Consider the logical operators when they are relativized to component implication. Let us call these operators component conjunctions, component hypotheticals, and so forth, when the implication relation is componential.

Component-style logical operators

Basically, the general picture of component-style logical operators is this: The logical operator Oc (the component-style version of the logical operator O) acts upon the elements A, B, C, …, all of which belong to the same component of the implication structure, and assigns those elements of that component that the operator O assigns – if it assigns anything at all. If A, B, C, … do not all belong to the same component, then there is no member of S that Oc assigns to them. In other words, the component-style logical operators stay in the component of the elements that it acts upon; they are not component-hopping. Later we shall see what the situation is for each of the operators when they act on elements of S that may not all be in the same component.

Component-style conjunctions

In our preliminary (unparameterized) description of the conjunction operator, C(A, B) was taken to be the weakest member of the structure that implied A as well as B.

Constant features of varied structures

Throughout the preceding chapters we have, from time to time, remarked that certain implications, A1, …, An ⇒ B, do not hold in all implication structures, and that other implications do. For example, we noted that in any structure I = 〈S, ⇒〉 in which NN(A) exists for A in S, one always has A ⇒ NN(A). However, there are structures that have members that are not implied by their double negation. We also noted that there are certain quantificational implications, such as Ui(P) ⇒*N*(EiN*(P)) [Theorem 20.26, the counterpart of “ ‘(∀x)Px → ¬(∃x)¬Px’ is a theorem”], that hold in all extended implication structures in which the relevant operators have values. On the other hand, we saw that N*(EiN*(P)) ⇒*Ui(P) [the counterpart of “ ‘¬(∃x)¬Px ⇒ (∀x)Px’ is a theorem”] does not hold in all extended implication structures (Exercise 20.5).

Thus, some results hold in all implication structures (when the relevant negations, hypothetical, and quantifications exist), and some do not. We noted in passing that, in general, it was the “intuitionistic” results that held in all implication structures, and it was the “classical” ones that held in all the classical structures.

We shall now provide a more precise description of this situation and sketch a proof of it. We do have to be careful in the description of what holds in all structures.

It will prove useful in this study to split the operators into two groups: the hypothetical, negation, conjunction, and disjunction in one group; quantification and identity in the other. In studying the features of these operators it will be convenient as well to begin with a simple characterization of each of them. Later (Chapter 15) we shall describe the official, parameterized forms for them. This division into the simple and the parameterized forms segregates the features of the logical operators in a way that makes for a more perspicuous picture, but separates the features in a way that has its own theoretical interest. We begin with the simple, unparameterized story.

The simple characterization

If I = 〈S, ⇒〉 is an implication structure, then the hypothetical operator H (on I) is the function H⇒ that assigns to each pair 〈A, B〉 of members of S a special subset H(A, B) of S. The special character of the subset is given by a condition that characterizes the function H– just as the special characters of the other logical operators are brought out by certain conditions that characterize them in turn. In fact, as we shall see, all the members (if there are any) of H(A, B) are equivalent to each other with respect to the implication relation “⇒” of S, so that often we shall simply refer to the hypothetical H(A, B) as if it were a member of S rather than a subset of S.

Another program for understanding the logical operators is due to N. D. Belnap, Jr. It is similar to that of Gentzen and our own, in that it involves a division of labor: a background theory of implication that is structural, and a theory of the operators characterized against that background.

In a witty and elegant essay, Belnap developed a view about the significance of the logical connectives that stressed the role of the connectives with respect to inference. Belnap's theory clearly has the Gentzen style: A general theory of inference (which he calls “deducibility”) is set forth. As he remarked, it makes no difference whether one chooses to use a syntactic notion of deducibility or a semantic concept of logical consequence to represent what he intended by “inference.” Against the theory of inference, a theory of the logical connectives is developed as an extension of the theory of inference. The meaning of a connective like “and” is, according to Belnap, given by its role in inference. Although the strategy of explanation is Gentzenesque, the resultant theory is somewhat different in detail from Gentzen's, and very different from our own use of the Gentzen framework.

Belnap's remarks were designed to answer a probing challenge by A. N. Prior to the claim that “the complete answer” to the question of what is the meaning or the definition of the logical particle “and” could be given by describing the role that “and” plays in a class of inferences that Prior called “analytic.”

In the preceding chapter we introduced the notion of an extended implication structure 〈E, Pr, S, ⇒〉 in order to study predication. The idea was to show how an account of predicates could be developed within the framework that we adopted for the logical operators. Once an account of predicates was in place, we then characterized the universal and existential quantifiers within the same framework. Predicates, it will be recalled, were taken to be mappings of E* to S, where E* consists of all the infinite sequences of the members of E. The theory of quantification was developed without singling out any predicates in particular.

There is one particular predicate, however, that is worth isolating for further study. The identity predicate has traditionally been regarded as a special predicate of logical theory, although it has sometimes been questioned that it is properly part of “logic.” In any case, the identity relation or predicate usually is regarded as a predicate of two arguments that has a characteristic connection with the predicates of the structure under study. We shall see that in the characterization we give for it, it has strong affinities with the characterization of the other logical operators. And it will be shown how its reflexivity, symmetry, and transitivity follow from that characterization. There is, in addition, a traditional connection, due principally to Leibniz, between the identity of e and e′ and the sharing of all the one-argument predicates of a structure (those whose support has only one element).

We have been considering various kinds of operators, some familiar and some foreign, as examples of modal operators. We now wish to consider the possible modal character of the concepts of knowledge and truth, to take two central philosophical notions. Our aim is not to establish either that they have or that they fail to have modal character. Our purpose is to reinforce the idea that although many familiar concepts are modal on our account, the claim that a particular concept is modal can be a strong one that may not be consonant with what we believe to hold for those concepts. Knowledge and truth provide a case in point. The claim for their modality places some very strong constraints upon them. These requirements go beyond the simple matter of attending to certain formal analogies with necessity or possibility operators of familiar modal systems. That alone will not settle the matter of the modal character of a concept.

Knowledge

In the few passing remarks we made in Sections 25.1 and 28.3 concerning knowledge, we were concerned with eliciting the conditions under which the operator “K” [K(A): “It is known that A”] is modal. Whether or not these conditions hold is not at issue. Similar remarks hold for the truth operator “T” [T(A): “It is true that A”], which we shall study in the following section.

In order that K should count as a modal operator, we have to specify an implication structure I = 〈S, ⇒〉, where S is a set of statements (or whatever it is that one knows), and K maps S to S.

The background theory against which the various logical operators are characterized requires that implications be relations on sets. These implication structures of the theory are at the heart of our account of the logical operators as well as the modal operators (Part IV). It is integral to our theory of the logical operators that any two implication relations that extend each other (are coextensional) will result in logical operators that are equivalent by either implication relation.

Recently, Peacocke (1981) has objected to the extensional treatment of deducibility relations in Hacking's account (1979) of the logical connectives. On Hacking's view, it makes no difference which of two coextensional deducibility relations is used to account for the logical connectives. Hacking's account is the target of Peacocke's objection, and the substance of the objection easily carries over to our own. In substance, Peacocke claims to have produced a case of two coextensional implication relations, such that conjunction relative to one is just the standard “&,” whereas conjunction with respect to the other is some nonstandard, partially defined connective.

Peacocke's remedy is this: Not all coextensional implication relations are on a par for a theory of the logical connectives. Instead of requiring that implication relations (he calls them “deducibility relations”) satisfy the conditions of Reflexivity, Dilution, and Transitivity, he requires instead that it is a priori that the relation satisfies those conditions (Peacocke, 1976, 1981).

Thus far we have suggested that our account of the logical operators is, in broad outline, a two-component theory. One component is a theory of implication structures consisting of a number of conditions that, taken together, characterize implication relations. Implication structures, which the first component studies, are sets together with implication relations defined over them. The second component consists in descriptions of the various logical operators with the aid of implication relations. The operators are taken to be functions that to any given implication structure assign members (or sets of members) of that structure.

There are two other programs that may be familiar to the reader, that of G. Gentzen and P. Hertz, on the one hand, and that of N. Belnap, Jr., on the other. Each of them, however, is different in scope and aim from the program described here.

The Gentzen–Hertz program

The allusion to a functional account of the logical operators, to their roles with respect to implication relations, will strike a familiar note for those who have studied the logical insights of Gentzen and Hertz. It was Gentzen (1933) who developed and extended the work of his teacher Hertz (1922, 1923, 1929) and adumbrated the view that the logical connectives were to be understood by their roles in inference (Gentzen, 1934). The Gentzen–Hertz insight, as it developed, isolated those aspects of inference against which the “meanings” of various logical connectives were to be explained or understood: There was a division of labor.

Two kinds of contraction

According to the view proposed in Chapter 3, one can never justify a deliberate expansion into inconsistency. Respect for the desideratum that error be avoided in changing states of full belief requires this. One can, however, legitimately though inadvertently expand into inconsistency via routine expansion.

Although routine expansion into such inconsistency may be legitimate in the sense that programs for routine expansion that can inject inconsistency are legitimately adopted even when they incur a risk of leading to inconsistency, it is, nonetheless, clear that an inconsistent result calls for contraction. I call such contraction coerced contraction because it is mandated by the inadvertent expansion into inconsistency and the urgency of retreating from inconsistency.

Contraction does not have to be coerced to be legitimate. One may be justified in ceasing to be certain that a doxastic proposition is true even though one is not retreating from inconsistency. Uncoerced contraction may be justified because someone has proposed a conjecture that, given the current state of full belief, is certainly false. The inquiring agent may be justified in incurring the loss of information that must result from such contraction in order to obtain a cognitive benefit. The contraction is not imposed as the remedy for the inadvertent expansion into inconsistency but is deliberately chosen as best for the purpose of realizing the inquirer's cognitive goals.

Uncoerced contraction is illustrated by situations where investigators are confronted with unexplained anomalies

This book elaborates, modifies, and extends some of the themes of the first three chapters of Enterprise of Knowledge. I was prompted to return to these matters by reading the excellent paper of Carlos Alchourròn, Peter Gärdenfors, and David Makinson, “On the Logic of Theory Change: Partial Meet Functions for Contraction and Revision,” and the substantial literature anticipating and emerging from this important paper. In addition, Gärdenfors presented me with an excellent target of criticism. Thanks to his book Knowledge in Flux, we have available a superb introduction to the contemporary discussion of belief change from the best-known student of the subject. To a very considerable degree, my book focuses on the points of disagreement between Gärdenfors and myself. I want to emphasize at the outset, therefore, my profound admiration for the quality of his work and the work of his colleagues Alchourròn and Makinson, and my gratitude to Gärdenfors for his collegial friendship and his generous assistance to me in writing this book. He has written extensive notes on all of the chapters, and I have relied on them heavily in preparing the final version.

As time goes by, I become more indebted to my former student, current colleague, and good friend, Teddy Seidenfeld. He sent me detailed comments on the first draft that have helped me clarify many obscurities and improve the final version.

While writing Chapter 2, I found enlightenment and stimulation in speaking with Akeel Bilgrami, Arnold Koslow, and Shaughan Lavine.

As inquiring agents modify their evolving doctrines, they come to believe where they were initially doubtful. Conjectures or hypotheses are thereby converted into settled assumptions free from serious doubt and, therefore, counted as certainly true. Such erstwhile conjectures are shifted to the status of evidence or knowledge and are deployed as premises in subsequent inquiries (pending future reconsideration).

Inquiries that terminate with the settling of an issue are provoked by the presence of doubt. To be sure, the presence of doubt does not automatically induce the doubter to engage in inquiry. We attach greater urgency to the solution of some problems than to the solution of others, and often disagree, and disagree intensely, as to the priority to be attached to various unsettled issues. Even so, when agents seek to justify the termination of inquiry by adding to the body of settled assumptions some proposed solution to the problem under investigation, their effort at justification is predicated on a distinction between those propositions taken for granted as settled and beyond reasonable doubt, and others that are not regarded as certainly true but as conjectural, more or less probable or improbable and, hence, as both possibly true and possibly false. And, given that distinction, the effort to justify the conversion of a conjecture into a certainty is an effort to justify a change in the set of assumptions accepted as evidence and as certainly true and, with this alteration, to institute a change in the way in which truth value-bearing claims are separated into those whose truth is seriously possible and those whose truth is not seriously possible.

Commitment and performance

My subject is changing states of full belief. Preliminary to that discussion, it is desirable to make some general remarks about the respects in which changes of states of full belief are under study.

I begin with the assumption that a useful distinction may be made between two kinds of change in full belief:

1. Changes in doxastic commitment.

2. Changes in doxastic performance.

At time t1, X fully believes that Albany is north of New York and that being north of is a transitive relation.

At time t2, X comes to believe fully, in addition to what he believes fully at t1, that Montreal is north of Albany.

At time t3 X comes to believe fully that Montreal is north of New York.

I consider the transition from t1 to t2 to be a change in doxastic commitment, whereas the change from t2 to t3 is a change in doxastic performance without a change in doxastic commitment. Att2, X is already committed to full conviction that Montreal is north of New York. But prior to t3, X does not recognize that his beliefs that Albany is north of New York, that Montreal is north of Albany, and that being north of is transitive commit him to believe that Montreal is north of New York. At t he recognizes his commitment and fulfills the obligation it entails.

In this volume as elsewhere, my attention is focused on changes in doxastic commitment (Levi, 1980s, ch. I, sect.5).

Routine and deliberate expansion

According to an influential tradition in epistemology that includes David Hume and W. V. Quine among its proponents, beliefs are either responses to stimuli or dispositions to such responses. As such they are not, in general, under the control of the believer. Inquirers cannot choose what to believe. Hence, suggesting that changes in belief ought to be evaluated according to principles of rational choice relative to appropriate cognitive goals rests on a misguided voluntarism. Epistemology should be engaged in explaining such responses to dispositions rather than in prescribing norms for choosing beliefs.

In Chapter 2, I indicated that I shall not be focusing on changes of beliefs construed as responses to stimuli or as dispositions to such responses. Such beliefs are doxastic performances that may or may not fulfill doxastic commitments and are not fully under the control of deliberating agents. The topic of this book is revision of doxastic commitments or undertakings. I contend that agents are able to choose how to revise their doxastic commitments and, in this sense, can deliberately change their beliefs.

Nonetheless, we should not conclude that agents always choose or ought to choose directly the changes in belief states (construed as doxastic commitments) they institute, even if they are able to do so. This is especially true in the case of expansions. Expansion may take place in one of two ways: deliberate and routine. In deliberate expansion, the inquirer chooses one of several expansions of his initial doctrine.