Proofs by structural induction

(a) Inductive generation. Aclass of objectsissaidtobe inductively generated if, given some finite collection of objects and operations on objects, each object in the class is obtained from the finite collection after some bounded number of operations.

The importance of inductive generation is that it gives us a systematic means of proving properties of infinite collections of objects: The principle is to first prove that each object in the given finite collection has the property, then to prove that each way of generating new objects by the operations maintains the property.

The above abstract description is best illustrated by examples. The first and best-known inductive class is that of the natural numbers. The given finite collection is just the number 0, and there is a single operation, what is called the successor of a number n, the application of which is written s (n). The standard notation is 1 for s (0), 2 for s(s (0)), etc.

The principle of proof that goes with an inductive generation of the natural numbers is called arithmetic induction, or ‘complete induction’ in the older literature: Given a property of natural numbers, written A(n) for n, prove that 0 (or sometimes 1) has the property and that if n has the property, also s(n) has. In other words, prove A(0) and A(n) ⊃ A(s(n)). If n is arbitrary, the conclusion is that every natural number has the property.

The construction of derivations in Gentzen's tree form is awkward: One would have to know, more or less, how the final tree has to look before one can start. So one looks at the assumptions, and one looks at the conclusion, and one tries to figure a way from the former to the latter. The good aspects of the tree form include that once you have the tree, its structure is transparent.

We shall modify a bit natural deduction, to get a logical calculus that supports proof search better. The starting point is the change in Section 3.4 in which all E-rules were written with an arbitrary conclusion. Consider any formula C in a derivation tree. The assumptions it depends on can be listed, and let them be A1,…, An. If you take the part of the tree that is determined by the chosen formula C, you geta subderivation of the original derivation. To be precise, to get a correct derivation tree you have to delete the labels from above those assumptions that have not been closed at the stage in which formula C is concluded. The subderivation establishes a derivability relation, namely A1,…, An ⊢ C.

It will be convenient to have a name for the thing a derivability relation establishes: Expressions of the form A1,…, An ⊢ C are called sequents. The name stems from the arrangement of formulas in a sequence in the antecedent, left part of a sequent.

Not all the systems mentioned in this book have been shown to be complete, only the ones for which a method has been described for converting trees into proofs. In this section, a more powerful strategy for showing completeness will be presented that applies to a wider range of propositional modal logics. It is a version of the so-called Henkin or canonical model technique, which is widely used in logic. This method is more abstract than the method of Chapter 8, and it is harder to adapt to systems that include quantifiers and identity, but a serious student of modal logic should become familiar with it. The fundamental idea on which the method is based is the notion of a maximally consistent set. Maximally consistent sets play the role of possible worlds. They completely describe the facts of a world by including either A or ~A (but never both) for each sentence A in the language.

The Lindenbaum Lemma

A crucial step in demonstrating completeness with such maximally consistent sets is to prove the famous Lindenbaum Lemma. To develop that result, some concepts and notation need to be introduced. When M is a (possibly) infinite set of sentences, ‘M, A’ indicates the result of adding A to the set M, and ‘M ⊢S C’ indicates that there is a finite list H formed from some of the members of M, such that H ⊢S C. Set M′ is an extension of M, provided that every member of M is a member of M′. We say that set M is consistent in S iff M ⊬S ⊥. M is maximal iff for every sentence A, either A or ~A is in M. M is maximally consistent for S (or mc for short) iff M is both maximal and consistent in S. When it is clear from the context what system is at issue, or if the results being discussed are general with respect to S, the subscript ‘S’ on ‘⊢’ will be dropped, and we will use ‘consistent’ in place of ‘consistent for S’. It should be remembered, however, that what counts as an mc set depends on the system S. We are now ready to state the Lindenbaum Lemma.

The completeness of quantified modal logics can be shown with the tree method by modifying the strategy used in propositional modal logic. Section 8.4 explains how to use trees to demonstrate the completeness of propositional modal logics S that result from adding one or more of the following axioms to K: (D), (M), (4), (B), (5), (CD). In this chapter, the tree method will be extended to quantified modal logics based on the same propositional modal logics. The reader may want to review Sections 8.3 and 8.4 now, since details there will be central to this discussion. The fundamental idea is to show that every S-valid argument is provable in S in two stages. Assuming that H / C is S-valid, use the Tree Model Theorem (of Section 8.3) to prove that the S-tree for H / C closes. Then use the method for converting closed S-trees into proofs to construct a proof in S of H / C from the closed S-tree. This will show that any S-valid argument has a proof in S, which is, of course, what the completeness of S amounts to.

The Quantified Tree Model Theorem

In order to demonstrate completeness for quantified modal logics, a quantified version of the Tree Model Theorem will be developed here. This will also be useful in showing the correctness of trees for the quantified systems. Proofs of the appropriate tree model theorems are complicated by the fact that there are so many different systems to be considered in quantified modal logic. Several different interpretations of the quantifiers have been developed. However, the transfer theorems proven in Sections 15.7 and 15.8 show how results for all these systems can be obtained from a proof of completeness for truth value models (tS-models). So a demonstration of the Tree Model Theorem will be given for tS-validity.

Preface

The main purpose of this book is to help bridge a gap in the landscape of modal logic. A great deal is known about modal systems based on propositional logic. However, these logics do not have the expressive resources to handle the structure of most philosophical argumentation. If modal logics are to be useful to philosophy, it is crucial that they include quantifiers and identity. The problem is that quantified modal logic is not as well developed, and it is difficult for the student of philosophy who may lack mathematical training to develop mastery of what is known. Philosophical worries about whether quantification is coherent or advisable in certain modal settings partly explain this lack of attention. If one takes such objections seriously, they exert pressure on the logician to either eliminate modality altogether or eliminate the allegedly undesirable forms of quantification.

Even if one lays those philosophical worries aside, serious technical problems must still be faced. There is a rich menu of choices for formulating the semantics of quantified modal languages, and the completeness problem for some of these systems is difficult or unresolved. The philosophy of this book is that this variety is to be explored rather than shunned. We hope to demonstrate that modal logic with quantifiers can be simplified so that it is manageable, even teachable. Some of the simplifications depend on the foundations – in the way the systems for propositional modal logic are developed. Some ideas that were designed to make life easier when quantifiers are introduced are also genuinely helpful even for those who will study only the propositional systems. So this book can serve a dual purpose. It is, I hope, a simple and accessible introduction to propositional modal logic for students who have had a first course in formal logic (preferably one that covers natural deduction rules and truth trees). I hope, however, that students who had planned to use this book to learn only propositional modal logic will be inspired to move on to study quantification as well.

Here we give completeness proofs for many quantified modal logics, using a variant of the method of maximally consistent sets. Although the previous chapter already established completeness for many quantified modal logics using the tree method, there are good reasons for covering the method of maximally consistent sets as well. First, this is the standard approach to obtaining completeness results, so most students of modal logic will want some understanding of the method. Second, the tree method applied only to those systems for which it was shown how to convert a tree into a proof. The method of maximally consistent sets applies to more systems, though it has limitations described below in Section 17.2.

How Quantifiers Complicate Completeness Proofs

One might expect that proving completeness of quantified modal logic could be accomplished by simply “pasting together” standard results for quantifiers with those for propositional modal logic. Unfortunately, it is not so easy. In order to appreciate the problems that arise, and how they may be overcome, let us first review the strategies used to show completeness for propositional modal logic with maximally consistent sets. Then it will be possible to outline the difficulties that arise when quantifiers are added.

Preface to the Second Edition

In the years since the first publication of Modal Logic for Philosophers, I have received many suggestions for its improvement. The most substantial change in the new edition is a response to requests for a chapter on logics for conditionals. This topic is widely mentioned in the philosophical literature, so any book titled “Modal Logic for Philosophers” should do it justice. Unfortunately, the few pages on the topic provided in the first edition did no more than whet the reader’s appetite for a more adequate treatment. In this edition, an entire chapter (Chapter 20) is devoted to conditionals. It includes a discussion of material implication and its failings, strict implication, relevance logic, and (so-called) conditional logic. Although this chapter still qualifies as no more than an introduction, I hope it will be useful for philosophers who wish to get their bearings in the area.

While the structure of the rest of the book has not changed, there have been improvements everywhere. Thanks to several classes in modal logic taught using the first edition, and suggestions from attentive students, a number of revisions have been made that clarify and simplify the technical results. The first edition also contained many errors. While most of these were of the minor kind from which a reader could easily recover, there were still too many where it was difficult to gather what was intended. A list of errata for the first edition has been widely distributed on the World Wide Web, and this has been of some help. However, it is time to gather these corrections together to produce a new edition where (I can hope) the remaining errors are rare.

Many different systems of quantified modal logic have been presented in this book, each one based on the minimal system fK. In the next few chapters, we will show the adequacy of many of these logics by showing both their soundness and completeness. When S is one of the quantified modal logics discussed, and the corresponding notion of an S-model has been defined, soundness and completeness together amount to the claim that provability-in-S and S-validity match.

1. (Soundness) If H ⊢S C then H ⊨S C.

2. (Completeness) If H ⊨S C then H ⊢S C.

This chapter will be devoted to soundness and to some theorems that will be useful for the completeness proofs to come. Some of these results are interesting in their own right, since they show how the various treatments of the quantifier are interrelated. Sections 15.4–15.8 will explain how notions of validity for the substitution, intensional, and objectual interpretations are shown equivalent to corresponding brands of validity on truth value models – the simplest kind of models. This will mean that the relatively easy completeness results for truth value models can be quickly transferred to substitution, intensional, and objectual forms of validity. Readers who wish to study only truth value models may omit those sections.

Two different strategies will be presented to demonstrate completeness for truth value models. Chapter 16 covers completeness using a variation on the tree method found in Chapter 8. The modifications needed to extend the completeness result to systems with quantifiers are fairly easy to supply. Chapter 17 presents completeness results using the canonical model technique of Chapter 9. This method is the standard technique found in the literature, but it requires fairly extensive modifications to the strategy used for propositional modal logic. Chapters 16 and 17 are designed to be read independently, so that one may be understood without the other.