, putting ‘it is not the case that’ in front of a sentence will result in a sentence which expresses a proposition which is true if the original sentence expresses one which is false, and a false proposition if it does not.

Similarly, from the proposition that Brutus killed Caesar and the proposition that Cassius killed Caesar we may form the proposition that either Brutus killed Caesar or Cassius killed Caesar. This proposition will be true iff (if and only if) at least one of the original propositions is true, and therefore false iff both of these are false.

‘It is not the case that’ and ‘either ... or ...’, when used in the way we have just described, may be said to be proposition-forming operators on propositions, because they make new propositions out of old ones. The propositions on which such an operator operates are called its arguments. If an operator requires only a single argument, as ‘it is not the case that’ does, it is said to be monadic; if, like ‘either ... or ...’, it requires two, it is said to be dyadic.

Our explanation of these operators, ‘it is not the case that’ and ‘either... or ...’, showed that the truth-value of a proposition formed by means of either of them depends in every case only on the truth-value of the operator’s argument or arguments. In other words, whenever we are given the truth-value of the argument or arguments, we can deduce the truth-value of the complex proposition. An operator which has this property is said to be a truth-functional operator, and the propositions it forms are said to be truth-functions of its arguments. Not all proposition-forming operators are of this kind. For example, given merely the truth or falsity of the proposition that Brutus killed Caesar we cannot deduce the truth or falsity of the proposition that Napoleon believed that Brutus killed Caesar; and given merely that two propositions are both true we cannot deduce from this either the truth or the falsity of the proposition that the first follows logically from the second (though if we are given that one proposition is false and another true, we can deduce from this that it is false that the first follows logically from the second). Hence although ‘Napoleon believed that’ and ‘follows logically from’ are proposition-forming operators on propositions (monadic and dyadic respectively), they are not truth-functional operators.

We interpret ˜ and ∨ as ‘it is not the case that’ and ‘either ... or ...’ respectively, in the senses we have explained, and we usually read them simply as ‘not’ and ‘or’. ˜ so interpreted is called the negation sign; ˜p is said to be the negation of p. Using 1 and 0 for the truth-values truth and falsity respectively, we can express ...

Table of contents

• Front Cover
• Half Title
• Title Page
• Copyright
• Contents
• Preface
• Part One: Basic Modal Propositional Logic
• Part Two: Normal Modal Systems
• Part Three: Modal Predicate Logic
• Axioms, Rules and Systems
• Solutions to Selected Exercises
• Bibliography
• Index