There is another kind of analogy, denominative analogy, that results from contrasts of predicate mode. It consists of differentiated denominations (relational naming: to plow/a plow; to smoke/smokes), of which there are several kinds.

Denominative analogy is particularly important in accounting (a) for the relatedness among paronyms, (b) for figurative discourse (metonymy, synecdoche, antonomasia), (c) for the transfer of thing-names and event-names into verbs, and vice versa, and (d) for the relatedness of the same words used contrastively to impute activity, proclivity, tendency, disposition, and the like. For example, ‘He cheated’ (once) as against ‘He cheated’ (characteristically); ‘He is a responsible young man’; ‘His was a responsible action under conditions of great uncertainty’.

Denominative analogy sometimes involves abstractive analogy of proportionality (cf. chapter 4) along with contrasting predicate modes: ‘Most of the employees at Newtown are healthy; Newtown, as the travel agents all say, is a healthy place.’ The second ‘healthy’ is both denominatively and abstractively differentiated from the first. And there can be metaphorically denominative pairs, too, that is the basis of personification (see chapter 6).

What Cajetan (1498: ch.2, sec. 10–15) called ‘analogy of attribution’ and Aquinas (De Veritate, 21, 4 ad 4 and Principles of Nature, ch.6, no. 38) ‘analogy because of diverse attributions’, is a species of denominative analogy. For example, the analogy of meaning between ‘healthy’, applied to the intrinsic state of a dog, and ‘healthy’, applied to the appearance of its coat.

Most philosophers think truth-conditional analysis tends to fail. From the paradigm itself, the Russellian contextual analysis of ‘The author of Waverly was Scott’ that treated ordinary proper names as definite descriptions, through Ayer's criteria of meaningfulness and verifiability, Lewis' analyses of modes of meaning, Hart's model definition for jurisprudence, Grice's analysis of speaker meaning, Hempel's analysis of explanation and Chisholm's (1966) analyses of knowledge, truth-conditional analyses founder on counterexamples and, when repaired, typically show more anomalies than did the originals. Besides, analyses that set out to clarify often become so complex (Schiffer 1972: 165; cf. Coady 1976: 102–9), abstract or artificial that they cannot convince. For example, Goodman's analysis of musical notation (1968:177–92); Field's (1978) and Fodor's (1975) account of beliefs as ‘language-like internal representations’ that are, or correspond to, ‘sentences or sentence analogues’. See also Harman (1974b: 57). Is the fault in our performance or in our expectations?

Truth-conditional analysis is misunderstood. Attention to the analogy phenomena will correct some misconceptions. Others persist from confusions about the inter-relationships of sentences, statements, propositions and truth conditions, from confusions about the notion of same logical form and from unrealistic expectations about what is to be accomplished by analysis. I touch incidentally on the latter matters but my main interest is in showing how analogy and equivocation place constraints upon the process of analysis.

I conclude that truth-conditional analysis is an articulation device, instrumental to encompassing strategies (like explanation, conceptual realignment, auditing a reasoning process, hypothesis establishment or rejection), that determine the units of appropriate logical and conceptual decomposition.

INTRODUCTION

Earlier chapters show how mere equivocation, analogy of proportionality and metaphor all result from the domination of susceptible words by contrasting, scheme-intransigent expressions.

‘Pen’ is indifferent to instrument schemes (that include ‘pencil’, ‘quill’, ‘stylus’, etc.) and to enclosure schemes (that include ‘stall’, ‘paddock’, ‘run’, etc.). The verb ‘wrote’ is indifferent to means expressing schemes (that include ‘telephoned’, ‘telegraphed’, ‘semaphored with’, ‘drew with’, ‘inscribed with’, ‘signed with’, etc.). Yet, concatenated into ‘He wrote with a pen’, ‘wrote’ is not indifferent to the enclosure scheme for ‘pen’; rather, it is resistant because it belongs to a means demanding scheme which, concatenated with that enclosure scheme, would in the supposed environment yield an unacceptable sentence, e.g. ‘He wrote with a barnyard enclosure’. ‘Wrote’ dominates ‘pen’. And ‘pen’, thus determined, reciprocally dominates ‘wrote’ to exclude the means expressing scheme (‘telegraphed’, etc.) because that result would be similarly unacceptable.

Of course, in a story about stock exchange trading, ‘He telegraphed with his pen’ could mean that he sent a message by a motion of his pen, and the context would be read so as to make ‘telegraphed’ means-demanding (‘with his pen’). And ‘ink’ doesn't always dominate ‘pen’. In a story about counterfeiters hiding their ink somewhere on a farm, ‘They put the ink into the pen’ might not have ‘ink’ dominating ‘pen’. Still, ‘He put the pen into the ink’ would usually have ‘ink’ dominating because of the tight syntactical link to ‘pen’ and the commonplace improbability of the other reading.

RECONCEIVING THE PROBLEM

The central issues in the dispute about the cognitive content of Judeo-Christian religious discourse are not peculiar to religion. Metaphysical, ethical, aesthetic, legal and scientific discourse raise generically the same issues. They are all craftbound and their vocabularies are similarly embedded by analogy in other craft discourse and in unbound discourse.

The business discourse of plumbers, football coaches, mechanics, philosophers, physicists, physicians and lawyers is paradigmatically craftbound. That name, ‘craftbound discourse’, connotes that skill in action is necessary for a full grasp of the discourse. Ordinarily you master a craftbound discourse only if you become an ‘insider’ to the doing that the discourse is about.

The basic vocabulary of these kinds of discourse is similarly anchored to benchmark situations (legal cases, Scripture stories, scientific experiments, particular professional observations and responses) that structure and stabilize the central meaning relationships.

So, although the immediate subject matter of this chapter is the religious discourse of relatively orthodox Christianity, the forms of argument, the general strategy and the specific analogy claims apply to any craftbound discourse whose cognitive content is challenged and, in particular, apply to explaining the relationship of theoretical predicates in science to ‘observational predicates in discourse that is independent’ of the theory. To illustrate that generality, I provide some examples from legal discourse and refer to further discussion of jurisprudence in chapter 8.

In this chapter we introduce the subject of modal logic by surveying some of the main features of the system of modal logic known as 55. This system is but one of many we shall study. Because it is one of the simplest, we choose it to begin with.

The system S5 is determined semantically by an account of necessity and possibility that dates to the philosopher Leibniz: a proposition is necessary if it holds at all possible worlds, possible if it holds at some. The idea is that different things may be true at different possible worlds, but whatever holds true at every possible world is necessary, while that which holds at at least one possible world is possible.

In section 1.1 we develop this semantic idea by means of a definition of truth at a possible world in a model for a language of necessity and possibility. This leads to a definition of validity, and we set out some valid sentences and principles governing validity, as well as some examples of invalidity.

The totality of valid sentences forms the modal logic S5. In terms of the principles set out in section 1.1 it is possible to deduce all the valid sentences. Some evidence of this appears in section 1.2, where we take the principles in section 1.1 as axioms and rules of inference, formulate S5 as a deductive system, and derive a number of further principles.

In this chapter we present and prove a number of determination theorems for normal modal logics with respect to classes of standard models. Section 5.1 contains the basic theorem for the soundness of such systems and a proof that the fifteen systems on the diagram in figure 4.1 are all distinct. In section 5.2 we return briefly to the topic of modalities in normal systems. In section 5.3 we define the idea of a canonical standard model for a normal system and prove some fundamental theorems about completeness. Section 5.4 contains determination theorems for the logics in figure 4.1, including a theorem to the effect that the system S5 is determined by the class of models of the sort described in chapter 1. In section 5.5 we generalize the ideas in sections 5.1 and 5.3 to obtain a very large class of determination results in one fell swoop, by proving that the system KGk,l,m,n is determined by the class of k,l,m,n-incestual models. Finally, in section 5.6 we prove the decidability of the fifteen systems in figure 4.1.

Soundness

The following theorem provides the basis for proofs of soundness for normal modal logics, with respect to classes of standard models.

Theorem 5.1. Let S1, …, Snbe schemas valid respectively in classes of standard models C1, …, Cn. Then the system of modal logic KS1 … Sn is sound with respect to the class C1 ∩ … ∩ Cn.

According to the account of necessity and possibility in chapter 1, a sentence of the form □A is true at a possible world just in case A itself is true at all possible worlds, and a sentence of the form ◇A holds at a possible world if and only if A holds at some possible world. This idea was modeled very simply in terms of a collection of possible worlds together with an assignment of truth values, at each world, to the atomic sentences. We saw that the ensuing notion of validity is quite strict, encompassing as it does a large assortment of principles.

In the present chapter we modify this leibnizian conception of necessity and possibility by introducing an element of relative possibility. The result is a much more supple notion of validity, one that greatly reduces the stock of principles that are bound to hold.

In section 3.1 we define the idea of a standard model, state the truth conditions for modal sentences at worlds in models of this sort, and prove a theorem about validity in classes of standard models. In section 3.2 we single out the schemas D, T, B, 4, and 5 for special attention, both because of their historical prominence (recall that they are all theorems of S5) and because the techniques required for their treatment are illuminating and instructive.

Conditionality affords a good example of a concept susceptible of analysis by means of the kinds of models and systems studied in this book. In section 10.1 we present the basic systems of conditional logic and the classes of models that determine them. In section 10.2 we return to the subject of deontic logic and define a minimal logic for conditional obligation. In section 10.3 we offer a definition of the conditional obligation operator in terms of simple obligation and non-deontic conditionality.

As with chapter 6, the purpose of this chapter is to illustrate the use of our semantic and deductive-theoretic techniques in the analysis of philosophically interesting concepts. Again, as in the earlier chapter, the reader will be the judge of the merit of the endeavor and the extent to which it is successful.

Conditionality

Into the language of propositional logic we introduce sentences of the form A => B. The operator => is meant to express a notion of conditionality – a notion in general distinct from that expressed by →.

In a standard conditional model ℳ = 〈W, f, P〉for the language of conditional logic/is a mapping that selects a proposition (set of worlds) f(α, X) for each world a and proposition, or condition, X.

This book is an introductory text in modal logic, the logic of necessity and possibility. It is intended for readers with the equivalent of a first course in formal logic, and it is designed to be used as a basic text in courses at the advanced undergraduate or beginning graduate level. The material in the book can easily be covered in a full-year course; with selectivity most of the material can be covered in a single term.

There are three parts to the book. Part I consists of two chapters, meant to introduce the reader to the subject of modal logic and to furnish a sufficient background for the parts that follow. Chapter 1 is a relatively informal examination of S5, one of the best-known systems of modal logic. Chapter 2 – ‘Logical preliminaries’ – contains almost everything needed for an understanding of the rest of the book. Some readers may prefer to go quickly through this chapter and then reread as necessary sections required in the context of succeeding chapters.

Part II comprises four chapters on standard models and normal systems of modal logic. The models, sometimes called ‘Kripke models’, are explained in chapter 3. In chapter 4 normal systems are presented from an axiomatic standpoint. Chapter 5 contains theorems on completeness and decidability, which bring together the model-theoretic and deductive-theoretic treatments of the preceding chapters. As an illustration of normal systems chapter 6 offers a discussion of deontic logic, the logic of obligation.

This chapter is an introduction to most of the concepts we shall use in studying modal logic.

In section 2.1 we set out most of the syntactic concepts. Section 2.2 introduces semantic concepts: the general idea of a model, truth conditions for non-modal sentences, and definitions of truth in a model and validity in a class of models. Filtrations of models are described in section 2.3. In section 2.4 the idea of a system of modal logic is explained, along with such relevant notions as theoremhood, deducibility, and consistency. Axiomatizability is discussed in section 2.5. Maximal sets of sentences and Lindenbaum's lemma occupy section 2.6. In section 2.7 we define determination and explain our approach, using canonical models, to proofs of determination. Finally in section 2.8 we outline our method of proving the decidability of systems of modal logic.

As the need arises the reader may wish to return to various sections of this chapter, for important definitions and theorems.

Syntax

This section is devoted to a recital of the basic syntactic concepts for the language of modal logic, many of which the reader has likely gleaned from chapter 1. The ideas are very simple. The few formal definitions we offer may be helpful, but they are not essential; we state them mainly for the sake of completeness and future reference.

This chapter is devoted to studying, from a purely deductive standpoint, a class of systems of modal logic we call normal.

In section 4.1 we first define the class of normal systems. Then we derive a number of theorems and rules of inference common to all normal modal logics and use some of them to formulate alternative deductive characterizations of such systems. Theorems on replacement, negation, and duality are proved in section 4.2 for normal modal logics (they hold more generally for all classical systems, as we discover in chapter 8). These results provide rules and theorems that serve to facilitate derivations.

The smallest normal system of modal logic we call K. Thus every normal system of modal logic is a K-system. (The converse is false; not all K-systems are normal.) To simplify naming normal systems we write

KS1 … Sn

to denote the normal modal logic obtained by taking the schemas S1 …, Sn as theorems. In other words:

KS1 … Sn = the smallest normal system of modal logic containing (every instance of) the schemas S1 …, Sn.

So, for example, KT4 is the smallest normal system produced by treating the schemas T and 4 as theorems in a normal modal logic. (It is also denoted by K4T; the order of the schema names is irrelevant.) As the limiting case, where there are no schemas, the definition yields K as the smallest normal system.

In this chapter we connect classical modal logics and classes of minimal models by way of determination theorems. Our method is much the same as in chapter 5 for normal systems and standard models. In section 9.1 we treat questions of soundness and prove the distinctness of the eight classical systems on the diagram in figure 8.1 having M, C, and N as theorems. Section 9.2 contains the definition of canonical minimal models and the fundamental theorems for completeness. We do not single out any particular canonical minimal model as ‘proper’, as we did in the case of normal systems. But we indicate a uniform way of describing canonical minimal models – a way that highlights two very useful such models, which we call the smallest and the largest, for any classical logic.

In section 9.3 there are determination theorems for the systems in figure 8.1. The idea of supplementation plays an important part in obtaining completeness results for the monotonic systems, and we use augmentation to reach, again, the conclusion that normal systems are complete with respect to the class of standard models. Then in section 9.4 we treat in an abridged fashion questions of completeness for classical systems having as theorems familiar schemas such as D, T, B, 4, and 5. Finally, in section 9.5 we prove the decidability of the systems E, M, R, and, once again, K.