Conditions on implication

In Part I we outlined a program for the logical operators that has two components, one consisting of implication relations, the other consisting of characterizations of the logical operators relative to these implication relations. If we think of an implication structure I = 〈S, ⇒〉 as a nonempty set S together with an implication relation “⇒” on it, then the logical operators are functions that on each implication structure assign (sets of) elements of it that count as hypothetical, conjunctions, negations, disjunctions, quantifications, and so forth, of that structure. Thus, the general picture is one in which the logical operators are relativized to implication relations, and each one is specified without recourse to any of the operators to be characterized. Otherwise the program would be circular from the outset.

Our present concern, then, is the study of implication structures, nonempty sets with implication relations on them. We have described implication relations (on a set S) as any finitary relation on S (which we write with a double arrow) that satisfies the six conditions specified in Chapter 1. Any relation will count for us as an implication relation if it satisfies these conditions. Some of these conditions follow from the others, but we shall retain the less elegant formulation for the sake of greater perspicuity.

Quantification, universal or existential, concerns operators that act on implication structures. More precisely, they act on the predicates of what we shall call extended implication structures, as well as the members of what we have called implication structures. In order to see how these operators can be characterized as logical operators, we need first to introduce a somewhat more complex kind of implication structure than the kind we have employed thus far.

Extended implication structures

The extended implication structures are like the implication structures considered thus far, with two additions: (1) a set E of “objects,” or “entities” (possibly empty), and (2) a set Pr (possibly empty) of special kinds of functions that we shall call “predicates.” We shall indicate an extended implication structure by I = 〈E, Pr, S, ⇒〉, where S is a nonempty set and “⇒” is an implication relation on S.

We think of a predicate P of the extended structure as a function that maps the infinite sequences of the set E to the set S. Thus, we shall let E* be the set of all sequences (s1, s2, …), where the si belong to E, and a predicate P of the extended structure is a mapping of E* to S.

The idea of a predicate as a function on infinite sequences of objects may seem a bit strange, despite its central use in such classic studies as A. Tarski's work on truth and A. Mostowski's study of quantifiers.

Our aim has been to study the modal operators as functions on implication structures. There is nothing mysterious or arcane about the kind of functions that they are, and there is nothing recherché about the particular examples that qualify as modal. It should come as no surprise, then, that some of the operators commonly studied in elementary logic and mathematics should turn out to have modal character. This should help reinforce the idea that some modals are quite familiar, ordinary, and uncontroversial. With this in mind, we turn to some elementary examples.

Power sets

The power set of a set A is the set of all the subsets of A. Suppose S is a set of sets (1) that is closed under finite intersections and unions of its members, and (2) for any set A in S, P(A) is in S, and (3) there are at least two members, A* and B*, of S such that neither is a subset of the other.

Let I = 〈S, ⇒〉 be an implication structure, where S is as described above, and for any sets A1, …, An and B in S, A1, …, An ⇒ B if and only if A1 ∩ … ∩ An ⊆ B.

There are examples of implication relations that are not quite “from scratch.” One kind of example uses the particular structure that the elements have in order to define an implication relation over them. Thus, there are implication relations that can be defined over sets, taking into account that it is sets over which the relation is to be defined. Another example uses theories or Tarskian systems in order to define an implication relation. A third example studies certain implication relations over individuals (or their names) and uses the fact about these individuals that they have parts that can enter into a whole-part relation. A fourth example defines an implication relation over interrogatives, exploiting the fact that these interrogatives are of a type that have “direct answers” (Chapter 23), and there is an interesting implication relation that can be defined for the integers when they are encoded in binary notation in the programming language called BASIC (see Appendix A). These examples are very different from the simple bisection relations. The latter are genuinely “topic-neutral” in that they are not sensitive to whatever structure the elements may have. These examples exploit the particular nature of the elements in the implication structure.

The second kind of example we shall describe involves the construction of implications on a set, by using an implication that is already in place. The most theoretically interesting of this type is the notion of a dual implication relation, although a second example, component implication, has some theoretical interest as well.

The dual of the negation operator on a structure is just the negation operator on the dual of that structure. This characterization of the dual of negation yields some simple results, as we noted in Chapter 12: If N̂ is the dual of N, then N̂N̂(A) ⇒ A for all A, but the converse does not hold in all structures; if A ⇒ B, then N̂(B) ⇒ N̂(A) in all structures, but not conversely; even if N(A) exists for some A, N̂(A) need not exist (if the structure is not classical); N̂ is a logical operator. These simple consequences are part of the story about dual negation, but only a part. There is an interesting story about this little-studied operator that deserves telling.

Theorem 17.1.Let I = 〈S, ⇒〉) be an implication structure for which disjunctions of its members always exist. Then D(A, N̂(A)) [that is, A ∨ N̂(A)] is a thesis of I for all A in S.

Proof. For every A in S, A, N̂(A) ⇒̂ B for all B in S, since N̂ is the negation operator on the dual of I. Since disjunctions always exist in the structure, it follows by Theorem 14.8 that B ⇒ D(A, N̂(A)) for all B in S. Consequently, D(A, N̂(A)) is a thesis of I for all A in S.

Since N and N̂ are logical operators on any structure, it is possible to compare N and N̂ with respect to their implication strengths.

Quantifiers as modals

It is a familiar observation that the universal quantifier of, say, standard first-order theory behaves in a way that is very similar to the necessity operator or box in most standard modal systems. The universal quantifier acts on predicates, open sentences, or formulas in such a way that the quantification of a conjunction implies the conjunction of the conjuncts, ∀x(Px) implies Px, and several other familiar truths of quantification all combine to suggest strongly that the quantifier “∀x” has modal character.

Similar observations have been noted for the relation of the existential quantifier “Ex” and the corresponding modal, the diamond. Thus, it seems natural to think that “∀x” corresponds to some kind of □, and “Ex” corresponds to its dual, ◊. The network of correspondences becomes even more compelling when it is observed that (at least classically), “Ex” can be defined as “¬∀x¬”.

It seems to be no accident that the quantifiers look very much like modals. We need not try to force our understanding of quantification to make it appear as if quantification is a type of necessity (or possibility). The reason that universal and existential quantifiers count as modal operators is the same as in all the other cases that we have considered: There is a certain implication structure containing the items that are quantified, and quantification over those items is modal when the corresponding operator distributes over the implication relation, but fails to distribute over its dual.

D modals

Definition 28.1. Let I = 〈S, ⇒〉 be an implication structure. We shall say that a modal operator φ on I satisfies the D condition if and only if for all A in S, φ(A) ⇒ ¬φ(¬A).

One reason for studying modals that satisfy this special condition can be traced to the interest in what is obligatory and what is permissible. We shall not worry, for the moment, whether it is acts, or sentences, or statements upon which the corresponding operators O and P are defined. It is suggested that “P(A)” (“it is permissible that A”) conveys that it is not obligatory that “not A” (or it is not obligatory that you do “not A”), and conversely. Thus it looks as if “P” might be defined simply as “¬O¬,” and it is then argued that the condition O(A) ⇒ P(A) holds: It is obligatory that A implies that it is permissible that A. Nothing can be both obligatory and not permissible. Together with the definition of “P,” we have the particular condition O(A) ⇒ ¬O(¬A) for all A's in the relevant structure. This is, then, if one is persuaded, a particular case of a D modal.

There is another way of looking at the D condition that is more logical in flavor and in turn sheds some light on the plausibility of the D condition holding for obligation.

The Montague lesson

Before we consider a host of examples of modal operators that draw upon traditional as well as nontraditional sources, we should say something about a bundle of theorems due to Montague (1963). The import of his results seems to be that syntactic treatments of modality for sufficiently rich languages are in deep trouble. It is true, of course, that our characterization of modality is not syntactical and is not restricted to structures that are syntactical. Nevertheless, it is worth looking at his results, for it might happen that there will be analogues to them that could create trouble for the more general account that we have been developing. It might just be that even when modals are taken as operators (so that modals of sentences are just certain kinds of functions that map sentences to sentences), analogues or counterparts to Montague's theorems might still hold.

By a syntactical characterization of modality Montague seems to have had in mind the idea that “□” or “N” (his notation) is predicated of sentences [i.e., N(┌A┐), rather than prefixed to the sentence itself as NA, as it would be were “N” regarded as a connective].

Consider one of Montague's theorems. It is the counterpart of the Tarski result that certain kinds of formal languages are inconsistent if they contain their own truth predicate and the diagonalization lemma is provable for the predicates of that language.

The discovery of the following underlying implication relation is due to D. T. Langendoen (private communication).

BASIC implication

In the BASIC programming language available on IBM personal computers (BASIC Compiler 2.00, Fundamentals, Boca Raton, Florida, 1985) there are many numeric functions called “logical operators.” These functions are defined over a finite set of integers that are encoded in binary notation as strings of sixteen bits of 0's and 1's. The first bit is 0 if the integer is positive, and 1 if negative. That leaves fifteen bits for encoding, so that the range of these functions consists of all the integers from −32768 to +32767 (−215 to +215). The representing string for any integer in this range, therefore, consists of an initial 0 or 1 according as it is positive or negative, and a tail end that depends on the binary representation of the integer. If the integer is positive, then the tail end consists of its binary representation; if the integer is negative, then the tail end is determined as follows: If the negative integer is −m, and the binary encoding of m has n digits, then calculate what integer together with −2(n−1) would sum to −m, and let its binary representation be at the tail end on the sixteen-bit string. Thus, 3, being positive, would be represented as 0000000000000011, the initial bit representing the positive character, and the tail end (11) being the binary representation of 3.

Recent studies have suggested ways in which the theory of implication relations might be weakened. There have not, to my knowledge, been any suggestions that would strengthen the theory. We have already referred to the suggestion that the Dilution condition be rejected – or at least not required. The theory of the logical operators that we shall develop in this study uses the full six conditions on implication relations. It is also possible to relativize the operators to nonmonotonic “implication” relations. The behaviors of the operators will, as a consequence, be different. The section later in this study that is devoted to component implication (relevance implication) is a case in point. But the general picture of the logical operators in the nonmonotonic case is largely unexplored, and thus far we have no systematic results to present.

The Cut condition has been included as a part of the characterization of an implication relation. When this condition and the others are taken as rules of inference of a system of logic, it can sometimes be shown that the Cut rule is eliminable – that is, it becomes, for such systems, a derived rule of inference.

One might suspect that the Cut condition is a consequence of the remaining conditions, but it is not.

There is a question that naturally arises when modal operators are thought of as operators on implication structures. Suppose that there are several modals on a structure. How are they related, if at all? What are their comparative strengths?

Definition 34.1. Let I = 〈S, ⇒〉 be an implication structure, and let φ and φ* be two modal operators on I. Then φ is stronger than φ* if and only if φ(A) ⇒ φ*(A) for all A in S. We shall say that φ is definitely stronger than φ* if and only if φ is stronger than φ* and there is some A* in S such that φ*(A*) ⇏ φ(A*).

If φ and φ* are modal operators on some structure I = 〈S, ⇒〉, then φ and φ* are comparable (on I) (Definition 28.3) if and only if at least one of them is stronger than the other.

We have already observed (Section 27.3.1) that there are implication structures on which there are noncomparable modals. Thus, the comparability of modal operators is not to be expected in general, not even for very simple structures. Nevertheless, there are a few results available concerning the comparative strengths of modal operators, provided they are comparable.

We have already noted two results on the strength of modals that are comparable. The first concerned necessitation modals: If there are two comparable modals on a structure, one of which is a necessitation modal, and the other not, then the necessitation modal is the weaker of the two (Theorem 28.11).

Models for structures

Let I = 〈S, ⇒〉 be an implication structure. Let T be the set of all strongly closed subsets of S – that is, those subsets U of S for which whenever A1, …, An ⇒ B, and all the Ai's belong to U, then B belongs to U. In this section, when we speak of theories, it will be these strong theories that are intended. Let R be any binary relation on the set T of all theories of the structure I. Under these conditions we shall say that F = 〈T, R〉 is a frame for the implication structure I.

We now wish to introduce the notion of a model of (or for) the structure I. Essentially this is given by specifying a function f, the model function, that associates to each A in S a collection f(A) of theories of I, that is, some subset of T, and satisfies certain additional conditions.

We do not always have connectives available in an implication structure. Consequently, the usual way of providing conditions for a model are not generally available. The use of “∨,” for example, in indicating disjunction, was a convenience for indicating the elements in the structure that are assigned by the disjunction operator (if it assigns an element at all). Similar remarks hold for our use of the other connective signs. Thus, the usual method of exploiting the syntax to define the notion of a model by saying what the model assigns to “A ∨ B,” “A ⇒ B,” “A & B,” “¬A,” and so forth, is not available generally.

Theorem 16.1 (expansion).If I = 〈S, ⇒〉 is an implication structure, and A and D are any elements of S, then H(A, B) ⇒ H(C(A, D), B) [that is, (A, → B) ⇒ ((A & D) → B)].

Proof. C(A, D), H(A, B)⇒A, since C(A, D) ⇒ A. By Projection, C(A, D), H(A, B) ⇒ H(A, B). Therefore C(A, D), H(A, B) ⇒ B. Consequently, H(A, B) ⇒ H(C(A, D), B).

Theorem 16.2 (importation).If I = 〈S, ⇒〉 is an implication structure, and A, B, and D are any members of S, then H(A, H(B, D)) ⇒ H(C(A, B), D) [that is, (A → (B → D)) ⇒ ((A & B) → D)].

Proof. C(A, B), H(A, H(B, D)) ⇒ H(B, D), since C(A, B), H(A, H(B, D)) ⇒ A, as well as H(A, H(B, D)). Moreover, since C(A, B) ⇒ B, C(A, B), H(A, H(B, D)) ⇒ D. Consequently, by H2, H(A, H(B, D)) ⇒ H(C(A, B), D).

Theorem 16.3.If I = 〈S, ⇒,〉 is an implication structure, and A and B are any members of S, then D(N(A), B) ⇒ H(A, B) [that is, (¬A ∨ B) ⇒ (A → B)].

Proof. B ⇒ H(A, B), and N(A) ⇒ H(A, B). Consequently, by D1D(N(A), B) ⇒ H(A, B)].

Theorem 16.4.It is not true that in every implication structure, for all A and B, H(A, B) ⇒ D(N(A), B) [that is, (A → B) ⇒ (¬A ∨ B)].

Modals as functions

In our characterization of modal operators we stressed the idea that they are certain kinds of functions or mappings on implication structures, I = 〈S, ⇒〉, mapping the set S of the structure to itself and preserving the implication relation “⇒,” but not its dual.

There are several types of questions that arise naturally if we focus on the modals as operators: (1) Given any implication structure I, are there any modals on it? That is, do any modals exist? We shall show that there is a simple necessary and sufficient condition for their existence. (2) If several modal operators exist on a structure, how are they related? For example, will any two, such as φ and φ*, have to be comparable in the sense that either φ(A) ⇒ φ*(A) for all A in S or else φ*(A) ⇒ φ(A) for all A in S. We shall see that the modals on some structures need not be comparable, but that there are interesting consequences when they are. (3) What about more refined descriptions of modals as special kinds of functions? That is, what can one say about those kinds of modals φ for which φ(A) ⇒ A for all A in S? Under what conditions do they exist? Furthermore, if φ is a modal operator, is its functional product with itself, φφ, always a modal operator as well? What can be said about those special modal operators that always map theses of the structure into theses?