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Theory and History of Ontology by Raul Corazzon | e-mail: rc@ontology.co

 

Complete and Annotated Bibliography of Nino Cocchiarella 1986 - 1990

ESSAYS

  1. Cocchiarella, Nino. 1986. "Frege, Russell and Logicism: A Logical Reconstruction." In Frege Synthesized: Essays on the Philosophical and Foundational Work of Gottlob Frege, edited by Haaparanta, Leila and Hintikka, Jaakko, 197-252. Dordrecht: Reidel.

    Reprinted as Chapter 2 in Logical Studies in Early Analytic Philosophy, pp. 64-118.

    "Logicism by the end of the nineteenth century was a philosophical doctrine whose time had come, and it is Gottlob Frege to whom we owe its arrival. “Often,” Frege once wrote, “it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, in stripping off the irrelevant accretions which veil it from the eyes of the mind” (Frege, The Foundations of Arithmetic, [Fd], xix). Prior to Frege logicism was just such a concept whose pure form was obscured by irrelevant accretions; and in his life’s work it was Frege who first presented this concept to humanity in its pure form and developed it as a doctrine of the first rank.

    That form, unfortunately, has become obscured once again. For today, as we approach the end of the twentieth century, logicism, as a philosophical doctrine, is said to be dead, and even worse, to be impossible. Frege’s logicism, or the specific presentation he gave of it in Die Grundgesetze der Arithmetik, ([Gg]), fell to Russell’s paradox, and, we are told, it cannot be resurrected. Russell’s own subsequent form of logicism presented in [PM], moreover, in effect gives up the doctrine; for in overcoming his paradox, Russell was unable to reduce classical mathematics to logic without making at least two assumptions that are not logically true; namely, his assumption of the axiom of reducibility and his assumption of an axiom of infinity regarding the existence of infinitely many concrete or nonabstract individuals.

    Contrary to popular opinion, however, logicism is not dead beyond redemption; that is, if logicism is dead, then it can be easily resurrected. This is not to say that as philosophical doctrines go logicism is true, but only that it can be logically reconstructed and defended or advocated in essentially the same philosophical context in which it was originally formulated. This is true especially of Frege’s form of logicism, as we shall see, and in fact, by turning to his correspondence with Russell and his discussion of Russell’s paradox, we are able to formulate not only one but two alternative reconstructions of his form of logicism, both of which are consistent (relative to weak Zermelo set theory).

    In regard to Russell’s form of logicism, on the other hand, our resurrection will not apply directly to the form he adopted in [PM] but rather to the form he was implicitly advocating in his correspondence with Frege shortly after the completion of [POM]. In this regard, though we shall have occasion to refer to certain features of his later form of logicism, especially in our concluding section where a counterpart to the axiom of reducibility comes into the picture, it is Russell’s early form of logicism that we shall reconstruct and be concerned with here.

    Though Frege’s and Russell’s early form of logicism are not the same, incidentally, they are closely related; and one of our goals will be to reconstruct or resurrect these forms with their similarity in mind. In particular, it is our contention that both are to be reconstructed as second order predicate logics in which nominalized predicates are allowed to occur as abstract singular terms. Their important differences, as we shall see, will then consist in the sort of object each takes nominalized predicates to denote and in whether the theory of predication upon which the laws of logic are to be based is to be extensional or intensional." (pp. 64-65 of the reprint)

    References

    Frege, Gottlob, [Fd] The Foundations of Arithmetic, trans, by J. L. Austin, Harper & Bros., N.Y. 1960.

    Frege, Gottlob, [Gg] Die Grundgesetze der Arithmetik, vols. 1 and 2, Hildesheim, 1962.

    Russell, Bertrand, [PM] Principia Mathematica, coauthor, A. N. Whitehead, Cambridge University Press, 1913.

    Russell, Bertrand, [POM] The Principles of Mathematics, 2nd ed., W. W. Norton & Co., N.Y., 1937.

  2. ———. 1986. "Conceptualism, Ramified Logic, and Nominalized Predicates." Topoi.An International Review of Philosophy no. 5:75-87.

    "The problem of universals as the problem of what predicates stand for in meaningful assertions is discussed in contemporary philosophy mainly in terms of the opposing theories of nominalism and logical realism. Conceptualism, when it is mentioned, is usually identified with intuitionism, which is not a theory of predication but a theory of the activity of constructing proofs in mathematics. Both intuitionism and conceptualism are concerned with the notion of a mental construction, to be sure, and both maintain that there can only be a potentially infinite number of such constructions. But whereas the focus of concern in intuitionism is with the construction of proofs, in conceptualism our concern is with the construction of concepts. This difference sets the two frameworks apart and in pursuit of different goals, and in fact it is not at all clear how the notion of a mental construction in the one framework is related to that in the other. This is especially true insofar as mathematical objects, according to intuitionism, are nothing but mental constructions, whereas in conceptualism concepts are anything but objects. In any case, whatever the relation between the two, our concern in this paper is with conceptualism as a philosophical theory of predication and not with intuitionism as a philosophy of mathematics.

    Now conceptualism differs from nominalism insofar as it posits universals, namely, concepts, as the semantic grounds for the correct or incorrect application of predicate expressions. Conceptualism differs from logical realism, on the other hand, insofar as the universals it posits are not assumed to exist independently of the human capacity for thought and representation. Concepts, in other words, are neither predicate expressions nor independently real properties and relations. But then, at least for the kind of conceptualism we have in mind here, neither are they mental images or ideas in the sense of particular mental occurrences. That is, concepts are not objects (saturated individuals) but are rather cognitive capacities, or cognitive structures otherwise based upon such capacities, to identify and classify or characterize and relate objects in various ways. Concepts, in other words, are intersubjectively realizable cognitive abilities which may be exercized by different persons at the same time as well as by the same person at different times. And it is for this reason that we speak of concepts as objective universals, even though they are not independently real properties and relations.

    As cognitive structures, however, concepts in the sense intended here are not Fregean concepts (which for Frege are independently real unsaturated functions from objects to truth values). But they may be modeled by the latter (assuming that there are Fregean concepts to begin with) -especially since as cognitive capacities which need not be exercized at any given time (or even ever for that matter), concepts in the sense intended here also have an unsaturated nature corresponding to, albeit different from, the unsaturated nature of Fregean concepts. Thus, in particular, the saturation (or exercise) of a concept in the sense intended here results not in a truth value but a mental act, and, if overtly expressed, a speech act as well. The un-saturatedness of a concept consists in this regard in its non-occurrent or purely dispositional status as a cognitive capacity, and it is the exercise (or saturation) of this capacity as a cognitive structure which informs particular mental acts with a predicable nature (or with a referential nature in the case of concepts corresponding to quantifier expressions)." (pp. 75-76)

  3. ———. 1987. "Rigid Designation." In Encyclopedic Dictionary of Semiotics. Vol. 2, edited by Sebeok, Thomas A., 834. Berlin: Mouton de Gruyter.

  4. ———. 1987. "Russell, Bertrand." In Encyclopedic Dictionary of Semiotics. Vol. 2, edited by Sebeok, Thomas A., 840-841. Berlin: Mouton de Gruyter.

  5. ———. 1988. "Predication Versus Membership in the Distinction between Logic as Language and Logic as Calculus." Synthese no. 75:37-72.

    Contents: 0. Introduction; 1. The problem with a set-theoretic semantics of natural language; 2. Intensional logic as a new theoretical framework for philosophy; 3.The incompleteness of intensional logic when based on membership; 4. Predication versus membership in type theory; 5. Second order predicate logic with nominalized predicates; 6. A set theoretic semantics with predication as fundamental; 7. Concluding remarks.

    "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 Aristotle's theory of the syllogism, or syllogistic logic, and in time evolved first into Boole's 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 ["Logic as Language and Logic as Calculus", Synthese 17,] 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 use 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 completeness 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." (p. 37)

    (...)

    "Concluding Remarks. The account we have given here of the view of logic as language should not be taken as a rejection of the set-theoretical approach or as defense of the metaphysics of possibilist logical realism. Rather, our view is that there are really two types of conceptual framework corresponding to our two doctrines of the nature of logic. The first type of framework is based on membership in the sense of the iterative concept of set; although extensionality is its most natural context (since sets have their being in their members), it may nevertheless be extended to include intensional contexts by way of a theory of senses (as in Montague's sense-denotation intensional logic). The second type of framework is based on predication, and in particular developments it is associated with one or another theory of universals. Extensionality is not the most natural context in this theory, but where it does hold and extensions are posited, the extensions are classes in the logical and not in the mathematical sense.

    Russell's paradox, as we have explained, has no real bearing on set-formation in a theory of membership based on the iterative concept of set, but it does bear directly on concept-formation or the positing or universals in a theory based on predication. As a result, our second type of framework has usually been thought to be incoherent or philosophically bankrupt, leaving us with the set-theoretical approach as, the only viable alternative. This is why so much of analytic philosophy in the 20th Century has been dominated by the set-theoretical approach. Set theory, after all, does seem to serve the purposes of a mathesis universalis.

    What is adequate as a mathesis universalis, however, need not also therefore be adequate as a l ingua philosophica or characteristica universalis. In particular, the set-theoretic approach does not seem to provide a philosophically satisfying semantics for natural language; this is because it is predication and not membership that is fundamental to natural language. An adequate semantics for natural language, in other words, seems to demand a conceptual framework based on predication and not on membership.

    (...)

    We do not maintain, accordingly, that we should give up the set-theoretic approach, especially when dealing with the philosophy and foundations of mathematics, or that only a theory of predication associated with possibilist logical realism will provide an adequate semantics for natural language. In both cases we may find a principle of tolerance, if not outright pluralism, the more appropriate attitude to take." (pp. 69-70)

  6. ———. 1989. "Philosophical Perspectives on Formal Theories of Predication." In Handbook of Philosophical Logic. Vol. 4. Topics in the Philosophy of Language, edited by Gabbay, Dov and Guenthner, Franz, 253-326. Dordrecht: Reidel.

    Contents: 1. Predication and the problem of universal 254; 2. Nominalism 256; 3. A nominalistic semantics for predicative second order logic 261; 4. Nominalism and modal logic 266; 5 . Conceptualism vs . nominalism 270; 6. Constructive conceptualism 273; 7. Ramification of constructive conceptualism 280; 8. Holistic conceptualism 286; 9. Logical realism vs holistic conceptualism 289; 10. Possibilism and actualism in modal logical realism 292; 11. Logical realism and cssentialism 301; 12. Possibilism and actualism within conceptualism 306; 13. Natural realism and conceptualism 313; 14. Aristotelian essentialism and the logic of natural kinds 318; References 325-326.

    "Predication has been a central, if not the central, issue in philosophy since at least the lime 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.

    Though each of these three types of theories of universals may be said to have many variants, we shall ignore their differences here to the extent that they do not characterize different theories of predication. This will apply especially to nominalism where but one formal theory of predication is involved. In both conceptualism and realism, however, the different variants of each type do not all agree and form two distinct subtypes each with its own theory of predication. For this reason we shall distinguish between a constructive and a holistic form of conceptualism on the one hand, and a logical and a natural realism on the other. Constructive conceptualism, as we shall see, has affinities with nominalism with which it is sometimes confused, and holistic conceptualism has affinities with logical realism with which it is also sometimes confused. Both forms of conceptualism may assume some form of natural realism as their causal ground; and natural realism in turn must presuppose some form of conceptualism as its background theory of predication. Both forms of realism may be further divided into their essentialist and non-essentialist variants (and in logical realism even a form of anti-essentialism), and though an essentialist logical realism is sometimes confused with Aristotelian essentialism, the latter is really a form of natural realism with natural kinds as the only essential properties objects can have." (pp. 253-254)

  7. ———. 1989. "Russell's Theory of Logical Types and the Atomistic Hierarchy of Sentences." In Rereading Russell: Essays on Bertrand Russell's Metaphysics and Epistemology, edited by Savage, C.Wade and Anderson, C.Anthony, 41-62. Minneapolis: University of Minnesota Press.

    Reprinted as Chapter 5 in Logical Studies in Early Analytic Philosophy, pp. 193-221.

    "Russell’s philosophical views underwent a number of changes throughout his life, and it is not always well-appreciated that views he held at one time came later to be rejected; nor, similarly, that views he rejected at one time came later to be accepted. It is not well-known, for example, that the theory of logical types Russell described in his later or post-[PM] philosophy is not the same as the theory originally described in [PM] in 1910-13; nor that some of the more important applications that Russell made of the theory at the earlier time cannot be validated or even significantly made in the framework of his later theory. What is somewhat surprising, however, is that Russell himself seems not to have realized that he was describing a new theory of logical types in his later philosophy, and that as a result of the change some of his earlier logical constructions, including especially his construction of the different kinds of numbers, were no longer available to him.

    In the original framework, for example, propositional functions are independently real properties and relations that can themselves have properties and relations of a higher order/type, and all talk of classes, and thereby ultimately of numbers, can be reduced to extensional talk of properties and relations as “single entities,” or what Russell in [POM] had called “logical subjects.” The Platonic reality of classes and numbers was replaced in this way by a more fundamental Platonic reality of propositional functions as properties and relations. In Russell's later philosophy, however, “a propositional function is nothing but an expression. It does not, by itself, represent anything. But it can form part of a sentence which does say something, true or false” (Russell, My Philosophical Development, ([MPD]), 69). Surprisingly. Russell even insists that this was what he meant by a propositional function in [PM]. “Whitehead and I thought of a propositional function as an expression containing an undetermined variable and becoming an ordinary sentence as soon as a value is assigned to the variable: ‘x is human’, for example, becomes an ordinary sentence as soon as we substitute a proper name for V. In this view . . . the propositional function is a method of making a bundle of such sentences” ([MPD], 124). Russell does realize that some sort of change has come about, however, for he admits, “I no longer think that the laws of logic are laws of things; on the contrary, I now regard them as purely linguistic” (ibid., 102).

    (...)

    Now it is not whether [PM] can sustain a nominalistic interpretation that is our concern in this essay, as we have said, but rather how it is that Russell came to be committed in his later philosophy to the atomistic hierarchy and the nominalistic interpretation of propositional functions as expressions generated in a ramified second order hierarchy of languages based on the atomistic hierarchy. We shall pursue this question by beginning with a discussion of the difference between Russell’s 1908 theory of types and that presented in [PM] in 1910. This will be followed by a brief summary of the ontology that Russell took to be implicit in [PM], and that he described in various publications between 1910 and 1913. The central notion in this initial discussion is what Russell in his early philosophy called the notion of a logical subject, or equivalently that of a “term” or “single entity”. (In [PM], this notion was redescribed as the systematically ambiguous notion of an “object.”) As explained in chapter 1 this notion provides the key to the various problems that led Russell in his early philosophy to the development of his different theories of types, including that presented in [PM]. This remains true, moreover, even when we turn to Russell’s later philosophy, i.e., to his post-[PM] views, only then it is described as the notion of what can and cannot be named in a logically perfect language. The ontology of these later views is what Russell called logical atomism, and it is this ontology that determines what Russell described as the atomistic hierarchy of sentences. In other words, it is the notion of what can and cannot be named in the atomistic hierarchy that explains how Russell, however unwittingly, came to replace his earlier theory of logical types by the theory underlying the atomistic hierarchy of sentences as the basis of a logically perfect language." (pp. 193-195 of the reprint)

    References

    POM] Russell, Bertrand, The Principles of Mathematics, 2d ed. (NY., Norton & Co., 1938).

    [PM] Russell, Bertrand and Alfred Whitehead, Principia Mathematica, vol. 1 (1910), vol. 2 (1912), and vol. 3 (1913) (London: Cambridge Univ. Press,).

  8. ———. 1989. "Conceptualism, Realism and Intensional Logic." Topoi.An International Review of Philosophy no. 7:15-34.

    Contents: 0. Introduction 15; 1. A conceptual analysis of predication 16; 2. Concept-correlates and Frege's double correlations thesis 17; 3. Russell's paradox in conceptual realism 18: 4. What are the natural numbers and where do they come from? 22; 5. Referential concepts and quantifier phrases 24; 5. Singular reference 24; 7. The intensions of refrential concepts as components of applied predicable concepts 26; 8. Intensional versus extensional predicable concepts 28; 9. The intentional identity of intensional objects 29; Notes 31; Reference 33-35.

    "0. Introduction

    Linguists and philosophers are sometimes at odds in the semantical analysis of language. This is because linguists tend to assume that language must be semantically analyzed in terms of mental constructs, whereas philosophers tend to assume that only a platonic realm of intensional entities will suffice. The problem for the linguist in this conflict is how to explain the apparent realist posits we seem to be committed to in our use of language, and in particular in our use of infinitives, gerunds and other forms of nominalized predicates. The problem for the philosopher is the old and familiar one of how we can have knowledge of independently real abstract entities if all knowledge must ultimately be grounded in psychological states and processes. In the case of numbers, for example, this is the problem of how mathematical knowledge is possible. In the case of the intensional entities assumed in the semantical analysis of language, it is the problem of how knowledge of even our own native language is possible, and in particular of how we can think and talk to one another in all the ways that language makes possible.

    I believe that the most natural framework in which this conflict is to be resolved and which is to serve as the semantical basis of natural language is an intensional logic that is based upon a conceptual analysis of predication in which what a predicate stands for in its role as a predicate is distinguished from what its nominalization denotes in its role as a singular term. Predicates in such a framework stand for concepts as cognitive capacities to characterize and relate objects in various ways, i.e. for dispositional cognitive structures that do not themselves have an individual nature, and which therefore cannot be the objects denoted by predicate nominalizations as abstract singular terms. The objects purportedly denoted by nominalized predicates, on the other hand, are intensional entities, e.g. properties and relations (and propositions in the case of zero-place predicates), which have their own abstract form of individuality, which, though real, is posited only through the concepts that predicates stand for in their role as predicates. That is, intensional objects are represented in this logic as concept-correlates, where the correlation is based on a logical projection of the content of the concepts whose correlates they are.

    (...)

    Before proceeding, however, there is an important distinction regarding the notion of a logical form that needs to be made when joining conceptualism and realism in this way. This is that logical forms can be perspicuous in either of two senses, one stronger than the other. The first is the usual sense that applies to all theories of logical form, conceptualist or otherwise; namely, that logical forms are perspicuous in the way they specify the truth conditions of assertions in terms of the recursive operations of logical syntax. In this sense, fully applied logical forms are said to be semantic structures in their own right. In the second and stronger sense, logical forms may be perspicuous not only in the way they specify the truth conditions of an assertion, but in the way they specify the cognitive structure of that assertion as well. To be perspicuous in this sense, a logical form must provide an appropriate representation of both the referential and the predicable concepts that underlie an assertion.

    Our basic hypothesis in this regard will be that every basic assertion is the result of applying just one referential concept and one predicable concept, and that such an applied predicable concept is always fully intensionalized (in a sense to be explained). This will place certain constraints on the conditions for when a complex predicate expression is perspicuous in the stronger sense — such as that a referential expression can occur in such a predicate expression only in its nominalized form. (A similar constraint will also apply to a defining or restricting relative clause of a referential expression.) In the cases where a relational predicable concept is applied, the assumption that there is still but one referential concept involved leads to the notion of a conjunctive referential concept, a notion that requires the introduction in intensional logic of special quantifiers that bind more than one individual variable. Except for briefly noting the need for their development, we shall not deal with conjunctive quantifiers in this essay." (pp. 15-16)

  9. ———. 1991. "Conceptualism." In Handbook of Metaphysics and Ontology, edited by Smith, Barry and Burkhardt, Hans, 168-174. Munich: Philosophia Verlag.

    Conceptualism is one of the three types of theories regarding the nature of universals described by Porphyry in his introduction to Aristotle's Categories. The other two are nominalism and realism. Because a universal, according to Aristotle, is that which can be predicated of things (De Int. 17a39), the difference between these three types of theories lies in what it is that each takes to be predicable of things. In this regard we should distinguish predication in language from predication in thought, and both from predication in reality, where there is no presumption that one kind of predication precludes the others.

    All three types of theories agree that there is predication in language, in particular that predicates can be predicated of things in the sense of being true or false of them. Nominalism goes further in maintaining that only predicates can be predicated of things, that is, that there are no universals other than the predicate expressions of some language or other. Conceptualism opposes nominalism in this regard and maintains that predicates can be true or false of things only because they stand for concepts, where concepts are the universals that are the basis of predication in thought. Realism also opposes nominalism in maintaining that there are real universals, viz. properties and relations, that are the basis of predication in reality." (p. 168)

    (...)

    "Conceptualism is by no means a monolithic theory, but has many forms, some more restrictive than others, depending on the mechanisms assumed as the basis for concept-formation. None of these forms, in themselves, precludes being combined with a realist theory, whether Aristotelian (as in conceptual natural realism) or Platonist (as in conceptual intensional realism), or both. Some conceptualists, such as Sellars, have made it a point to disassociate conceptualism from any form of realism regarding abstract entities, but that disassociation has nothing to do with conceptualism as a theory about the nature of predication in thought. Conceptualism’s shift in emphasis from metaphysics to psychology, in other words, while important in determining what kind of theory is needed to explain predication in thought, should not be taken as justifying a restrictive form of conceptualism that precludes both a natural and an intensional realism." (p. 174)

  10. ———. 1991. "Logic V: Higher Order Logics." In Handbook of Metaphysics and Ontology, edited by Smith, Barry and Burkhardt, Hans, 466-470. Munich: Philosophia Verlag.

    "Higher-order logic goes beyond first-order logic in allowing quantifiers to reach into the predicate as as well as the subject positions of the logical forms it generates. A second feature, usually excluded in standard formulations of second-order logic, allows nominal-ized forms of predicate expressions (simple or complex) to occur in its logical forms as abstract singular terms. (E.g., ‘Socrates is wise’, in symbols W(s), contains ‘is wise’ as a predicate, whereas ‘Wisdom is a virtue’, in symbols V(W), contains ‘wisdom’ as a nominalized form of that predicate. ‘Being a property is a property’, in symbols P(P), or with λ-abstracts, PλxP(x)), where λχΡ(χ) is read ‘to be an x such that x is a property’, contains both the predicate ‘is a property’ and a nominalized form of that predicate, viz. ‘being a property’. Frege’s well-known example, ‘The concept Horse is not a concept’, contains ‘the concept Horse’ as a nominalized form of the predicate phrase ‘is a horse’.)" (p. 466)

  11. ———. 1991. "Ontology, Fomal." In Handbook of Metaphysics and Ontology, edited by Smith, Barry and Burkhardt, Hans, 640-647. Munich: Philosophia Verlag.

    "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. modes, and aspects of being, or of entities in general, and the method of modern symbolic logic is the rigorous construction of formal, axiomatic systems, formal ontology, the result of combining these two methods, is the systematic, formal, axiomatic development of the logic of all forms and modes of being. As such, formal ontology is a science prior to all others in which particular forms, modes, or kinds of being are studied." (p. 641)

  12. ———. 1991. "Russell, Bertrand." In Handbook of Metaphysics and Ontology, edited by Smith, Barry and Burkhardt, Hans, 796-798. Munich: Philosophia Verlag.

    "Russell held a number of different metaphysical positions throughout his career, with the idea of logic as a logically perfect language being a common theme that ran through each.

    (...)

    "A fundamental notion of Russell’s logical realism, sometimes also called ontological logicism, was that of a propositional function, the extension of which Russell took to be a class as many. Initially, as part of his response to the problem of the One and the Many, Russell had assumed that each propositional function was a single and separate entity over and above the many propositions that were its values, and, similarly, that to each class as many there corresponded a class as one. Upon discovering his paradox, Russell maintained that we must distinguish a class as many from a class as one, and that a class as one might not exist corresponding to a class as many. He also concluded that a propositional function cannot survive analysis after all, but ‘lives’ only in the propositions that are its values, i.e. that propositional functions are nonentities."

    (...)

    "As a result of arguments given by Ludwig Wittgenstein in 1913, Russell, from 1914 on, gave up the Platonistic view that properties and relations could be logical subjects. Predicates were still taken as standing for properties and relations, but only in their role as predicates; i.e., nominalized predicates were no longer allowed as abstract singular terms in Russell’s new version of his logically perfect language. Only particulars could be named in Russell's new metaphysical theory, which he called logical atomism, but which, unlike his earlier 1910-13 theory, is a form of natural realism, and not of logical realism, since now the only real properties and relations of his ontology are the simple material properties and relations that are the components of the atomic facts that make up the world. Complex properties and relations in this framework are simply propositional functions, which, along with propositions, are now merely linguistic expressions. (Russell remained unaware that as a result of the change in his metaphysical views from logical to natural realism his original theory of types was restricted to the much weaker sub-theory of ramified second-order logic, and that he could no longer carry through his logicist programme. This reinforced the confusion of nominalists into thinking that Russell’s earlier theory of types could be given a nominalistic interpretation, since such an interpretation is possible for ramified second-order logic.)" (pp. 797-798)

  13. ———. 1991. "Quantification, Time and Necessity." In Philosophical Applications of Free Logic, edited by Lambert, Karel, 242-256. New York: Oxford University Press.

    Contents: 0. Introduction; 1. A Logic a Actual and Possible Objects; 2. A Completeness Theorem for Tense Logic; 3. Modality Within Tense Logic; 4. Some Observations on Quantifiers in Modal and Tense Logic; 5. Concluding Remarks.

    Abstract: "A logic of actual and possible objects is formulated in which "existence" and "being", as second-level concepts represented by first-order (objectual) quantifiers, are distinguished. A free logic of actual objects is then distinguished as a subsystem of the logic of actual and possible object. Several complete first-order tense logics are then formulated in which temporal versions of possibilism and actualism are characterized in terms of the free logic of actual objects and the wide logic of actual and possible objects. It is then shown how a number of different modal logics can be interpreted within quantified tense logic, with the latter providing a paradigmatic framework in which to distinguish the interplay between quantifiers, tenses and modal operators and within which we can formulate different temporal versions of actualism and possibilism."

    "The fundamental assumption of a logic of actual and possible objects is that the concept of existence is not the same as the concept of being. Thus, even though necessarily whatever exists has being, it is not necessary in such a logic that whatever has being exists; that is, it can be the case that there be something that does not exist. No occult doctrine is needed to explain the distinction between existence and being, for an obvious explanation is already at hand in a framework of tense logic in which being encompasses past, present, and future objects (or even just past and present objects) while existence encompasses only those objects that presently exist. We can interpret modality in such a framework, in other words, whereby it can be true to say that some things do not exist. Indeed, as indicated in Section 3, infinitely many different modal logics can be interpreted in the framework of tense logic. In this regard, we maintain, tense logic provides a paradigmatic framework in which possibilism (i.e., the view that existence is not the same as being, and that therefore there can be some things that do not exist) can be given a logically perspicuous representation.

    Tense logic also provides a paradigmatic framework for actualism as the view that is opposed to possibilism; that is, the view that denies that the concept of existence is different from the concept of being. Indeed, as we understand it here, actualism does not deny that there can be names that have had denotations in the past but that are now denotationless, and hence that the statement that some things do not exist can be true in a semantic metalinguistic sense (as a statement about the denotations, or lack of denotations, of singular terms). What is needed, according to actualism, is not that we should distinguish the concept of existence from the concept of being, but only that we should modify the way that the concept of existence (being) is represented in standard first-order predicate logic (with identity). A first-order logic of existence should allow for the possibility that some of our singular terms might fail to denote an existent object, which, according to actualism, is only to say that those singular terms are denotationless rather than what they denote are objects (beings) that do not exist. Such a logic for actualism amounts to what nowadays is called free logic." (pp. 242-243)

  14. ———. 1992. "Conceptual Realism Versus Quine on Classes and Higher-Order Logic." Synthese no. 90:379-436.

    Contents: 0. Introduction; 1. Predication versus Membership; 2. Old versus New Foundations; 3. Concepts versus ultimate Classes; 4. Frege versus Quine on Higher-Order Logic; 5. Conceptualism versus Nominalism as Formal Theories of predication; 6. Conceptualism Ramified versus Nominalism Ramified; 7. Constructive Conceptual Realism versus Quine's view of Conceptualism as a Ramified Theory of Classes; 8. Holistic Conceptual Realism versus Quine's Class Platonism.

    Abstract: "The problematic features of Quine's 'set' theories NF and ML are a result of his replacing the higher-order predicate logic of type theory by a first-order logic of membership, and can be resolved by returning to a second-order logic of predication with nominalized predicates as abstract singular terms. We adopt a modified Fregean position called conceptual realism in which the concepts (unsaturated cognitive structures) that predicates stand for are distinguished from the extensions (or intensions) that their nominalizations denote as singular terms. We argue against Quine's view that predicate quantifiers can be given a referential interpretation only if the entities predicates stand for on such an interpretation are the same as the classes (assuming extensionality) that nominalized predicates denote as singular terms. Quine's alternative of giving predicate quantifiers only a substitutional interpretation is compared with a constructive version of conceptual realism, which with a logic of nominalized predicates is compared with Quine's description of conceptualism as a ramified theory of classes. We argue against Quine's implicit assumption that conceptualism cannot account for impredicative concept-formation and compare holistic conceptual realism with Quine's class Platonism."

    "According to Quine, in one of his later works, the pioneers in modern logic, such as Frege and Russell, overestimated the kinship between membership and predication and in that way came to view set theory as logic (Quine 1970, p. 65). Such a claim, we maintain, is both false and misleading. Frege and Russell did assume a logical kinship between predication and membership, but what they meant by membership was membership in a class as the extension of a concept (where a concept is a predicable entity, i.e., a universal in the traditional sense) and not membership in a set. Sets, unlike classes, as we have said, have their being in their members, and in that regard there need be no kinship at all between predication and membership in a set. Classes in the logical sense, on the other hand, have their being in the concepts whose extensions they are, which means that any theory of membership in a class presupposes a superseding theory of predication. (3) Frege and Russell did not view set theory as logic, but they each did develop a theory

    of classes and they each did so based on a superseding higher-order theory of predication." (p. 382)

  15. ———. 1992. "Cantor's Power-Set Theorem Versus Frege's Double-Correlation Thesis." History and Philosophy of Logic no. 13:179-201.

    Abstract: "Frege’s thesis that second-level concepts can be correlated with first-level concepts and that the latter can be correlated with their value-ranges is in direct conflict with Cantor’s power-set theorem, which is a necessary part of the iterative, but not of the logical, concept of class. Two consistent second-order logics with nominalised predicates as abstract singular terms are described in which Frege’s thesis and the logical notion of a class are defended and Cantor’s theorem is rejected. Cantor’s theorem is not incompatible with the logical notion of class, however. Two alternative similar kinds of logics are also described in which Cantor’s theorem and the logical notion of a class are retained and Frege’s thesis is rejected."

    "There is another problem with Russell’s solution, however, in addition to that of the relativisation of classes to each logical type. This problem has to do with the fact that the particular theory of types that Russell adopted is a theory of ramified types, which, unlike the theory of simple types, is based on a constructive (i.e. ‘predicative’) comprehension principle. Such a constructive approach is not without merit, but it does affect the logical notion of a class in a fundamental way. In particular, because of the kind of constructive constraints imposed by the theory on the comprehension principle, Cantor’s theorem, which involves objects of different types, cannot be proved in such a framework (cf. Quine 1963, 265). That is not objectionable in itself, but it does not get at the root of the matter of the real conflict between Cantor’s power-set theorem and the logical notion of class as represented by an impredicative comprehension principle.

    An impredicative comprehension principle is provable in the theory of simple types. But in this framework, as in the theory of ramified types as well, Russell’s paradox cannot even be stated (because of the gramatical constraints on the conditions of well-formedness), which means that the description of the class upon which Russell’s paradox is based is meaningless. Thus, not only must the universal class be relativised and duplicated, potentially, infinitely many times in order to avoid Russell’s paradox on this approach, but the paradox must itself be ruled as meaningless. The theory of types, whether simple or ramified, is not really a solution of the problem so much as a way of avoiding it altogether.

    There is another way in which we can preserve our logical intuitions and not give up the logical notion of a class in favor of the mathematical (i.e. in favor of set theory), and yet in which not only is Cantor’s theorem formulable but so is Russell’s paradox—though, of course, the latter will no longer be provable. Indeed, there is not just one such way, but at least two (both of which themselves have two alternatives). On the first, it is not the logical notion of a class that must be rejected as the way of resolving Russell’s paradox, but Cantor’s theorem instead. This rejection is not ad hoc or arbitrary on this approach, but is based on a more general principle, which we refer to as Frege’s double-correlation thesis. It is this approach that we shall turn to first. On the second and alternative approach, which we shall turn to later, the trouble lies in neither Cantor’s theorem nor in the assumption that there is a universal class (both of which can be retained without contradiction on this approach), but rather in how the logic of identity is to be applied in certain contexts. On this approach, the claim that a contradiction results by combining Cantor’s theorem with the assumption that the universal class exists is not a ‘truism’ after all but is outright false."

    References

    Quine, W. V. 1963 Set theory and its logic, Cambridge, Mass. (Harvard University Press).

  16. ———. 1993. "On Classes and Higher-Order Logic: A Critique of W.V.O. Quine." Philosophy and the History of Science.A Taiwanese Journal no. 2:23-50.

    Abstract: "The problematic features of Quine's set theories NF and ML result from compressing the higher-order predicate logic of type theory into a first-order logic of membership, and can be resolved by turning to a second-order predicate logic with nominalized predicates as abstract singular terms. A modified Fregean position, called conceptual realism, is described in which the concepts (unsaturated cognitive structures) that predicates stand for are distinguished from the extensions (or intensions) that their nominalizations denote as abstract singular terms. Quine's view that conceptualism cannot account for impredicative concept-formation is rejected, and a holistic conceptual realism is compared with Quine's class Platonism."

  17. ———. 1995. "Knowledge Representation in Conceptual Realism." International Journal of Human-Computer Studies no. 43:697-721.

    "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." (p. 697)

    (...)

    " Concluding remarks. We have given here only an overview or sketch of conceptual realism as a formal ontology, i.e. as a theory of logical form having both conceptual and ontological categories - but in which the latter are represented in terms of the former. The categories of natural kinds and of natural properties and relations, for example, are represented in terms of the categories of sortals and predicable concepts, respectively, and the category of abstract objects is represented in terms of the process of conceptual nominalization (reification) as a subcategory of objects. Not all of these categories or parts of this formal ontology will be relevant in every domain of knowledge representation, but each is relevant at least to some domains and is needed in a comprehensive framework for knowledge representation. In those domains where certain categorial distinctions are not needed - such as that between predicative and impredicative concepts, or that between predicable concepts and natural properties and relations, or between sortal concepts and natural kinds, etc. - we can simply ignore or delete the logical forms in question. What must remain as the core of the system is the intensional logic around which all of the other categories are built - namely, the second-order predicate logic with nominalized predicates as abstract singular terms that we call HST*-lambda. It is this core, I believe, that can serve as a universal standard by which to evaluate other representational systems." (p. 721)

  18. ———. 1996. "Conceptual Realism as a Formal Ontology." In Formal Ontology, edited by Poli, Roberto and Simons, Peter, 27-60. Dordrecht: Kluwer.

    Contents: 1. Introduction; 2. Substitutional versus Ontological Interpretations of Quantifiers; 3. The Importance of the Notion of Unsaturedness in Formal Ontology; 4. Referential and Predicable Concepts Versus Immanent Objects of Reference; 5. Conceptual Natural Realism and the Analogy of Being Between Natural and Intelligible Universals; 6. Conceptual Natural Realism and Aristotelian Essentialism; 7. Conceptual Intensional Realism versus Conceptual Platonism and the Logic of Nominalized Predicates

    8. Concluding Remarks.

    Abstract: "Conceptualism simpliciter, wheter constructive or holistic, provides an account of predication only in thought and language, and represents in that regard only a truncated formal ontology. But conceptualism can be extended to an Aristotelian conceptual natural realism in which natural properties and relations (and natural kinds as well) can be analogically posited corresponding to some of Our concepts, thereby providing an account of predication in the space-time causal Order as well. In addition, through a pattern of reflexive abstraction corresponding to the process of nominalization in language (and in which abstract objects are hypostatized corresponding to our concepts as unsaturated cognitive structures), conceptualism can also be extended to a conceptual Platonism or intensional realism that can provide an account of both the intensional objects of fiction and the extensional objects of mathematics. Conceptual realism is thus shown to be a paradigm formal ontology in which the distinctions between abstract reality, natural reality, and thought and language are properly represented, and in which the traditional opposition between Platonism and Aristotelianism is finally overcome by properly locating their different functions, and the way each should be rep resented, in formal ontology."

    "Concluding Remarks. As this informal sketch indicates, conceptual realism, by which we mean conceptual natural realism and conceptual intensional realism together, provides the basis of a general conceptual-ontological framework, within which, beginning with thought and language, a comprehensive formal ontology can be developed. Not only does conceptual realism explain how, in naturalistic terms, predication in thought and language is possible, but, in addition, it provides a theory of the nature of predication in reality through an analogical theory of properties and relations. In this way, conceptual realism can be developed into a reconstructed version of Aristotelian realism, including a version of Aristotelian essentialism. In addition, through the process of nominalization, which corresponds to a reflexive abstraction in which we attempt to represent our concepts as if they were objects, conceptualism can be developed into a conceptual intensional realism that can provide an account not only of the abstract reality of numbers and other mathematical objects, but of the intensional objects of fiction and stories of all kinds, both true and false, and including those stories that we systematically develop into theories about the world. In this way, conceptual realism provides a framework not only for the conceptual and natural order, but for the mathematical and intensional order as well. Also, in this way, conceptual realism is able to reconcile and provide a unified account both of Platonism and Aristotelian realism, including Aristotelian essentialism - and it does so by showing how the ontological categories, or modes of being, of each of these ontologies can be explained in terms a conceptualist theory of predication and its analogical extensions." (p. 60)

  19. ———. 1997. "Formally Oriented Work in the Philosophy of Language." In Routledge History of Philosophy. Vol. X - The Philosophy of Meaning, Knowledge and Value in the 20th Century, edited by Canfield, John, 39-75. New York: Routledge.

    Contents: 1. The notion of a Characteristica Universalis as a philosophical language; 2. The notion a a logically perfect language as a regulating ideal; 3. The theory of logical types; 4. Radical empiricism and the logical construction of the world; 5. The logical empiricist theory of meaning; 6. Semiotic and the trinity of syntax, semantics and pragmatics; 7. Pragmatics from a logical point of view; 8. Intensional logic; 9. Universal Montague grammar; 10. Speech-act theory and the return to pragmatics.

    Abstract: "One of the perennial issues in philosophy is the nature of the various relationships between language and reality, language and thought, and language and knowledge. Part of this issue is the question of the kind of methodology that is to be brought to bear on the study of these relationships. The methodology that we shall discuss here is based on a formally oriented approach to the philosophy of language, and specifically on the notion of a logically ideal language as the basis of a theory of meaning and a theory of knowledge."

  20. ———. 1997. "Conceptual Realism as a Theory of Logical Form." Revue Internationale de Philosophie:175-199.

    "The central notion in the philosophy of logic is the notion of a logical form, and the central issue is which theory of logical form best represents our scientific (including our mathematical) and commonsense understanding of the world. Here, by a theory of logical

    form, we mean not only a logical grammar in the sense of a system of formation rules characterizing the well-formed expressions of the theory, but also a logical calculus characterizing what is valid (i.e., provable or derivable) in the theory. The representational role of the logical forms of such a theory consists in the fact that they are perspicuous in the way they specify the truth conditions, and thereby the validity, of formulas in terms of the recursive operations of logical syntax. In conceptualism we also require that logical forms be perspicuous in the way they represent the cognitive structure of our speech and mental acts, including in particular the referential and predicable concepts underlying those acts.

    The purpose of a theory of logical form, accordingly, is that it is to serve as a general semantical framework by which we can represent in a logically perspicuous way our commonsense and scientific understanding of the world, including our understanding of ourselves

    and the cognitive structure of our speech and mental acts. So understood, the logical forms of such a theory are taken to be semantic structures in their own right, relative to which the words, phrases, and (declarative) sentences of a (fragment of) natural language, or of a scientific or mathematical theory, are to be represented and interpreted. The process by which the expressions of a natural language or scientific theory are represented and interpreted in such a theory — relative to the aims, purposes and pragmatic considerations regarding a given context or domain of discourse — amounts to a logical analysis of those expressions. (A different group of aims, purposes, etc., might give a finer- or a coarser-grained analysis of the domain.)

    Ideally, where the syntax of a target language or theory has been recursively characterized, such an analysis can be given in terms of a precisely characterized theory of translation (1). Usually, however, the analysis is given informally.

    In what follows I will briefly describe and attempt to motivate some (but not all) aspects of a theory of logical form that I associate with the philosophical system I call conceptual realism. The realism involved here is really of two types, one a natural realism (amounting to a modem form of Aristotelian essentialism) and the other an intensional realism (amounting to a modem, but also mitigated, form of Platonism). The core of the theory is a second-order logic in which predicate expressions (both simple and complex) can be nominalized and treated as abstract singular terms (in the sense of being substituends of individual variables). In this respect the core is really a form of conceptual intensional realism, which is the only part of the system we will discuss here (2)." (pp. 175-176)

    (1) See Montague (1969) for a description of such a theory of translation (for Montague’s type-theoretical intensional logic).

    (2) See Cocchiarella (1996), §§ 5-6, for a description of conceptual natural realism as a modem form of Aristotelian Essentialism.

    References

    Cocchiarella, N.B. (1996), “ Conceptual Realism as a Formal Ontology”, in Formal Ontology, R. Poli and P. Simons, eds., Kluwer Academic press, Dordrecht, pp. 27-60.

    Montague, R.M. (1969), “ Universal Grammar”, in Formal Philosophy, Selected papers of Richard Montague, edited by R.H. Thomason, Yale University Press, New Haven, 1974.

  21. ———. 1998. "Property Theory." In Routledge Encyclopedia of Philosophy - Vol. 7, edited by Craig, Edward, 761-767. New York: Routledge.

    Abstract: "Traditionally, a property theory is a theory of abstract entities that can be predicated of things. A theory of properties in this sense is a theory of predication -just as a theory of classes or sets is a theory of membership. In a formal theory of predication, properties are taken to correspond to some (or all) one-place predicate expressions. In addition to properties, it is usually assumed that there are n-ary relations that correspond to some (or all) n-place predicate expressions (for n > 2). A theory of properties is then also a theory of relations.

    In this entry we shall use the traditional labels 'realism' and 'conceptualism' as a convenient way to classify theories. In natural realism, where properties and relations are the physical, or natural, causal structures involved in the laws of nature, properties and relations correspond to only some predicate expressions, whereas in logical realism properties and relations are generally assumed to correspond to all predicate expressions.

    Not all theories of predication take properties and relations to be the universals that predicates stand for in their role as predicates. The universals of conceptual ism, for example. are unsaturated concepts in the sense of cognitive capacities that are exercised (saturated) in thought and speech. Properties and relations in the sense of intensional Platonic objects may still correspond to predicate expressions, as they do in conceptual intensional realism, but only indirectly as the intensional contents of the concepts that predicates stand for in their role as predicates. In that case, instead of properties and relations being what predicates stand for directly, they are what nominalized predicates denote as abstract singular terms. It is in this way that concepts - such as those that the predicate phrases 'is wise', 'is triangular' and 'is identical with' stand for - are distinguished from the properties and relations that are their intensional contents - such as those that are denoted by the abstract singular terms 'wisdom', 'triangularity' and 'identity, respectively. Once properties are represented by abstract singular terms, concepts can be predicated of them, and, in particular, a concept can be predicated of the property that is its intensional content. For example, the concept represented by 'is a property' can be predicated of the property denoted by the abstract noun phrase 'being a property', so that 'being a property is a property' (or, 'The property of being a property is a property') becomes well-formed. In this way, however, we are confronted with Russell's paradox of (the property of) being a non-self-predicable property, which is the intensional content of the concept represented by ' is a non-self-predicable property'. That is, the property of being a non-self- predicable property both falls and does not fall under the concept of being a non-self-predicable property (and therefore both falls and does not fall under the concept of being self-predicable)." (p. 761)

  22. ———. 1998. "The Theory of Types (Simple and Ramified)." In Routledge Encyclopedia of Philosophy - Vol. 9, edited by Craig, Edward, 359-362. New York: Routledge.

    Abstract: "The theory of types was first described by Bertrand Russell in 1908. He was seeking a logical theory that could serve as a framework for mathematics and, in particular, a theory that would avoid the so-called 'vicious-circle' antinomies, such as his own paradox of the property of those properties that are not properties of themselves - or, similarly, of the class of those classes that are not members of themselves. Such paradoxes can be thought of as resulting when logical distinctions are not made between different types of entities and, in particular, between different types of properties and relations that might be predicated of entities, such as the distinction between concrete objects and their properties, and the properties of those properties, and so on. In 'ramified' type theory, the hierarchy of properties and relations is, as it were, two-dimensional, where properties and relations are distinguished first by their order, and the by their level within each order. In 'simple' type theory properties and relations are distinguished only by their orders." (p. 359)

  23. ———. 1998. "Reference in Conceptual Realism." Synthese no. 114:169-202.

    Contents: 1. The core of Conceptual Intensional Realism; 2. Referential concepts, simple and complex; 3. Geach's negation and complex predicate arguments; 4. Active versus deactivated referential concepts; 5. Deactivation and Geach's arguments; 6. Relative pronouns and referential concepts; 7. Relative pronouns as referential expressions; 8. Concluding remarks.

    Abstract: "A conceptual theory of the referential and predicable concepts used in basic speech and mental acts is described in which singular and general, complex and simple, and pronominal and non-pronominal, referential concepts are given a uniform account. The theory includes an intensional realism in which the intensional contents of predicable and referential concepts are represented through nominalized forms of the predicate and quantifier phrases that stand for those concepts. A central part of the theory distinguishes between active and deactivated referential concepts, where the latter are represented by nominalized quantifier phrases that occur as parts of complex predicates. Peter Geach's arguments against theories of general reference in "Reference and Generality" are used as a foil to test the adequacy of the theory. Geach's arguments are shown to either beg the question of general as opposed to singular reference or to be inapplicable because of the distinction between active and deactivated referential concepts."

    "Concluding Remarks. We do not claim that the theory of relative pronouns as referential expressions proposed in Section 7 is unproblematic, it should be noted. If it should turn out that it cannot be sustained, then we still have the theory proposed in Section 6, where relative pronouns are taken as anaphoric proxies for non-pronominal referential expressions. In other words, whether the proposal of Section 7 is sustained or not, we maintain that Geach's arguments against complex names and general reference do not work against the kind of conceptualist theory we have presented here.

    We also do not claim to have proved that our conceptualist theory of reference resolves all problems about reference, but only that it has passed an initial test of adequacy as far as Geach's arguments in (Geach Reference and Generality third edition, 1980) are concerned. It is our view that a conceptualist theory is what is needed to account for reference and predication in our speech and mental acts, and that only a theory of the referential and predicable concepts that underlie the basic forms of such acts will suffice. Such a theory, we maintain, must provide a uniform account of general as well as singular reference, and, in terms of the referential and predicable concepts involved in a speech or mental act, it must distinguish the logical forms that represent the cognitive structure of that act from the logical forms that only represent its truth conditions. That, in any case, is the kind of conceptualist theory we have attempted to describe and defend here." (p. 198)

  24. ———. 2000. "Russell's Paradox of the Totality of Propositions." Nordic Journal of Philosophical Logic no. 5:25-37.

    Abstract: "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."

    "In Appendix B of his 1903 Principles of Mathematics (PoM), Russell described a ‘‘new contradiction’’ about ‘‘the totality of propositions’’ that his ‘‘doctrine of types’’ (as described in Appendix B) was unable to avoid. (1)

    In recent years this ‘‘new contradiction’’ has been connected with a number of modal paradoxes, some purporting to show that there cannot be a totality of true propositions, (2) or that even the idea of quantifying over the totality of propositions leads to contradiction. (3) A number of these claims have been discussed recently by Mika Oksanen and shown to be spurious relative to the set theory known as NFU. (4) In other words, if NFU is used instead of ZF as the semantical metalanguage for modal logic, the various ‘‘paradoxes’’ about the totality of propositions (usually construed as the totality of sets of possible worlds) can be seen to fail (generally because of the existence of a universal set and the failure of the general form of Cantor’s power-set theorem in NFU). It is not clear, however, how Russell’s own paradox about the totality of propositions is resolved on this analysis, and although Oksanen quoted Russell’s description of the paradox in detail, he did not show how it is explained in NFU after his resolution of the other related modal paradoxes; in fact, it is not at all clear how this might be done in NFU.

    One reason why Russell’s argument is difficult to reconstruct in NFU is that it is based on the logic of propositions, and implicitly in that regard on a theory of predication rather than a theory of membership. A more appropriate medium for the resolution of these paradoxes, in other words, would be a formal theory of predication that is a counterpart to NFU.

    Fortunately, there are two such theories, λHST* and HST*λ, that are equiconsistent with NFU and that share with it many of the features that make it a useful framework within which to resolve a number of paradoxes, modal or otherwise. (5)" (pp. 25-26)

    (1) PoM, p. 527.

    (2) See, e.g., Grim 1991, pp. 92f.

    (3) See, e.g., Grim 1991, p. 119 and Jubien 1988, p. 307.

    (4) See Oksanen 1999. NFU is a modified version of Quine’s system NF. It was first described in Jensen 1968 and recently has been extensively developed in Holmes 1999.

    (5) See Cocchiarella 1986, chapters IV and VI for proofs of the connection of NFU with these systems. Also, see Cocchiarella 1985 for how these systems are related to Quine’s systems NF and ML. For a discussion of the refutation of Cantor’s power-set theorem in

    these systems, see Cocchiarella 1992.

    References

    Cocchiarella, N. B. 1985. Frege’s double-correlation thesis and Quine’s set theories NF and ML. Journal of Philosophical Logic, vol. 4, pp. 1–39.

    Cocchiarella, N. B. 1986. Logical Investigations of Predication Theory and the Problem of Universals. Bibliopolis Press, Naples, 1985.

    Cocchiarella, N. B. 1992. Cantor’s Power-Set Theorem Versus Frege’s Double-Correlation Thesis, History and Philosophy of Logic, vol. 13, 179–201.

    Holmes, R. 1999. Elementary Set Theory with a Universal Set. Cahiers du Centre de Logique, Bruylant-Academia, Louvain-la-Neuve, Belgium.

    Grim, P. 1991. The Incomplete Universe. MIT Press, Cambridge, MA.

    Jensen, R. 1968. On the consistency of a slight (?) modification of Quine’s New Foundations. Synthese, vol. 19, pp. 250–263.

    Oksanen, M. 1979. The Russell-Kaplan paradox and other modal paradoxes; a new solution. Nordic Journal of Philosophical Logic, vol. 4, no. 1, pp. 73–93.

    Russell, B. 1937. The Principles of Mathematics, 2nd edition. W. W. Norton & Co., N.Y.

  25. ———. 2001. "A Logical Reconstruction of Medieval Terminist Logic in Conceptual Realism." Logical Analysis and History of Philosophy no. 4:35-72.

    Abstract: "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 (the fitcum theory) to his later view of a concept as a mental act (the intellectio theory), is reconstructed in this framework in terms of the idea of concepts as unsaturated cognitive structures. Intentional objects (ficta) are not rejected but are reconstructed as the objetified 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."

    "Conclusion. 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 (the f ictum theory) to his later view of a concept as a mental act (the intellectio theory), is reconstructed in this framework in terms of the notion of a concept as an unsaturated cognitive structure. Referential and predicable concepts in particular are unsaturated cognitive structures that mutually saturate each other in mental acts, analogous to the way that quantifier phrases and predicate expressions mutually saturate each other in language. Intentional objects (ficta) are not rejected but are reconstructed as the objectified intensional contents of concepts, i.e., as intentional objects obtained through the process of nominalization — and in that sense as products of the evolution of language and thought. Their reconstruction as intensional objects is an essential part of the theory of predication of conceptual realism. In particular, the truth conditions determined by predicable concepts based on relations — including the relation the copula stands for — are characterized in part in terms of these objectified intensional contents. It is by means of this conceptualist theory of predication 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, based on the exercise and mutual saturation of referential and predicable concepts, is not the same as supposition in terminist logic. Nevertheless, the various “modes” or types of personal supposition are accounted for in a natural and intuitive way in terms of the theory of reference of conceptual realism. Ockham’s application of merely confused supposition to common names occurring within the scope of an intensional verb is rejected, as it should be, but its rejection is grounded on the notion of a deactivated referential concept—a deactification that, because of the intensionality of the context in question, cannot be “activated,” the way it can be in extensional contexts." (p. 71)

  26. ———. 2001. "Logic and Ontology." Axiomathes.An International Journal in Ontology and Cognitive Systems no. 12:127-150.

    Contents: 1. Logic as Language versus Logic as Calculus; 2. Predication versus Membership; 3. The vagaries of Nominalism; 4. The Vindication (Almost) of Logical Realism; 5. Conceptualism Without a Transcendental Subject; 6. Conceptual Natural realism and the Analogy of being Between Natural and Conceptual Universals; 7. Conceptual Intensional Realism; 8. Concluding Remarks.

    Abstract: "A brief review of the historical relation between logic and ontology and of the opposition between the views of logic as language and logic as calculus is given. We argue that predication is more fundamental than membership and that different theories of

    predication are based on different theories of universals, the three most important being nominalism, conceptualism, and realism. These theories can be formulated as formal ontologies, each with its own logic, and compared with one another in terms of their respective explanatory powers. After a brief survey of such a comparison, we argue that an extended form of conceptual realism provides the most coherent formal ontology and, as such, can be used to defend the view of logic as language."

    "Concluding Remarks: Despite our extended discussion and defense of conceptual realism, the fact remains that this is a formal ontology that can be described and compared with other formal ontologies in the set-theoretic framework of comparative formal ontology. Set theory, as we have said, provides a convenient mathematical medium in which both the syntax and an extrinsic semantics of different formal ontologies can be formulated, which then can be compared and contrasted with one another in their logical and descriptive powers. This is the real insight behind the view of logic as calculus. But membership is at best a pale shadow of predication, which underlies thought, language and the different categories of reality. Set theory is not itself an adequate framework for general ontology, in other words, unless based on a theory of predication (as in Quine's nominalist-platonism). Only a formal theory of predication based on a theory of universals can be the basis of a general ontology. This is the real insight behind the view of logic as language. But there are alternative theories of universals, and therefore alternative formal theories of predication, each with its own logic and theory of logical form. A rational choice can be made only by formulating and comparing these alternatives in comparative formal ontology, a program that can best be carried out in set theory. Among the various alternatives that have been formulated and investigated over the years, the choice we have made here, for the reasons given, is what we have briefly described above as conceptual realism, which includes both a conceptual natural realism and a conceptual intensional realism. Others may make a different choice. As Rudolf Carnap once said: "Everyone is at liberty to build up his own logic, i.e. his own form of language, as he wishes." But then, at least in the construction of a formal ontology, we all have an obligation to defend our choice and to give reasons why we think one system is better than another. In this regard, we do not accept Carnap's additional injunction that in logic, there are no morals." (pp. 145-146)

    Translated in Italian by Flavia Marcacci with revision by Gianfranco Basti, as: Logica e Ontologia in Aquinas. Rivista Internazionale di Filosofia, 52, 2009.

  27. ———. 2001. "A Conceptualist Interpretation of Leśniewski's Ontology." History and Philosophy of Logic no. 22:29-43.

    Contents: 1. Introduction 29; 2. Leśniewski’s Ontology as a First-Order Theory 29; 3. The Logic of Names in Conceptual Realism 31; 4. A Conceptualist Interpretation of Leśniewski’s System 35; 5. Reduction of Leśniewski’s Theory of Definitions 39; 6. Consistency and Decidability 40; References 43.

    Abstract: "A first-order formulation of Leśniewski’s ontology is formulated and shown to be interpretable within a free first-order logic of identity extended to include nominal quantification over proper and common-name concepts. The latter theory is then shown to be interpretable in monadic second-order predicate logic, which shows that the first-order part of Leśniewski’s ontology is decidable."

    "Introduction. One of the important applications of Leśniewski’s system of ontology, sometimes also called the logic of (proper and common) names, (1) has been as a logistic framework that can be used in the reconstruction of medieval terminist logic. (2) This is especially so because the basic relation of Leśniewski’s system, singular inclusion, amounts to a version of the two-name theory of the copula. (3) An alternative reconstruction of medieval terminist logic can also be given within the logistic framework of conceptual

  28. ———. 1986. "Frege, Russell and Logicism: A Logical Reconstruction." In Frege Synthesized: Essays on the Philosophical and Foundational Work of Gottlob Frege, edited by Haaparanta, Leila and Hintikka, Jaakko, 197-252. Dordrecht: Reidel

    Reprinted as Chapter 2 in Logical Studies in Early Analytic Philosophy, pp. 64-118.

    "Logicism by the end of the nineteenth century was a philosophical doctrine whose time had come, and it is Gottlob Frege to whom we owe its arrival. “Often,” Frege once wrote, “it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, in stripping off the irrelevant accretions which veil it from the eyes of the mind” (Frege, The Foundations of Arithmetic, [Fd], xix). Prior to Frege logicism was just such a concept whose pure form was obscured by irrelevant accretions; and in his life’s work it was Frege who first presented this concept to humanity in its pure form and developed it as a doctrine of the first rank.

    That form, unfortunately, has become obscured once again. For today, as we approach the end of the twentieth century, logicism, as a philosophical doctrine, is said to be dead, and even worse, to be impossible. Frege’s logicism, or the specific presentation he gave of it in Die Grundgesetze der Arithmetik, ([Gg]), fell to Russell’s paradox, and, we are told, it cannot be resurrected. Russell’s own subsequent form of logicism presented in [PM], moreover, in effect gives up the doctrine; for in overcoming his paradox, Russell was unable to reduce classical mathematics to logic without making at least two assumptions that are not logically true; namely, his assumption of the axiom of reducibility and his assumption of an axiom of infinity regarding the existence of infinitely many concrete or nonabstract individuals.

    Contrary to popular opinion, however, logicism is not dead beyond redemption; that is, if logicism is dead, then it can be easily resurrected. This is not to say that as philosophical doctrines go logicism is true, but only that it can be logically reconstructed and defended or advocated in essentially the same philosophical context in which it was originally formulated. This is true especially of Frege’s form of logicism, as we shall see, and in fact, by turning to his correspondence with Russell and his discussion of Russell’s paradox, we are able to formulate not only one but two alternative reconstructions of his form of logicism, both of which are consistent (relative to weak Zermelo set theory).

    In regard to Russell’s form of logicism, on the other hand, our resurrection will not apply directly to the form he adopted in [PM] but rather to the form he was implicitly advocating in his correspondence with Frege shortly after the completion of [POM]. In this regard, though we shall have occasion to refer to certain features of his later form of logicism, especially in our concluding section where a counterpart to the axiom of reducibility comes into the picture, it is Russell’s early form of logicism that we shall reconstruct and be concerned with here.

    Though Frege’s and Russell’s early form of logicism are not the same, incidentally, they are closely related; and one of our goals will be to reconstruct or resurrect these forms with their similarity in mind. In particular, it is our contention that both are to be reconstructed as second order predicate logics in which nominalized predicates are allowed to occur as abstract singular terms. Their important differences, as we shall see, will then consist in the sort of object each takes nominalized predicates to denote and in whether the theory of predication upon which the laws of logic are to be based is to be extensional or intensional." (pp. 64-65 of the reprint)

    References

    Frege, Gottlob, [Fd] The Foundations of Arithmetic, trans, by J. L. Austin, Harper & Bros., N.Y. 1960.

    Frege, Gottlob, [Gg] Die Grundgesetze der Arithmetik, vols. 1 and 2, Hildesheim, 1962.

    Russell, Bertrand, [PM] Principia Mathematica, coauthor, A. N. Whitehead, Cambridge University Press, 1913.

    Russell, Bertrand, [POM] The Principles of Mathematics, 2nd ed., W. W. Norton & Co., N.Y., 1937.

  29. ———. 1986. "Conceptualism, Ramified Logic, and Nominalized Predicates." Topoi.An International Review of Philosophy no. 5:75-87.

    The problem of universals as the problem of what predicates stand for in meaningful assertions is discussed in contemporary philosophy mainly in terms of the opposing theories of nominalism and logical realism. Conceptualism, when it is mentioned, is usually identified with intuitionism, which is not a theory of predication but a theory of the activity of constructing proofs in mathematics. Both intuitionism and conceptualism are concerned with the notion of a mental construction, to be sure, and both maintain that there can only be a potentially infinite number of such constructions. But whereas the focus of concern in intuitionism is with the construction of proofs, in conceptualism our concern is with the construction of concepts. This difference sets the two frameworks apart and in pursuit of different goals, and in fact it is not at all clear how the notion of a mental construction in the one framework is related to that in the other. This is especially true insofar as mathematical objects, according to intuitionism, are nothing but mental constructions, whereas in conceptualism concepts are anything but objects. In any case, whatever the relation between the two, our concern in this paper is with conceptualism as a philosophical theory of predication and not with intuitionism as a philosophy of mathematics.

    Now conceptualism differs from nominalism insofar as it posits universals, namely, concepts, as the semantic grounds for the correct or incorrect application of predicate expressions. Conceptualism differs from logical realism, on the other hand, insofar as the universals it posits are not assumed to exist independently of the human capacity for thought and representation. Concepts, in other words, are neither predicate expressions nor independently real properties and relations. But then, at least for the kind of conceptualism we have in mind here, neither are they mental images or ideas in the sense of particular mental occurrences. That is, concepts are not objects (saturated individuals) but are rather cognitive capacities, or cognitive structures otherwise based upon such capacities, to identify and classify or characterize and relate objects in various ways. Concepts, in other words, are intersubjectively realizable cognitive abilities which may be exercized by different persons at the same time as well as by the same person at different times. And it is for this reason that we speak of concepts as objective universals, even though they are not independently real properties and relations.

    As cognitive structures, however, concepts in the sense intended here are not Fregean concepts (which for Frege are independently real unsaturated functions from objects to truth values). But they may be modeled by the latter (assuming that there are Fregean concepts to begin with) -especially since as cognitive capacities which need not be exercized at any given time (or even ever for that matter), concepts in the sense intended here also have an unsaturated nature corresponding to, albeit different from, the unsaturated nature of Fregean concepts. Thus, in particular, the saturation (or exercise) of a concept in the sense intended here results not in a truth value but a mental act, and, if overtly expressed, a speech act as well. The un-saturatedness of a concept consists in this regard in its non-occurrent or purely dispositional status as a cognitive capacity, and it is the exercise (or saturation) of this capacity as a cognitive structure which informs particular mental acts with a predicable nature (or with a referential nature in the case of concepts corresponding to quantifier expressions)." (pp. 75-76)

  30. ———. 1987. "Rigid Designation." In Encyclopedic Dictionary of Semiotics. Vol. 2, edited by Sebeok, Thomas A., 834. Berlin: Mouton de Gruyter.

  31. ———. 1987. "Russell, Bertrand." In Encyclopedic Dictionary of Semiotics. Vol. 2, edited by Sebeok, Thomas A., 840-841. Berlin: Mouton de Gruyter.

  32. ———. 1988. "Predication Versus Membership in the Distinction between Logic as Language and Logic as Calculus." Synthese no. 75:37-72

    Contents: 0. Introduction; 1. The problem with a set-theoretic semantics of natural language; 2. Intensional logic as a new theoretical framework for philosophy; 3.The incompleteness of intensional logic when based on membership; 4. Predication versus membership in type theory; 5. Second order predicate logic with nominalized predicates; 6. A set theoretic semantics with predication as fundamental; 7. Concluding remarks.

    "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 Aristotle's theory of the syllogism, or syllogistic logic, and in time evolved first into Boole's 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 ["Logic as Language and Logic as Calculus", Synthese 17,] 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 use 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 completeness 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." (p. 37)

    (...)

    "Concluding Remarks. The account we have given here of the view of logic as language should not be taken as a rejection of the set-theoretical approach or as defense of the metaphysics of possibilist logical realism. Rather, our view is that there are really two types of conceptual framework corresponding to our two doctrines of the nature of logic. The first type of framework is based on membership in the sense of the iterative concept of set; although extensionality is its most natural context (since sets have their being in their members), it may nevertheless be extended to include intensional contexts by way of a theory of senses (as in Montague's sense-denotation intensional logic). The second type of framework is based on predication, and in particular developments it is associated with one or another theory of universals. Extensionality is not the most natural context in this theory, but where it does hold and extensions are posited, the extensions are classes in the logical and not in the mathematical sense.

    Russell's paradox, as we have explained, has no real bearing on set-formation in a theory of membership based on the iterative concept of set, but it does bear directly on concept-formation or the positing or universals in a theory based on predication. As a result, our second type of framework has usually been thought to be incoherent or philosophically bankrupt, leaving us with the set-theoretical approach as, the only viable alternative. This is why so much of analytic philosophy in the 20th Century has been dominated by the set-theoretical approach. Set theory, after all, does seem to serve the purposes of a mathesis universalis.

    What is adequate as a mathesis universalis, however, need not also therefore be adequate as a lingua philosophica or characteristica universalis. In particular, the set-theoretic approach does not seem to provide a philosophically satisfying semantics for natural language; this is because it is predication and not membership that is fundamental to natural language. An adequate semantics for natural language, in other words, seems to demand a conceptual framework based on predication and not on membership.

    (...)

    We do not maintain, accordingly, that we should give up the set-theoretic approach, especially when dealing with the philosophy and foundations of mathematics, or that only a theory of predication associated with possibilist logical realism will provide an adequate semantics for natural language. In both cases we may find a principle of tolerance, if not outright pluralism, the more appropriate attitude to take." (pp. 69-70)

  33. ———. 1989. "Philosophical Perspectives on Formal Theories of Predication." In Handbook of Philosophical Logic. Vol. 4. Topics in the Philosophy of Language, edited by Gabbay, Dov and Guenthner, Franz, 253-326. Dordrecht: Reidel

    Contents: 1. Predication and the problem of universal 254; 2. Nominalism 256; 3. A nominalistic semantics for predicative second order logic 261; 4. Nominalism and modal logic 266; 5 . Conceptualism vs . nominalism 270; 6. Constructive conceptualism 273; 7. Ramification of constructive conceptualism 280; 8. Holistic conceptualism 286; 9. Logical realism vs holistic conceptualism 289; 10. Possibilism and actualism in modal logical realism 292; 11. Logical realism and cssentialism 301; 12. Possibilism and actualism within conceptualism 306; 13. Natural realism and conceptualism 313; 14. Aristotelian essentialism and the logic of natural kinds 318; References 325-326.

    "Predication has been a central, if not the central, issue in philosophy since at least the lime 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.

    Though each of these three types of theories of universals may be said to have many variants, we shall ignore their differences here to the extent that they do not characterize different theories of predication. This will apply especially to nominalism where but one formal theory of predication is involved. In both conceptualism and realism, however, the different variants of each type do not all agree and form two distinct subtypes each with its own theory of predication. For this reason we shall distinguish between a constructive and a holistic form of conceptualism on the one hand, and a logical and a natural realism on the other. Constructive conceptualism, as we shall see, has affinities with nominalism with which it is sometimes confused, and holistic conceptualism has affinities with logical realism with which it is also sometimes confused. Both forms of conceptualism may assume some form of natural realism as their causal ground; and natural realism in turn must presuppose some form of conceptualism as its background theory of predication. Both forms of realism may be further divided into their essentialist and non-essentialist variants (and in logical realism even a form of anti-essentialism), and though an essentialist logical realism is sometimes confused with Aristotelian essentialism, the latter is really a form of natural realism with natural kinds as the only essential properties objects can have." (pp. 253-254)

  34. ———. 1989. "Russell's Theory of Logical Types and the Atomistic Hierarchy of Sentences." In Rereading Russell: Essays on Bertrand Russell's Metaphysics and Epistemology, edited by Savage, C.Wade and Anderson, C.Anthony, 41-62. Minneapolis: University of Minnesota Press

    Reprinted as Chapter 5 in Logical Studies in Early Analytic Philosophy, pp. 193-221.

    "Russell’s philosophical views underwent a number of changes throughout his life, and it is not always well-appreciated that views he held at one time came later to be rejected; nor, similarly, that views he rejected at one time came later to be accepted. It is not well-known, for example, that the theory of logical types Russell described in his later or post-[PM] philosophy is not the same as the theory originally described in [PM] in 1910-13; nor that some of the more important applications that Russell made of the theory at the earlier time cannot be validated or even significantly made in the framework of his later theory. What is somewhat surprising, however, is that Russell himself seems not to have realized that he was describing a new theory of logical types in his later philosophy, and that as a result of the change some of his earlier logical constructions, including especially his construction of the different kinds of numbers, were no longer available to him.

    In the original framework, for example, propositional functions are independently real properties and relations that can themselves have properties and relations of a higher order/type, and all talk of classes, and thereby ultimately of numbers, can be reduced to extensional talk of properties and relations as “single entities,” or what Russell in [POM] had called “logical subjects.” The Platonic reality of classes and numbers was replaced in this way by a more fundamental Platonic reality of propositional functions as properties and relations. In Russell's later philosophy, however, “a propositional function is nothing but an expression. It does not, by itself, represent anything. But it can form part of a sentence which does say something, true or false” (Russell, My Philosophical Development, ([MPD]), 69). Surprisingly. Russell even insists that this was what he meant by a propositional function in [PM]. “Whitehead and I thought of a propositional function as an expression containing an undetermined variable and becoming an ordinary sentence as soon as a value is assigned to the variable: ‘x is human’, for example, becomes an ordinary sentence as soon as we substitute a proper name for V. In this view . . . the propositional function is a method of making a bundle of such sentences” ([MPD], 124). Russell does realize that some sort of change has come about, however, for he admits, “I no longer think that the laws of logic are laws of things; on the contrary, I now regard them as purely linguistic” (ibid., 102).

    (...)

    Now it is not whether [PM] can sustain a nominalistic interpretation that is our concern in this essay, as we have said, but rather how it is that Russell came to be committed in his later philosophy to the atomistic hierarchy and the nominalistic interpretation of propositional functions as expressions generated in a ramified second order hierarchy of languages based on the atomistic hierarchy. We shall pursue this question by beginning with a discussion of the difference between Russell’s 1908 theory of types and that presented in [PM] in 1910. This will be followed by a brief summary of the ontology that Russell took to be implicit in [PM], and that he described in various publications between 1910 and 1913. The central notion in this initial discussion is what Russell in his early philosophy called the notion of a logical subject, or equivalently that of a “term” or “single entity”. (In [PM], this notion was redescribed as the systematically ambiguous notion of an “object.”) As explained in chapter 1 this notion provides the key to the various problems that led Russell in his early philosophy to the development of his different theories of types, including that presented in [PM]. This remains true, moreover, even when we turn to Russell’s later philosophy, i.e., to his post-[PM] views, only then it is described as the notion of what can and cannot be named in a logically perfect language. The ontology of these later views is what Russell called logical atomism, and it is this ontology that determines what Russell described as the atomistic hierarchy of sentences. In other words, it is the notion of what can and cannot be named in the atomistic hierarchy that explains how Russell, however unwittingly, came to replace his earlier theory of logical types by the theory underlying the atomistic hierarchy of sentences as the basis of a logically perfect language." (pp. 193-195 of the reprint)

    References

    POM] Russell, Bertrand, The Principles of Mathematics, 2d ed. (NY., Norton & Co., 1938).

    [PM] Russell, Bertrand and Alfred Whitehead, Principia Mathematica, vol. 1 (1910), vol. 2 (1912), and vol. 3 (1913) (London: Cambridge Univ. Press,).

  35. ———. 1989. "Conceptualism, Realism and Intensional Logic." Topoi.An International Review of Philosophy no. 7:15-34

    Contents: 0. Introduction 15; 1. A conceptual analysis of predication 16; 2. Concept-correlates and Frege's double correlations thesis 17; 3. Russell's paradox in conceptual realism 18: 4. What are the natural numbers and where do they come from? 22; 5. Referential concepts and quantifier phrases 24; 5. Singular reference 24; 7. The intensions of refrential concepts as components of applied predicable concepts 26; 8. Intensional versus extensional predicable concepts 28; 9. The intentional identity of intensional objects 29; Notes 31; Reference 33-35.

    "0. Introduction

    Linguists and philosophers are sometimes at odds in the semantical analysis of language. This is because linguists tend to assume that language must be semantically analyzed in terms of mental constructs, whereas philosophers tend to assume that only a platonic realm of intensional entities will suffice. The problem for the linguist in this conflict is how to explain the apparent realist posits we seem to be committed to in our use of language, and in particular in our use of infinitives, gerunds and other forms of nominalized predicates. The problem for the philosopher is the old and familiar one of how we can have knowledge of independently real abstract entities if all knowledge must ultimately be grounded in psychological states and processes. In the case of numbers, for example, this is the problem of how mathematical knowledge is possible. In the case of the intensional entities assumed in the semantical analysis of language, it is the problem of how knowledge of even our own native language is possible, and in particular of how we can think and talk to one another in all the ways that language makes possible.

    I believe that the most natural framework in which this conflict is to be resolved and which is to serve as the semantical basis of natural language is an intensional logic that is based upon a conceptual analysis of predication in which what a predicate stands for in its role as a predicate is distinguished from what its nominalization denotes in its role as a singular term. Predicates in such a framework stand for concepts as cognitive capacities to characterize and relate objects in various ways, i.e. for dispositional cognitive structures that do not themselves have an individual nature, and which therefore cannot be the objects denoted by predicate nominalizations as abstract singular terms. The objects purportedly denoted by nominalized predicates, on the other hand, are intensional entities, e.g. properties and relations (and propositions in the case of zero-place predicates), which have their own abstract form of individuality, which, though real, is posited only through the concepts that predicates stand for in their role as predicates. That is, intensional objects are represented in this logic as concept-correlates, where the correlation is based on a logical projection of the content of the concepts whose correlates they are.

    (...)

    Before proceeding, however, there is an important distinction regarding the notion of a logical form that needs to be made when joining conceptualism and realism in this way. This is that logical forms can be perspicuous in either of two senses, one stronger than the other. The first is the usual sense that applies to all theories of logical form, conceptualist or otherwise; namely, that logical forms are perspicuous in the way they specify the truth conditions of assertions in terms of the recursive operations of logical syntax. In this sense, fully applied logical forms are said to be semantic structures in their own right. In the second and stronger sense, logical forms may be perspicuous not only in the way they specify the truth conditions of an assertion, but in the way they specify the cognitive structure of that assertion as well. To be perspicuous in this sense, a logical form must provide an appropriate representation of both the referential and the predicable concepts that underlie an assertion.

    Our basic hypothesis in this regard will be that every basic assertion is the result of applying just one referential concept and one predicable concept, and that such an applied predicable concept is always fully intensionalized (in a sense to be explained). This will place certain constraints on the conditions for when a complex predicate expression is perspicuous in the stronger sense — such as that a referential expression can occur in such a predicate expression only in its nominalized form. (A similar constraint will also apply to a defining or restricting relative clause of a referential expression.) In the cases where a relational predicable concept is applied, the assumption that there is still but one referential concept involved leads to the notion of a conjunctive referential concept, a notion that requires the introduction in intensional logic of special quantifiers that bind more than one individual variable. Except for briefly noting the need for their development, we shall not deal with conjunctive quantifiers in this essay." (pp. 15-16)

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On the website "Theory and History of Ontology"

Frege's Ontology: Being, Existence, and Truth

Bertrand Russell's Ontological Development

The Ontology of Wittgenstein's Tractatus

Edmund Husserl: Formal Ontology and Transcendental Logic

Stanislaw Lesniewski's Logical Systems: Protothetic, Ontology, Mereology