Francesco Orilia

Abstract Objects: An Introduction to Axiomatic Metaphysics.

1. Theory, Data, and Explanation.- 2. The Origins of the Theory.- I. Elementary Object Theory.- 1. The Language.- 2. The Semantics.- 3. The Logic.- 4. The Proper Axioms.- 5. An Auxiliary Hypothesis.- II. Applications of the Elementary Theory.- 1. Modelling Platos Forms.- 2. Modelling the Round Square, etc..- 3. The Problem of Existence 50 Appendix.- III. The Modal Theory of Abstract Objects (With Propositions).- 1. The Language.- 2. The Semantics.- 3. The Logic.- 4. The Proper Axioms.- IV. The Applications of the Modal Theory.- 1. Truth.- 2. Modelling Possible Worlds.- 3. Modelling Leibnizs Monads.- 4. Modelling Stories and Native Characters.- 5. Modality and Descriptions.- V. The Typed Theory of Abstract Objects.- 1. The Language.- 2. The Semantics.- 3. The Logic.- 4. The Proper Axioms.- VI. Applications of the Typed Theory.- 1. Modelling Freges Senses (I).- 2. Modelling Freges Senses (II).- 3. Modelling Impossible and Fictional Relations.- 4. Modelling Mathematical Myths and Entities.- Conclusion.- Appendices.- A. Modelling the Theory Itself.- B. Modelling Notions.- Notes.

Property Theory and the Revision Theory of Definitions

§1. Introduction. Russells type-theory can be seen as a theory of properties, relations, and propositions (PRPs) (in short, a property theory). It relies on rigid type distinctions at the grammatical level to circumvent the property theorists major problem, namely Russells paradox, or, more generally, the paradoxes of predication. Type theory has arguably been the standard property theory for years, often taken for granted, and used in many applications. In particular, Montague [27] has shown how to use a type-theoretical property-theory as a foundation for natural language semantics. In recent years, it has been persuasively argued that many linguistic and ontological data are best accounted for by using a type-free property theory. Several type-free property theories, typically with fine-grained identity conditions for PRPs, have therefore been proposed as potential candidates to play a foundational role in natural language semantics, or for related applications in formal ontology and the foundations of mathematics (Bealer [6], Cocchiarella [18], Turner [35], etc.). Attempts have then been made to combine some such property theory with a Montague-style approach in natural language semantics. Most notably, Chierchia and Turner [15] propose a Montague-style semantic analysis of a fragment of English, by basically relying on the type-free system of Turner [35]. For a similar purpose Chierchia [14] relies on one of the systems based on homogeneous stratification due to Cocchiarella. Cocchiarellas systems have also been used for applications in formal ontology, inspired by Montagues account of quantifier phrases as, roughly, properties of properties (see, e.g., Cocchiarella [17], [19], Landini [25], Orilia [29]).

Argument deletion, thematic roles, and Leibniz's logico-grammatical analysis of relations

I present a formal framework historically faithful to Leibnizs analysis of relational sentences, which: (i) engrafts thematic roles and the non-truth-functional connective insofar as (quatenus) into the monadic fragment of first-order logic; (ii) suggests a plausible ontological picture of thematic roles and relational facts; (iii) supports argument deletion and related inferential patterns that are not taken into account by standard first-order logic.

A Contingent Russell's Paradox

It is shown that two formally consistent type-free second-order systems, due to Cocchiarella, and based on the notion of homogeneous strat- ification, are subject to a contingent version of Russells paradox.

Singular reference : a descriptivist perspective

Introduction: Referentialism vs. Descriptivism.- Background Notions.- Why Descriptivism Was So Successful.- Why Referentialism Is So Successful.- Definite Descriptions and Proper Names.- Indexicals.- Tense, Temporal Indexicals and Other Miscellaneous Issues.- Conclusion: Accounting for the Referentialist DataReferentialist data .

Knowledge Representation, Exemplification, and the Gupta-Belnap Theory of Circular Definitions

Guptas and Belnaps revision theory of circular definitions (RTD) provides a general technique for specifying circular definitions in a way that at worst gives rise to some “vacuous” uses of definienda, but never to contradiction. This is a first step in applying RTD to the problem of constructing a type-free theory of properties, relations and propositions (in short, PRPs). To this end, exemplification is viewed as a circular concept analyzed in terms of RTD. This yields a formal semantics system, P*, wherein the generality of lambda-conversion is circumscribed so as to avoid, e.g., Russells paradox. The construction of P* is motivated by showing how it can provide a foundation for a knowledge representation system capable of dealing with belief (or more generally intensional) contexts and with inferences involving PRPs in subject position.

Belief Representation in a Deductivist Type-free Doxastic Logic

Konoliges technical notion of belief based on deduction structures is briefly reviewed and its usefulness for the design of artificial agents with limited representational and deductive capacities is pointed out. The design of artificial agents with more sophisticated representational and deductive capacities is then taken into account. Extended representational capacities require in the first place a solution to the intensional context problems. As an alternative to Konoliges modal first-order language, an approach based on type-free property theory is proposed. It considers often neglected issues, such as the need for a more general account of thede dicto-de re distinction, and quasi-indicators. Extended deductive capacities require a subdivision of Konoliges notion of belief into two distinct technical notions,potential anddispositional belief. The former has to do with what an artificial agent could in principle come to actively believe, given enough time and its specific logical competence; the latter with what an agent can be assumed to believe with respect to a specific goal to be fulfilled.

Meaning and circular definitions

Guptas and Belnaps Revision Theory of Truth defends the legitimacy of circular definitions. Circularity, however, forces us to reconsider our conception of meaning. A readjustment of some standard theses about meaning is here proposed, by relying on a novel version of the sense–reference distinction.

Positions, Ordering Relations and O‐Roles

This paper first discusses how Russell and Hochberg have addressed some phenomena of relatedness, notably relational order, in a similarly ‘positionalist’ way, yet by appealing to different sorts of formal relations: “positions” in Russells case and “ordering relations” in Hochbergs. After pointing out some shortcomings of both approaches, the paper then proposes an alternative view based on ‘o-roles’, which are, roughly speaking, ontological counterparts of the thematic roles postulated in linguistics. It is argued that o-roles are sort of middle-of-the-road entities in the sense that they have the virtues of positions and those of ordering relations, without having their respective vices. Some tentative ideas on which o-roles should be acknowledged are also put forward.

Quantum-mechanical Statistics and the Inclusivist Approach to the Nature of Particulars

There have been attempts to derive anti-haeccetistic conclusions from the fact that quantum mechanics (QM) appeals to non-standard statistics. Since in fact QM acknowledges two kinds of such statistics, Bose-Einstein and Fermi-Dirac, I argue that we could in the same vein derive the sharper anti-haeccetistic conclusion that bosons are bundles of tropes and fermions are bundles of universals. Moreover, since standard statistics is still appropriate at the macrolevel, we could also venture to say that no anti-haecceitistic conclusion is warranted for ordinary objects, which could then tentatively be identified with substrates. In contrast to this, however, there has been so far no acknowledgement of the possibility of inclusivism, according to which ontological accounts of particulars as widely different as those can possibly coexist in one world picture. The success of the different statistics in physics at least calls for a revision in this respect.