Nino Cocchiarella

A semantics for logical necessity, based on Carnap's criterion of adequacy, is given with respect to the ontology of logical atomism. A calculus for sentential (propositional) modal logic is described and shown to be complete with respect to this semantics. The semantics is then modified in terms of a restricted notion of 'all possible worlds' in the interpretation of necessity and shown to yield a completeness theorem for the modal logic S5. Such a restricted notion introduces material content into the meaning of necessity so that, in addition to atomic facts, there are "modal facts" that distinguish one world from another.

A minimal second order modal logic of natural kinds is formulated. Concepts are distinguished from properties and relations in the conceptual-logistic background of the logic through a distinction between free and bound predicate variables. Not all concepts (as indicated by free predicate variables) need have a property or relation corresponding to them (as values of bound predicate variables). Issues pertaining to identity and existence as impredicative concepts are examined and an analysis of mass terms as nominalized predicates for kinds of stuff is proposed. The minimal logic is extendible through a summum genus, an infima species or a partition principle for natural kinds.

Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed in terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many.

Conceptual realism begins with a conceptualist theory of the nexus of predication in our speech and mental acts, a theory that explains the unity of those acts in terms of their referential and predicable aspects. This theory also contains as an integral part an intensional realism based on predicate nominalization and a reflexive abstraction in which the intensional contents of our concepts are “object”-ified, and by which an analysis of predication with intensional verbs can be given. Through a second nominalization of the common names that are part of conceptual realism’s theory of reference (via quantifier phrases), the theory also accounts for both plural reference and predication and mass noun reference and predication. Finally, a separate nexus of predication based on natural kinds and the natural properties and relations nomologically related to those natural kinds, is also an integral part of the framework of conceptual realism.

The framework of conceptual realism provides a logically ideal language within which to reconstruct the medieval terminist logic of the 14th century. The terminist notion of a concept, which shifted from Ockham's early view of a concept as an intentional object to his later view of a concept as a mental act, is reconstructed in this framework in terms of the idea of concepts as unsaturated cognitive structures. Intentional objects are not rejected but are reconstructed as the objectified intensional contents of concepts. Their reconstruction as intensional objects is an essential part of the theory of predication of conceptual realism. It is by means of this theory that we are able to explain how the identity theory of the copula, which was basic to terminist logic, applies to categorical propositions. Reference in conceptual realism is not the same as supposition in terminist logic. Nevertheless, the various "modes" of personal supposition of terminist logic can be explained and justified in terms of this conceptualist theory of reference.

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.

Russell's "new contradiction" about "the totality of propositions" has been connected with a number of modal paradoxes. M. Oksanen has recently shown how these modal paradoxes are resolved in the set theory NFU. Russell's paradox of the totality of propositions was left unexplained, however. We reconstruct Russell's argument and explain how it is resolved in two intensional logics that are equiconsistent with NFU. We also show how different notions of possible worlds are represented in these intensional logics