# A realist theory of categories an essay on ontology

Annotated Bibliography of His Writings: Cocchiarella proved the first completeness theorems in tense logic and second-order modal logic. He was the first to develop several second-order logics with nominalized predicates as abstract singular terms and then to use those systems in a consistent logical reconstruction of both Frege's and Russell's early logics and in the application of those reconstructions to the semantic analysis of natural language.

This work also led to Cocchiarella's development of formal theories of predication and comparative formal ontology, including especially logical reconstructions of nominalism, conceptualism, logical realism, and the logic of natural kinds.

Cocchiarella also showed how logical atomism is compatible with logical necessity as a modality, and that it is the only ontology in which logical necessity, as opposed to other kinds of modalities, makes sense. Cocchiarella's own preferred ontological framework is conceptual realism, which he has been formally developing for many years, and which contains a logic of both actualism and possibilism in terms of a distinction between concepts that entail concrete existence and those that do not.

It also contains a logic of classes as many as plural objects, which is the basis of Cocchiarella's semantics for plurals and mass nouns in natural language, and in which the Leonard-Goodman calculus of individuals and therefore Lesniewski's mereology as well is reducible. Cocchiarella has also shown that Lesniewski's ontology, which is also called a logic of names, is reducible to his theory of reference in conceptual realism, and that the medieval supposition theories of Ockham, Buridan, and other medieval logicians can be logically reconstructed in terms of this theory of reference.

Cocchiarella is currently continuing his work on different subsystems of conceptual realism, including in particular a logic of events as truth-makers.

Cocchiarella has taught introductory, intermediate, and advanced courses in logic, semantics, set theory and Montague Grammar, as well as seminars on some of the most recent areas of research in logic. He has placed an emphasis in his teaching on the logical analysis of natural language and the ontological interpretations of both scientific and mathematical language.

Cocchiarella sees logic as a powerful tool for the analysis of our scientific theories and the structures that underlie natural language and our commonsense understanding of the world. The study of logical categories in particular provides an important way to study the semantic and ontological categories underlying our scientific and commonsense world views.

Gabbay, John Woods eds. Who's Who in Logic, London: College Publications 2009, pp. FORMAL ONTOLOGY "Formal ontology is the result of combining the intuitive, informal method of classical ontology with the formal, mathematical method of modern symbolic logic, and ultimately identifying them as different aspects of one and the same science.

That is, where the method of ontology is the intuitive study of the fundamental properties. As such, formal ontology is a science prior to all others in which particular forms, modes, or kinds of being are studied. Logic can be distinguished from formal ontology, but only in the sense of logic as an uninterpreted calculus, Le.

A formal system in which logical or syncategorematic constants can be distinguished from non-logical or categorematic constants and in which the axioms and rules are assumed to be logically valid is not an uninterpreted calculus, however, but a logistic system in which logic is a language with content in its own right. The defining characteristic of a logistic system is that it propounds a theory of logical form.

The purely formal or non-descriptive content of the existence of any and all physically real individuals or of the natural properties and relations that such individuals might have in nature, is not independent of the different modes of being of such entities, and in fact presupposes such modes in its very articulation.

### INTRODUCTION

For Husserl, logic has both an apophantic assertional aspect, which he called formal apophantics and which amounts to a theory of logical form as characterized aboveand an ontological aspect, which he called formal ontology.

It is in this way, for example, that propositional forms and their predicative components as generated in the theory of logical forms are transformed into nominal forms that stand for states of affairs and properties and relations, respectively. The important connection of ontology with a logistic system is that the logico-grammatical distinctions made in the latter are based ultimately on a distinction between different modes of being, even if that distinction is initially described in terms of different modes of significance.

More is required by way of comprehensive grasp, however, before a logistic system can be taken as a system of logic or a formal ontology in its fullest sense. In particular, such a logistic system must be rich enough to contain, when suitable non-logical constants, axioms, and meaning postulates regarding such constants are added to it, every scientific theory and the logical analysis of every meaningful declarative sentence of any natural language.

In that case such a logistic system can be taken as a lingua philosophica, or what Leibniz also called a characteristica universalis, and as such it is also none other than a comprehensive system of formal ontology. Beginning with Aristotle, the standard assumption in pre-formal ontology has been that being is not a genus, i. This raises the problem of how the different categories or modes of being fit together, and of whether one of the senses or modes of being is preeminent and the others somehow dependent on that sense or mode of being.

The differential categorial analyses that have been pro-posed as a resolution of this problem have all turned in one way or another on a theory of predication, i. In formal ontology, the resolution of this problem involves the construction of a formal theory of predication. Aristotle's moderate realism regarding species, genera, and universals is a form of natural realism and not of logical realism, and a formal theory of predication constructed as an Aristotelian formal ontology must respect that distinction as well as give an adequate representation of the two ontological con-figurations underlying the Aristotelian analysis of predication.

In particular, such a theory must contain a logic of natural kinds and must impose the constraint of moderate realism that every natural property or relation is instantiated i.

Such a formal ontology, needless to say, will contain a modal logic for natural necessity and possibility, as well as a logic of natural kinds that is to be described in terms of that modal logic.

A Platonist theory of predication in contemporary formal ontology a realist theory of categories an essay on ontology the basis of logical realism where it is assumed that a property or relation exists corresponding to each well-formed predicate expression of logical grammar, regardless of whether or not it is even logically possible that such a property or relation has an instance. Comparative formal ontology, as our remarks have indicated throughout, is the proper domain of many issues and disputes in metaphysics, epistemology, and the methodology of the deductive sciences.

Just as the construction of a particular formal ontology lends clarity and precision to our informal categorial analyses and serves as a guide to our intuitions, so too comparative formal ontology can be developed so as to provide clear and precise criteria by which to judge the adequacy of a particular system of formal ontology and by which we might be guided in our comparison and evaluation of different proposals for such systems.

It is only by constructing and comparing different formal ontologies that we can make a rational decision about which such system we should ourselves ultimately adopt, and that is a decision that can be made only in comparative formal ontology. Nino Cocchiarella, "Ontology II: Philosophia Verlag 1991, pp. On the one hand, it is a section of Mathematics treating of classes, relations, combinations of symbols, etc. On the other hand, it is a science prior to all others, which contains the ideas and principles underlying all sciences" 1 In the former case, mathematical logic is principally, though not only, a calculus ratiocinator.

Under that aspect, beyond consistency, no special heed need be paid a formal system regarding the philosophical significance of its grammatical forms and the viability of the primitive concepts and assumptions expressed by means of these forms.

In the latter case, however, it is quite otherwise. Under this second aspect, mathematical logic, or formal ontology, is concerned with the adequacy of formal systems as alternative formulations of the deepest structural maps of reality. Different metaphysical schools, of course, will be interested in different ways of understanding a formal system as a map of reality.

Conceptualists, for example, would view the grammar of a formal system together with its logistic behaviour as a proposed formal map of the structuring powers of human cognition, a proposed map, that is, of the structure of constructive cognitive processes of the human mind. Operations of the system must then be devised with limitations built into them that reflect in an appropriate manner the limitations of these same constructive powers of the mind.

It is much in this sort of way that the constructivist attitude in the philosophy of mathematics must be understood. Realists, on the other hand, would construe the operations and elements of the formal map as having ontological significance independently of the constructive power of the human mind.

Limitations built into the system, whether they apply to the notion of grammatical well-formedness or to the logistic behaviour of the ontological grammar, are evaluated then on grounds other than the nature of thought and its inherent limitations. Some of the most obvious of such grounds for limitation pertain to the way the implicit metaphysical scheme underlying the system proposes to resolve the known antinomies. Leibniz himself referred to such a formal system as a characteristica universalis.

But it was not until Frege and Peano that any significant attempt at the construction of a formal ontology was made. I mention it in part because it is an ambiguity which Russell apparently shared or perhaps even inherited from Frege in the construction of his own formal ontology, the ramified theory of types, and which he never himself adequately resolved.

Adjoining the axiom of reducibility to this formal ontology, however, can be justified only by taking a realistic attitude, an attitude which Russell clearly accepted in at least some of his writings. Let us note that where Myhill speaks to the question of a coherent philosophy of mathematics, I have referred instead to the problem of a a realist theory of categories an essay on ontology coherent formal ontology.

Certainly, since the problem of the nature of mathematical existence is an ontological problem par excellence, every philosophically coherent formal ontology must contain a coherent philosophy of mathematics.

A a realist theory of categories an essay on ontology acquires philosophical import only if its author claims that it is an ideal language Begriffsschrifti. The importance of formally constructing such partial or fragmentary ontologies is not at issue here.

Nor is it being suggested that a fragmentary ontology which defers the question of mathematical existence is for that reason incoherent. For this reason, metaphysical programmes should not long defer the question of containing a coherent philosophy of mathematics.

It is in its way the locus of a number of philosophical issues both in metaphysics and epistemology, not the least of which is the problem of universals. The latter problem, sometimes all too simply put as the question of whether there are universals or not, is especially germane to the notion of predication since a theory of universals is at least in part a semantic theory of predication; and it is just to such a theory that we must turn in any philosophical investigation of the notion of predication.

In doing so, however, we need not assume the truth or superiority of any one theory of universals over another.

Indeed, an appropriate preliminary to any such assumption might well consist of a comparative analysis of some of the different formal theories of predication that can be semantically associated with these different theories of universals: That, in any case, is the principal methodological assumption for the approach to the problem of universals we shall undertake in the present monograph where we will be more concerned with the construction and comparison of the abstract logical systems that may be associated with different theories of universals than with the metaphysical or epistemological issues a realist theory of categories an essay on ontology which they were originally designed.

It is our hope and expectation, however, that these comparative formal analyses will be instrumental toward any philosophical decision as to whether to adopt a given theory of universals or not. We shall retain the core of this notion throughout this essay and assume that whatever else it may be a universal has a predicable nature and that it is this predicable nature which is what constitutes its universality. Nothing follows from that assumption, however, regarding whether a universal is 1 merely a predicate expression nominalism of some language or other; 2 a concept conceptualism in the sense of a socio-biologically based cognitive ability or capacity to identify, collect or classify, and characterize or relate things in various ways; or 3 a real property or relation existing independently of both language and the natural capacity humans have for thought and representation realism.

We propose to take each of these interpretations or theories of universals seriously in what follows at least to the extent that we are able to associate each with a formal theory of predication. Our particular concern in this regard, moreover, will be with the explanation each provides of the predicable nature of universals, i. Our discussion and comparison of nominalism, conceptualism and realism, accordingly, will not deal with the variety of arguments that have been given for or against each of them, but with how each as a theory of universals may be semantically associated with a formal theory of predication.

Our assumption here, as indicated above, is that insofar as such an associated formal theory of predication provides a logically perspicuous medium for the articulation of the predicable nature of universals as understood by the theory of universals in question, then to that extent the formal theory may itself be identified with the explanation which that theory of universals provides of the predicable nature of universals.

It is in the sense of this assumption, moreover, that we understand a philosophical theory of predication to be a formal theory of predication together with its semantically associated theory of universals. Some forms of nominalism, more-over, are even more restrictive in this regard than others. We shall not concern ourselves with these variations here, however, but shall identify generic nominalism with three general semantical theses instead.

## The Conceptual Realism of Nino Cocchiarella

The first general thesis of nominalism is that universals have only a formal mode of existence, i. Predicate expressions, in other words, do not designate any universals beyond themselves; and therefore predicate expressions are the only entities according to nominalism that have a predicable nature.

For this reason, we shall occasionally refer to predicate expressions as nominalistic universals. We do not dispute here, it will be noted, that there are universals. That is, the problem of universals as we understand it here is not the problem whether there are universals; for indeed all theories of universals acknowledge that there are at least nominalistic universals, though some will assert that there are other universals as well.

The problem of universals, as we have already said, is the problem of providing a philosophically coherent explanation of the predicable nature of universals, i. And in nominalism, this problem concerns the sense in which predicate expressions may be predicated of individuals.

The second general thesis of nominalism is the thesis of extensionality, i. This means in particular that only an extensional logic is appropriate to nominalism, a corollary of which is anti-essentialism, i.

It is sometimes claimed, we should note, that only the latter thesis, or a suitable reconstruction of it, is really necessary to nominalism and that in fact a nominalistic formal theory of predication may contain a modal, and therefore non-extensional, logic after all.

We shall evaluate, and reject, this claim at a later section of this chapter. For now, however, we simply assume that nominalism requires the stronger thesis of extensionality.

The third general thesis of nominalism is that there are only individuals in the logical sensei. Whether, in addition, all and only concrete particulars are individuals, as has been maintained in more traditional variants of nominalism, we shall leave unspecified.

The third general thesis, it should be noted, does not follow from the first. Nevertheless, while it is consistent to maintain this, i. Indeed, standard articulation of the predicable nature of universals as understood by the theory of universals in question. We are in almost complete agreement a realist theory of categories an essay on ontology this association of nominalism with standard first-order predicate logic with identity.

Our one reservation concerns the fact that the latter, strictly speaking, occurs properly only as the logical component of applied first-order theories. That is, except for the possible use of dummy schema predicate letters, first- order predicate logic cannot be described as a pure formal theory of predication.