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Property theory and the revision theory of definitions


§1. Introduction. Russell's 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 theorist's major problem, namely Russell's 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. Cocchiarella's systems have also been used for applications in formal ontology, inspired by Montague's account of quantifier phrases as, roughly, properties of properties (see, e.g., Cocchiarella [17], [19], Landini [25], Orilia [29]).

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[1] P. Aczel , Frege structures and the notions of propositions, truth and set, The Kleene symposium ( J. Barwise , H. J. Keisler , and K. Kunen , editors), North-Holland, Amsterdam, 1980, pp. 3139.

[6] G. Bealer , Quality and concept, Oxford University Press, London, 1982.

[9] N. Belnap , Gupta's rule of revision theory of truth, Journal of Philosophical Logic, vol. 11 (1982), pp. 103116.

[11] H.-N. Castañeda , Ontology and grammar: I. Russell's paradox and the general theory of properties in natural language, Theoria, vol. 42 (1976), pp. 4492.

[13] A. Chapuis , Alternative revision theories of truth, Journal of Philosophical Logic, vol. 25 (1996), pp. 399423.

[15] G. Chierchia and R. Turner , Semantics andproperty theory, Linguistics and Philosophy, vol. 11 (1988), pp. 261302.

[17] N. Cocchiarella , Meinong reconstructed versus early Russell reconstructed, Journal of Philosophical Logic, vol. 11 (1982), pp. 183215.

[19] N. Cocchiarella , Conceptualism, realism, and intensional logic, Topoi, vol. 8 (1989), pp. 1534.

[21] A. Gupta , Truth and paradox, Journal of Philosophical Logic, vol. 11 (1982), pp. 160.

[23] H. G. Herzberger , Notes on naive semantics, Journal of Philosophical Logic, vol. 11 (1982), pp. 61102.

[24] P. Kremer , The Gupta-Belnap systems S# and S* are not axiomatisable, Notre Dame Journal of Formal Logic, vol. 34 (1993), pp. 583596.

[25] G. Landini , HOW to Russell another Meinongian: a Russellian theory of fictional objects versus Zalta's theory of abstract objects, Grazer Philosophische Studien, vol. 37 (1990), pp. 93122.

[29] F. Orilia , Belief representation in a type-free doxastic logic, Minds and Machines, vol. 4 (1994), pp. 163203.

[33] R. Smullyan , First-order logic, Springer-Verlag, Berlin, 1968.

[34] R. Thomason , A note on syntactical treatments of modality, Synthese, vol. 44 (1980), pp. 391395.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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