A cornucopia of applications

Modal logic has been used by logicians, philosophers, the “artificial intelligentsia” and computer scientists for many purposes. There is a cornucopia of applications of modal logic to problems and concepts in each of these discipline areas. We shall be looking at a very small number of those applications.

Logicians have used modal logic to try to produce logics that capture the notions of possibility, logical necessity and proof. These applications consider modal logic from an alethic point of view: from the point of view of issues related to truth. We turn to these in Chapter 9.

Philosophers, quantum physicists, and artificial intelligence researchers who work in the area of planning all have a deep interest in questions about time. Applications of modal logic to questions of time used to be known as “tense” (from past, present and future tense) applications. But in these days of disdain for grammar, such applications are called temporal. In Chapter 10 we introduce temporal applications for modal logic. Dynamic logic is an application of modal logic that is closely related to temporal logic.

Dynamic logic was first generated by computer scientists who applied modal logic to the study of processes in machines. It was first seen as a logic for state changes. Philosophers have begun to use dynamic logic in the study of action, and of belief change. In Chapter 11 we consider some elementary dynamic logic.

Introduction

Modal logic has long been studied from a proof theoretic perspective. The three main kinds of proof theory are well represented in the literature: natural deduction, sequent systems, and axiomatic systems. We shall set out both natural deduction systems and axiomatic systems for propositional modal logics.

Since most of you will have met some sort of natural deduction system in introductory logic, we begin with natural deduction. We set out a Fitch style system. His are in Symbolic Logic: An Introduction (Fitch 1952). We then move to axiomatic systems.

Natural deduction

We assume that you are already familiar with a natural deduction system for propositional logic. There are several well known systems. Fitch's and Lemmon's are two of the most well known. We base our exposition on Fitch and Hughes and Cresswell. An introductory exposition is to be found in Fitch, Copi or Kahane.

We take it that we have the standard set of replacement rules. These include the commutativity, associativity and idempotence for both & and ∨; double negation and DeMorgan's laws; contraposition and exportation/ importation for ⊃; and the interdefinability of ⊃ and ∨, often called material implication.

We assume that there are at least the following inference rules: simplification, conjunction, disjunctive addition, disjunctive syllogism, modus ponens, modus tollens, hypothetical syllogism, conditional proof, and some form of reductio ad absurdum. A summary of these is to be found at the end of this chapter.

Introduction

In this chapter we set out a propositional modal logic. The logic is known as S5. It was given its name by one of the most important modal logicians of the early twentieth century; C. I. Lewis (1883–1964). Lewis constructed five axiomatic systems of modal logic and named them S1 to S5 (System 1 to System 5). It turns out that the simplest of the logics based on possible worlds is the same as Lewis's S5.

In this chapter I set out S5 in terms of modal truth trees, or modal semantic tableaux. The trees for S5 make the simplest possible use of the idea of possible worlds. I will not set out S5 in axiomatic form in this chapter, but will look at an axiomatic formulation in a later chapter.

S5 is often seen as a system capturing the idea of logical possibility. The diamond and box symbols can be used to translate as follows:

This supposition, that S5 sets out the logic of the notions of logical possibility and logical necessity, while intuitively reasonable, is not without difficulties and is discussed in Chapter 9.

Propositional modal logic

I begin with propositional modal logic.

Introduction

Conditionals invite us to use our imagination and to consider possibilities. The antecedent “If A” turns us towards hypothetical circumstances and prepares us to consider what follows from them. The consequent “then C” expresses the claim that C follows from A or is implied by A. When we know that A is false, contrary to fact or counterfactual, then the speculations invite contention. There is a “what if ” dimension to conditionals. Public figures are often asked conditional questions such as: what would you do if A? They will often decline to answer such questions. The element of speculation raises all sorts of difficulties in giving answers. These difficulties are not just for public figures. Logicians and philosophers face the same questions about speculation and possibility when discussing conditionals and what makes for their being true or false.

If we set aside conditional or hypothetical questions for the moment, then the key question about conditional statements is what makes a conditional true. One quite standard answer has been that a conditional is true when either the antecedent is false or the consequent is true. This is the material conditional interpretation. This is problematic for several reasons, not least because it makes all counterfactual conditionals true, and all conditionals with true consequents true.

Introduction

Two of the normal modal logics are S4 and S5. These logics were created by C. I. Lewis (1883–1964) as part of a sequence of modal logics. The logics started with System 1, S1, and go on to S2, S3, S4, and S5. The first three of these are not normal modal logics, and are referred to as either non-normal or sub-normal. We shall use the term “non-normal” to refer to them, and to other such logics.

There are various ways in which the distinction between the normal and the non-normal logics can be described or defined. Some descriptions focus on the differences between the sets of valid formulas. Some focus on differences in the logic apparatus. In our case we shall focus on systematic differences in the tree rules.

The tree rules we have given so far are rules that apply to formulas in all worlds. The definition of validity applies across all worlds. In non-normal logics we depart from this uniformity. We need different modal rules depending on the world the formula is in. The definition of validity is also qualified.

Two sorts of worlds

In any tree for modal logic the tree begins at some arbitrary world. We have always begun with n. This world can be any world in a system of worlds for a normal modal logic. But, in the non-normal modal logics the worlds are divided into two sets.

Introduction

If we interpret the ☐ as “It is known that”, then we have an epistemic interpretation of modal logic. If we interpret the ☐ as “It is believed that”, then we have a doxastic interpretation of modal logic. Epistemic logic gives us a logic for knowledge, and doxastic logic gives us a logic for belief.

Since knowledge and belief both involve some knower or believer, many epistemic and doxastic logics use a subscript with the modal operator to indicate the agent. If the agent were a, then we would have ☐a. If several agents were to be considered, then we would have a logic for each agent.

It is usual to distinguish epistemic from doxastic logic by replacing the ☐ with K for knowledge, and with B for belief. So we translate:

Kap as a knows that p

and Bap as a believes that p

Epistemic and doxastic logics were proposed by a variety of people. One classic early paper was Lemmon's “Is there only one correct system of modal logic?” (1959). The classic summary and complete proposal for epistemic and doxastic logic is to be found in Hintikka's Knowledge and Belief (1962).

We shall focus mainly on the propositional part of Hintikka's epistemic and doxastic logics. We then discuss some of the issues generated by these logics. There will be a brief discussion of epistemic free logic.

S4 Knowledge

Hintikka's epistemic logic is straightforwardly an epistemic interpretation of S4.

Introduction

We shall consider three main topics in this chapter. They concern the use of modal logic to translate “possibility” and “necessity”, the GL model for the notion of proof, and the de dicto-de re distinction.

We begin with the application of modal logic to the notions of possibility and necessity, especially as they are expressed in ordinary language. It seems intuitively obvious that the formal notion of possible worlds should elucidate the notion of possibility and its close relative, necessity. But let the reader beware! Intuitive obviousness is sometimes misleading. Care is needed.

The S5 modal logic is often suggested as the system for logical possibility and necessity. If we are going to use S5 to assess the validity of arguments couched in English and containing possibility and necessity terms, then we need to know about the reliability of the S5 account of logical possibility and logical necessity. We have to consider the question of the relationship between S5 possibility and necessity, and the concepts of possibility and necessity embedded in ordinary English. So, without further ado we turn to the possibility and necessity terms of English.

Two kinds of possibility?

It has been suggested that there are two sorts of possibility embedded in ordinary English. First, there is a qualifiable possibility expressed by the phrase “possible for”. Second, there is a variable possibility expressed by the phrase “possible that”.

Introduction

If we interpret the ☐ as “It is obligatory to bring it about that”, then we have a deontic interpretation of modal logic. The ◊ is then interpreted as “It is permissable to bring it about that”.

Some deontic logicians have insisted that the deontic modal operators operate not on propositions but on acts. In that case the translation of the ☐ is “It is obligatory to do”, and the ◊ is translated as “It is permissable to do”. This means that the p and q of standard logic become the names of acts, not the descriptions of states of affairs. We shall try to present the deontic interpretation in such a way as to be neutral about the act versus proposition approaches. In what follows we shall assume that the alternative to “is true” of “is done” can be used instead of “is true”, and so on. In the next paragraph only we shall insert the alternatives. After that we use the states of affairs locutions.

Just as ◊p generates a world in which p is true (or is done), a permission to bring it about that p generates a world in which p is true (or is done), and all the truths (or actions) that it is obligatory to bring about (or do) are true (or are done). In deontic logic, the accessible worlds, the worlds we see into, are the worlds in which at least one permitted state of affairs (or action) occurs, and all obligations are fulfilled.

The second edition was prompted by two things. First there was the feedback from several readers asking for something about conditionals, especially since many modern conditional logics use possible worlds semantics. So there are two new chapters, one in Part 1 presenting the formalities of a range of conditional logics, and one in Part 2 with philosophical discussion of some of the issues raised by conditionals.

Secondly, the use of the volume in teaching has suggested some revisions and corrections and re-ordering of content in the first part of the text.

I wish to thank all those who have sent me comments, questions and corrections.

Preface to the first edition

This text is a second level logic text. It introduces students to modal logic as an extension of classical first-order logic. The emphasis is on introducing the object language and some of the applications of modal logic in philosophy and artificial intelligence.

This text is not intended to be a metatheory text for modal logic. There are several excellent texts in that area. (For example: Brian F. Chellas, Modal Logic: an introduction, Cambridge, London, 1980; and G. E. Hughes and M. J. Cresswell, A New Introduction to Modal Logic, Routledge, London, 1996.) Our main focus will be on presenting the logics at an object language level, with a minimum of metatheory. The emphasis will be on the possible worlds semantics. There will be only a brief mention of axiom systems.

Introduction

Modal logic has been used for purposes other than providing formal languages for necessity and possibility. In fact, at the very beginning of the creation of modern modal logic one of the main aims was to provide a formal language for conditional propositions, the propositions usually expressed by “If … then …” sentences. The major systematic development of early modal logic resulted in the five Lewis systems: S1 to S5. These were first developed as conditional logics. The whole enterprise grew out of a dissatisfaction with the use of the material conditional as the standard translation for “If … then …” and “… only if …”.

One early alternative idea was that “If p then q” should be understood not as “Either not p or q” but as “p is incompatible with not q” or as “It is not possible that p and not q”. Modal logic enters at once.

In this chapter we focus on a few of the formal systems of conditional logic. We shall look at a range of conditional logics and see what distinguishes one from the other. We shall not discuss directly the merits of any of the logics but simply note which formulas are theorems and which argument forms are valid. The question of which logic, if any, might be better than which will be discussed in Chapter 14 as part of a general philosophical discussion of conditionals. In this chapter we shall just look at formal features.