Nino B. Cocchiarella

Logic and Ontology

A brief review of the historicalrelation between logic and ontologyand of the opposition between the viewsof logic as language and logic as calculusis given. We argue that predication is morefundamental than membership and that differenttheories of predication are based on differenttheories of universals, the three most importantbeing nominalism, conceptualism, and realism.These theories can be formulated as formalontologies, each with its own logic, andcompared with one another in terms of theirrespective explanatory powers. After a briefsurvey of such a comparison, we argue that anextended form of conceptual realism provides themost coherent formal ontology and, as such, canbe used to defend the view of logic as language.

Conceptual Realism as a Formal Ontology

A formal ontology is both a theory of logical form and a metaphysical theory about the ontological structure of the world. What makes it a theory of logical form is that different ontological categories or modes of being are represented in it by different logico-grammatical categories. It is specified in this regard by what might be called an ontological grammar that determines how the expressions of those logico-grammatical categories can be meaningfully combined so as to represent different ontological aspects of the world.

On the primary and secondary semantics of logical necessity

The semantical development of modal logic over the past fifteen years has incorporated a particular model-theoretic artifice which has received little or no critical attention. It is our contention that this artifice introduces, at least within conceptual frameworks typified by logical atomism, a subtle form of descriptive as opposed to merely formal content into the semantics of modal operators. This is particularly noteworthy at least for systems containing operators for the so-called logical modalities, e.g., logical necessity or possibility, or their cognate binary modality, strict implication; for, if any modal operators or connectives had ever been conceptually ordained to represent merely logical or formal operations with no material or descriptive content, it is such as these. Yet, as a result of this model-theoretic artifice, that is precisely what they fail to do. Relative to a given non-empty universe of objects and a set of predicates of arbitrary (finite) addicity (representing the nexuses of atomic or basic states of affairs), the artifice in question concerns allowing modal operators to range (in their semantical clauses) over arbitrary non-empty subsets of the set of UN the possible worlds (models) based upon the given universe of objects and the set of predicates in question. The intuitive and natural interpretation of modal operators for logical modalities, however, is that they range over all the possible worlds (models) of a logical space (as determined by a universe of objects and a set of predicate-nexuses) and not some arbitrary non-empty subset of that totality. The latter interpretation, by allowing the exclusion of some of the worlds (models) of a logical space, imports material conditions into the semantics of modal operators. This exclusion, however appropriate for the representation of non-logical (e.g., causal or temporal) modalities, is quite inappropriate for the representation of what are purported to be merely formal or logical modalities. This model-theoretic artifice of allowing the exclusion of some of the worlds (models) of a logical space goes back to Kripke [5] where the notion of universal validity is used instead of the intuitive and primary notion of logical truth. Later semantical developments, by Kripke and others, retained

Frege's double correlation thesis and quine's set theories NF and ML

There are two fundamentally different notions of a class, which, following tradition, we might call the mathematical and the logical notions, respectively. The logical notion is essentially the notion of a class as the extension of a concept, and, following Frege, we shall assume that a class in this sense “simply has its being in the concept, not in the objects which belong to it” ([8], p. 183) regardless of whether or not concepts themselves differ, as Frege assumed, “only so far as their extensions are different” (ibid., p. 118). The mathematical notion of a class, on the other hand, is essentially the notion of a class as composed of its members, i.e., of a class which has its being in the objects which belong to it. This notion of a class, we claim, is none other than the iterative concept of set or at least that is what it comes to upon analysis. Note that although what accounts for the being of a class under the one notion is not the same as what accounts for the being of a class under the other, nevertheless the axiom of extensionality applies equally to both notions. This means that the axiom of extensionality does not of itself account for the being of a class.’ Of course the logical notion of a class, especially as developed in Frege’s form of logicism, is usually thought to be bankrupt as a result of Russell’s paradox. This assessment, however, is erroneous. In particular, in [5] I have explained how Frege’s view of classes in the logical sense can be reconstructed without paradox by modifying in either of two ways what I there referred to as Frege’s double correlation thesis. The two systems that result from these modifications, it turns out, have certain structural similarities with Quine’s two set theories NF and ML, especially when the latter are themselves modified so as to include urelements other than the empty set. This is significant because both NF and ML are commonly said to “lack a motivation” (cf. [ 11, p. 2 19). But this is because as theories of sets in the sense of classes which are composed of their members, which is really the only sense to which Quine is willing to commit himself, both NF and ML are incompatible with the iterative concept of set. As theories of classes in the logical sense, however, and in particular of the classes which Frege

The development of the theory of logical types and the notion of a logical subject in Russell's early philosophy

The development of the theory of logical types in Russells early philosophy proceeds along a difficult and rather involuted path; and even the final product, the theory as adumbrated in [PM], remains unclear as to its syntax and problematic as to its semantics. Indeed, one might well be left with the impression that Russell himself, in the end, remained unsure of which parts of the different views he had held along the way are finally to be adopted. In what follows, we shall attempt to describe the development of Russells early views, at least to the extent to which they are available in published form today, from the perspective of the development in those views of the notion of a logical subject. It is the development of this notion in Russells early philosophy, we believe, which holds the key to many of the problems confronting Russell in the development of his theory of logical types and which led to the various, and sometimes conflicting, proposals that he made along the way. It should be noted here, however, that in referring to the development of the theory of logical types in Russells early philosophy we have in mind only the views developed by Russell up to, but not subsequent to, the 1910-1913 publication of the first edition of [PM]. The subsequent views developed by Russell from 1913-1925, i.e., between the first and second editions of [PM], and summarized to some extent in his introduction (and added appendices) to the second edition, fall in what might best be called Russells middle philosophy. We shall not examine these views here both because of the limitations of space and because of the complexity of the issues involved. It is our position, moreover, that the views adumbrated in Russells middle philosophy do not accord well with the theory of logical types (of third or higher order).

Properties as Individuals in Formal Ontology

Russells supposed paradox of predication has occasionally been cited as a source for lessons in ontology. So, for example, Grossmann in [6] has argued that one of the lessons of Russells paradox is that there are no complex properties. A recent reevaluation of the supposed paradox, however, has led me to the conclusion that there is no paradox (cf. [3]). And of course where there is no paradox, there are no lessons of paradox. There may, however, be lessons of non-paradox, especially if instead of contradiction what results is a highly instructive ontological oddity. In what follows I shall briefly review the considerations that led me to conclude that there is no paradox but instead only this ontological oddity with instructive lessons of its own, relative of course to the ontological framework within which it occurs. I shall then briefly consider several ways of responding to this oddity, where each response presupposes an alternative ontological framework relative to which the response accounts for the oddity by either showing it to rest on an ontological error, as with Grossmanns response, or by mitigating its effect through what purports to be a deeper or wider framework than the original one in which the oddity occurs.

Philosophical Perspectives on Formal Theories of Predication

Predication has been a central, if not the central, issue in philosophy since at least the time of Plato and Aristotle. Different theories of predication have in fact been the basis of a number of philosophical controversies in both metaphysics and epistemology, not the least of which is the problem of universals. In what follows we shall be concerned with what traditionally have been the three most important types of theories of universals, namely, nominalism, conceptualism, and realism, and with the theories of predication which these theories might be said to determine or characterize.

Logical atomism, nominalism, and modal logic

universals. 15 Nominalism, however, or so it would seem, cannot resort to this criterion since nexuses are supposedly not part of the world and cannot therefore be quantified over for the determination or representation of their number. Instead, as metalinguisticaUy distinguishable syncategorematic ties of a formal ontology nexuses would seem to be limited in number, as would all the syncategorematic or formal elements of the system, so as not to exceed ~loFor this reason in what follows we shall assume that both the number of objects in the world and the number of nexuses as metalinguistically distinguishable syncategorematic ties between these objects do not exceed blo.

Predication versus membership in the distinction between logic as language and logic as calculus

There are two major doctrines regarding the nature of logic today. The first is the view of logic as the laws of valid inference, or logic as calculus. This view began with Aristotles theory of the syllogism, or syllogistic logic, and in time evolved first into Booles algebra of logic and then into quantificational logic. On this view, logic is an abstract calculus capable of various interpretations over domains of varying cardinality. Because these interpretations are given in terms of a set-theoretic semantics where one can vary the universe at will and consider the effect this has on the validity of formulas, this view is sometimes described as the set-theoretic approach to logic (see van Heijenoort 1967, p. 327). The second view of logic does not eschew set-theoretic semantics, it should be noted, and it may in fact utilize such a semantics as a guide in the determination of validity. But to u^e such a semantics as a guide, on this view, is not the same as to take that semantics as an essential characterization of validity. Indeed, unlike the view of logic as calculus, this view of logic rejects the claim that a set-theoretic definition of validity has anything other than an extrinsic significance that may be exploited for certain purposes (such as proving a com pleteness theorem). Instead, on this view, logic has content in its own right and validity is determined by what are called the laws of logic, which may be stated either as principles or as rules. Because one of the goals of this view is a specification of the basic laws of logic from which the others may be derived, this view is sometimes called the axiomatic approach to logic. There is no uniformity among the advocates of either view of logic, incidentally, as to what theory of logical form should be taken as a definitive system of logic. First order logic, for example, might be the favored system for an advocate of either view; but then so might a form of higher order logic as well. Indeed, even the distinction between intensional and extensional logic, as we shall see, fails to mark a clear line between the two views. A set-theoretic semantics for intensional logic is also called possible worlds semantics. We shall in

Knowledge representation in conceptual realism

Abstract Knowledge representation in Artificial Intelligence (AI) involves more than the representation of a large number of facts or beliefs regarding a given domain, i.e. more than a mere listing of those facts or beliefs as data structures. It may involve, for example, an account of the way the properties and relations that are known or believed to hold of the objects in that domain are organized into a theoretical whole—such as the way different branches of mathematics, or of physics and chemistry, or of biology and psychology, etc., are organized, and even the way different parts of our commonsense knowledge or beliefs about the world can be organized. But different theoretical accounts will apply to different domains, and one of the questions that arises here is whether or not there are categorial principles of representation and organization that apply across all domains regardless of the specific nature of the objects in those domains. If there are such principles, then they can serve as a basis for a general framework of knowledge representation independently of its application to particular domains. In what follows I will give a brief outline of some of the categorial structures of conceptual realism as a formal ontology. It is this system that I propose we adopt as the basis of a categorial framework for knowledge representation.