Background

It is part of the lore of logical theory that sooner or later one comes to the study of implication. This study of the modal operators is a consequence of taking the old advice to heart by considering implication sooner rather than later. Modality, on our account, is a way of studying the question whether or not implication continues to be preserved when the elements related by implication are transformed by an operator. The basic idea is that a modal operator is any operator or function φ that transforms or maps the set S of an implication structure I = 〈S, ⇒〉 to itself in such a way that if A1, …, An ⇒ B, then φ(A1), φ(A2), …, φ(An) ⇒ φ(B). There is a second condition concerning the relation of φ to the dual implication relation “⇒̂,” which we shall introduce shortly. The two conditions will then specify the kind of functions that count as having modal character.

If we are correct about this, then the study of modal operators is a natural continuation of the study of implication itself. Whatever reservations a philosopher might have about the philosophical merit of such concepts as “necessity” and “possibility”, there is every reason for studying the modal operators, since they are among the operators that preserve implication. The key idea, then, is to think of a modal operator as modal relative to some implication relation.

The simple characterization

Our aim here, as in the preceding chapter, is to characterize the negation operator as a special kind of function that can act on all implication structures, general and special.

Let I = 〈S, ⇒〉 be an implication structure. For any A in S, we shall say that N(A) is a negation of A if and only if it satisfies the following two conditions:

1. N1. A, N(A) ⇒ B for all B in S, and

2. N2. N(A) is the weakest member of the structure to satisfy the first condition. That is, if T is any member of S such that, A, T ⇒ B for all B in S, then T = ⇒ N(A).

As with the hypothetical, the negation operator on the structure is supposed to sort out, for any A, those elements in the structure that are the negations of A. Strictly speaking, then, the negation operator assigns to each A of S a set of members of S that will satisfy the preceding conditions. That set may be empty, for there is, as we shall see later, no guarantee that negations always exist. However, if there are several members, then they will be equivalent under the implication relation of the structure. Thus, as long as there is only one implication relation that is being studied on a set S, there will not be any confusion if we treat N(A) as if it were an element of S.

Parametric conditions for the operators

The preceding chapters describe the various logical operators, each without reference to any of the others. The descriptions are uniform in format: the weakest to satisfy a certain condition that is characteristic for the operator under discussion. We now wish to supplement that account with an additional requirement, a “parametric” requirement. The description of each operator still will be independent of the others, and the uniform format will be preserved. However, without the additional condition, the system would be weaker than the system studied thus far; without parameterization, some of the logical operators would fail to have some of the familiar features we normally expect them to have. The need for the additional condition needs a bit of explanation.

When we described the conjunction operator, we required that C(A, B) imply A as well as B and that it be the weakest member of the structure to imply A as well as B. It certainly seems as if that is the entire story to be told about conjunctions. Yet, surprisingly enough, that description will not guarantee that in all implication structures, A together with B implies their conjunction.

Let I = 〈S, ⇒〉 be an implication structure.

Accessibility relations and the diversity of modals

We have already observed (Section 28.3) how the nonempty theories of an implication structure can be used to provide a necessary and sufficient condition for an operator to be a necessitation modal. There we introduced the idea of an accessibility relation Rφ for each modal operator on the implication structure I (and, where σ is the dual of φ, Rσ is the appropriate accessibility relation). They are binary relations on the theories of the structure such that for all theories U, V, …,

1. (1) URφV if and only if φ−1U ⊆ V [where A is in φ−1U if and only if φ(A) is in U], and

2. (2) URσV if and only if V ⊆ σ−1U (where σ is the dual of φ).

Each accessibility relation is, according to these two conditions, tailored to the particular modal operator under study. The tightness of the relation between a modal and its accessibility relation was discussed, and need not be repeated here. The theories, as used in this chapter, are those subsets of the structure that are (weakly) closed under implication in this sense: If A is in U, and A ⇒ B, then B is in U as well.

Anyone who is even remotely aware of S. Kripke's ground-breaking work on the semantics of modal logic knows how the idea of accessibility relations and the possible worlds that they relate were used with startling effect to provide a series of completeness proofs of familiar modal systems and to distinguish the variety of modal laws from each other in a very simple and coherent manner.

It would be implausible to think of implication relations as not including the familiar notions of deducibility and logical consequence, and it would be misleading to think that implication relations were limited to cases of just these two types. A concept of (syntactic) deducibility is usually introduced over a well-defined set of sentences, according to which A1, …, An ⊢ B holds if and only if there is a sequence of sentences c1, …, Cm such that Cm is B, and every Ci is either an Aj (or an axiom if we are concerned with axiomatic formulations of the theory) or follows from one or more preceding sentences of the sequence by any member of a finite collection of rules R. It is easy to verify that “⊢” satisfies the condition for being an implication relation.

The semantic concept of consequence is also familiar. Starting with a well-defined set of sentences of, say, the propositional calculus, the notion of an interpretation is defined, and for any A1 …, An and B, we say that B is a logical consequence of A1 …, An that is, A1, …, An ╞ B if and only if every interpretation that assigns “true” to all the Ai's also assigns “true” to B. It is easy to see that “╞” is an implication relation. The details are familiar and need not be reviewed here.

This project is an attempt to give an account of the logical operators in a variety of settings, over a broad range of sentential as well as nonsentential items, in a way that does not rely upon any reference to truth conditions, logical form, conditions of assertability, or conditions of a priori knowledge. Furthermore, it does not presuppose that the elements upon which the operators act are distinguished by any special syntactic or semantic features. In short, it is an attempt to explain the character of the logical operators without requiring that the items under consideration be “given” or regimented in some special way, other than that they can enter into certain implication relations with each other. The independence of our account from the thorny questions about the truth conditions of hypotheticals, conjunctions, disjunctions, negations, and the other logical operators can be traced to two sources. Our account of the logical operators is based upon the notion of an implication structure. Such a structure consists of a nonempty set together with a finitary relation over it, which we shall call an implication relation. As we shall see, implication relations include the usual syntactic and semantic kinds of examples that come to mind. However, implication relations, as we shall describe them, are not restricted to relations of deducibility or logical consequence. In fact, these relations are ubiquitous: Any nonempty set can be provided with an implication relation.

This study has emerged in much longer form than I had ever intended. In part this happened because I was unsure how familiar or foreign its central ideas might be. I was concerned that enough detail and examples be provided so that the project would come to look more familiar and less foreign than it otherwise might appear initially. That concern, thanks to the encouragement of good friends and patient audiences, now seems to me to have been exaggerated. Still, there is the need to illustrate the power and the range of what are essentially several simple ideas that enable one to provide a uniform picture and some perspective on the enterprise of logic, standard and nonstandard. The work has also taken longer to appear than I would care to quantify. The resources used are far fewer than those normally used in logical studies, and it was not always clear what could be established and what could not. Theorem-proving aside, I must say that it took a long time to gain some understanding of the significance of the results that were obtained. It seems to me that there is still much more that needs to be understood.

Some of these ideas are found in one form or another in the literature, and others are part of the folklore. The intellectual inspiration for much of what follows derives from the seminal writings of G. Gentzen, P. Hertz, and A. Tarski. I have tried to indicate the sources of this project in Part I (Introduction) and to provide some comparison with related ideas.

The Conditions for Modal Character

Let I = 〈S, ⇒〉 be an implication structure. We shall say that any function φ that maps S to S is a modal operator on I if and only if the following conditions are satisfied:

1. M1. For any A1, …, An and B in S, if A1, …, An ⇒ B, then φ(A1), …, φ(An) ⇒ φ(B).

2. M2. For some A1, …, An and B in S, A1, …, An ⇒ ̂ B, but φ(A1), …, φ(An) ⇏ ̂ φ(B).

Briefly expressed, the first condition for being a modal operator with respect to the implication relation “⇒” is that the operator distributes over the implication relation. The second condition states that the operator does not distribute over “⇒̂,” the dual of the implication relation. According to the first condition, the operator φ preserves all implications of the structure I. This in itself is a very natural “homomorphism-style” condition. But natural or not, the condition would be beside the point if most of the operators associated with familiar modal systems did not satisfy it. This, as we shall see later, they certainly do. In fact, for all so-called normal modal systems L, those for which a distribution axiom over conditionals holds [i.e., ⊢L□(A → B) → (□A → □B)] yield an associated operator that satisfies the first condition of modality. The difference is that a distribution axiom such as the one just described concerns the behavior of an operator with respect to conditionals, and neither of our conditions refers to any of the logical operators.

We have characterized the modal operators as a special type of function that maps the set of an implication structure to itself. Certainly there are such operators, as the various examples show. It is also true that these operators behave in ways that modals are supposed to. They appear, then, to be familiar, not foreign. They count as modal. One question remains: How do some of the familiar kinds of modals figure in this scheme of things? One of our aims is to capture most, if not all, of what has usually been regarded as modal. Have we done so? Are there any blatant omissions? We turn now to a review of some traditional and nontraditional sources for modal operators.

Modal Operators of Normal Modal Systems

Normal modal systems consist of the language of the classical sentential calculus, with the addition of the box, □, to its vocabulary, and appropriate formation rules. In addition to the distribution axiom □(p → q) → (□p → □q), they have the rules modus ponens, necessitation, and, in some formulations, substitution. These systems differ from each other in the addition (beyond the distribution axiom) of axioms involving the box. The simplest normal system is (K), which has only the distribution axiom. (The usual normal systems were described in Chapter 25, notes 1 and 2.)

Let (L) be a normal modal system.

Introduction

It has been a recurrent theme in this study that the logical operators are functions on implication structures, simple or extended, and that these structures are so varied in their objects that the study of the logical operators is truly topic-neutral.

The conjunction and negation operators, for example, are defined over all implication structures, so that one can think of conjunctions of sets, statements, properties, individuals, and so forth. The subject in each case is only as interesting as the implication relations with respect to which the operators are relativized. Any set, and in particular a set of objects (or a set of names of those objects), can be thought of as having an implication relation on it. The question is whether or not there is an implication relation on a set of individuals, or the set of names of those individuals, that has theoretical interest. We shall call implication relations on individuals objectual implication relations (and implication on names of individuals, nominal implication).

It might be thought that the whole idea of an implication relation holding between individuals is incoherent, since implication is a relation between truth-bearers. On the more general view that we have advocated, the fact that individuals and names are not generally truthbearers is no impediment to the development of a theory of the logical operators on them. All that is required, since all the logical operators are relativized to implication relations, is that an implication relation be specified for them.

The distinctness of the logical operators

In studying the formal languages of the classical or intuitionistic calculus, there is a concern that the notation ensure that there be “unique readability.” If a sequence of expressions of the system is to count as a sentence of the system, then it should not happen that it counts as two sentences. Thus, where “∨” is the sign for the disjunction connective, a sequence such as “A ∨ B ∨ C” is not counted as a sentence, because it could be thought of as “(A ∨ B) ∨ C” or as “A ∨ (B ∨ C),” and these sequences are intended to count as distinct sentences, no matter how closely related they are logically. One way of expressing the concern is to think of various operators associated with sentence-building. For example, let E∨ and E→ be the operators that assign to the sentences A and B the sentences “(A ∨ B)” and “(A → B),” respectively. Then the condition for unique readability can be expressed by the requirement that each of these E's be one-to-one functions and that no two of them have the same range. Thus, in particular, if E∨(A, B) = E∨(C, D) for any sentences A, B, C, and D, then A = C, and B = D. And E∨(A, B) ≠ E→(C, D) for any A, B, C, and D.

In this study, logical operators such as conjunction C and disjunction D do not yield unique readability results.

Although the particular Introduction and Elimination rules for each connective are easily understood, the overall characterization is not easy to comprehend, nor is the general connection that each has to the other. Gentzen thought (1) that the Introduction rules provided a “definition” of the connectives, (2) that they also provided a sense of those connectives, (3) that it should be possible to display the E-inferences as unique functions of their corresponding I-inferences, and (4) that “in the final analysis” the Elimination rules were consequences of the corresponding Introduction rules. What he said about the character of each type of rule and their connections with one another is very much worth quoting in full:

Elementary questions about interrogatives

There has been some penetrating work on the logic of interrogatives, but it is not the aim of this study to review those findings. It is well known that there are interestingly different kinds of questions that count as interrogatives, and it has also been suggested that an account of questions can profitably be seen as part of the study of a more general notion. That is, if, roughly speaking, the logic of questions (erotetic logic) is seen as the study of wanted answers, then the somewhat more general setting would consider the notion of a relevant answer. And still other, more general settings for the logic of questions have been suggested. Perhaps the most complex issue that has to be faced is the typology of questions, since each study seems to use a different classification. No doubt this reflects, in part, the variety of theoretical devices that are brought to bear on the subject.

In this chapter we want to show how the use of a special implication relation among certain kinds of questions can be used to answer some elementary problems about questions. We shall not introduce a special typology to sort out different kinds of questions from each other. The problems that we shall consider lie at a rather elementary level and do not seem to need a more subtle classification for their discussion. A more detailed classification will eventually be necessary, because it is only with the aid of more refined structures that anything interesting can be proved and that anything accurate can be said about whether specific sentences in English are examples of one type of interrogative rather than another.

The loss of modal character

On our account, one may sometimes come across operators that seem to have modal character, but do not distribute over implication. Indeed, one may, to take a parallel example, come across hypotheticals that do not seem to satisfy the condition that corresponds to modus ponens. How is this possible? It is one of our conditions on the modals that they distribute over implication, and it is part of our conditions on hypothetical operators that A, A → B ⇒ B.

The answer lies in the recognition that modal operators as well as the logical operators are relativized to implication relations. That feature is an integral part of our account, and one of its deepest resources. A structure consists of a base set A, with an implication relation on it. There can, as we have seen, be various implication relations on a given set. However, with, say, two implication relations on a given set, the possibility arises that an operator may be modal with respect to one implication relation, but fail to be modal with respect to the other. Similarly, an operator may be a hypothetical with respect to one implication relation, but not with respect to another over the very same set. The result of this shift from one implication relation to another can be dramatic: Modus ponens will seem to fail in the case of hypotheticals, and modals will seem to violate the conditions even of monotonicity [if A ⇒ B, then φ(A) ⇒ φ(B)], or even seem to fail to be classical [if A ⇔ B, then φ(A) ⇔ φ(B)].

Disjunction: the simple characterization

We saw in preceding chapters how the duals of various logical operators can be defined. In the case of the hypothetical, H, its dual, Ĥ, has rarely been studied, if at all. In the case of negation, as we pointed out, the dual N̂ has an important role to play in the study of generalizations of classical results within nonclassical implication structures. Likewise with conjunctions: The dual of conjunction, Ĉ, is just conjunction on the dual structure Î and is, as we noted, a logical operator on I as well. But contemporary logicians think of the dual of conjunction not merely as the dual of an operator, but as an important logical operator in its own right. There is a direct characterization of Ĉ as a logical operator on I, since Ĉ is, as we indicated in the preceding chapter, a well-known logical operator on I in its own right, namely, disjunction on I.

Let I = 〈S, ⇒〉 be an implication structure. For any A and B in S, the logical operator of disjunction is a function that satisfies the following two conditions:

1. D1. For any T in S, if A ⇒ T and B ⇒ T, then D(A, B) ⇒ T, and

2. D2. D(A, B) is the weakest member in S to satisfy the first condition. That is, for any U in S, if [for all T in S, if A ⇒ T and B ⇒ T, then U⇒ T], then U ⇒ D(A, B).

Projective implication

One of the very simplest implication relations can be obtained by regarding an element B of a set S as implied by finitely many members listed as A1, …, An, just in case B is Aj for some j in {1, …, n}. The verification of the conditions for implication relations is straightforward.

Millean implication

This is a kind of implication relation for which A1, …, An implies B if and only if some Aj implies B for some j in {1, …, n} (see Chapter 4, note 2).

Exercise 8.1. Let “#” any reflexive and transitive binary relation on a set S of at least two members. Let “⇒#” hold between A1, …, An and B if and only if Aj # B for some j in {1, …, n}. Show that “⇒#” is a Millean implication relation.

Bisection implication

A much more theoretically interesting example whose properties we shall now study in some detail arises from a consideration of the various ways in which a nonempty set S can be partitioned into two nonempty, mutually exclusive subsets.