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Abstract object theory

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For the general concept of objecthood in philosophy, see Object (philosophy).

Abstract object theory (AOT) is a branch of metaphysics regarding abstract objects.[1] Originally devised by metaphysician Edward Zalta in 1981,[2] the theory was an expansion of mathematical Platonism.

Overview

Abstract Objects: An Introduction to Axiomatic Metaphysics (1983) is the title of a publication by Edward Zalta that outlines abstract object theory.

AOT is a dual predication approach (also known as "dual copula strategy") to abstract objects[3][4] influenced by the contributions of Alexius Meinong[5][6] and his student Ernst Mally.[7][6] On Zalta's account, there are two modes of predication: some objects (the ordinary concrete ones around us, like tables and chairs) exemplify properties, while others (abstract objects like numbers, and what others would call "non-existent objects", like the round square, and the mountain made entirely of gold) merely encode them.[8] While the objects that exemplify properties are discovered through traditional empirical means, a simple set of axioms allows us to know about objects that encode properties.[9] For every set of properties, there is exactly one object that encodes exactly that set of properties and no others.[10] This allows for a formalized ontology.

A notable feature of AOT is that several notable paradoxes in naive predication theory (namely Romane Clark's paradox undermining the earliest version of Héctor-Neri Castañeda's guise theory,[11][12][13] Alan McMichael's paradox,[14] and Daniel Kirchner's paradox)[15] do not arise within it.[16] AOT employs restricted abstraction schemata to avoid such paradoxes.[17]

In 2007, Zalta and Branden Fitelson introduced the term computational metaphysics to describe the implementation and investigation of formal, axiomatic metaphysics in an automated reasoning environment.[18][19]

Round square cupola

In ontology, the round square cupola is an example from Quine's 1948 paper On What There Is of the dual copula strategy used in reference to the problem of nonexistent objects.[20] Please note that the subject of this example is a type of architectural structure. A copula, on the other hand, is the linguistic concept as discussed in this article.

A similar example to Quine's cupola was the round square. The issue arose, most notably, between the theories of contemporary philosophers Alexius Meinong (see Meinong's 1904 book Investigations in Theory of Objects and Psychology)[21] and Bertrand Russell (see Russell's 1905 article "On Denoting").[22] Russell's critique of Meinong's theory of objects, also known as the Russellian view, became the established view on the problem of nonexistent objects.[23]

In late modern philosophy, the concept of the "square circle" (Template:Lang-de) had also been discussed before in Gottlob Frege's The Foundations of Arithmetic (1884).[24]

The dual copula strategy

The strategy employed is the dual copula strategy,[3] also known as the dual predication approach,[25] which is used to make a distinction between relations of properties and individuals. It entails creating a sentence that is not supposed to make sense by forcing the term "is" into ambiguous meaning.

The dual copula strategy was originally brought to prominence in contemporary philosophy by Ernst Mally.[26][27] Other proponents of this approach include: Héctor-Neri Castañeda, William J. Rapaport, and Edward N. Zalta.[4]

By borrowing Zalta's notational method (Fb stands for b exemplifies the property of being F; bF stands for b encodes the property of being F), and using a revised version of Meinongian object theory which makes use of a dual copula distinction (MOTdc), we can say that the object called "the round square" encodes the property of being round, the property of being square, all properties implied by these, and no others.[3] But it is true that there are also infinitely many properties being exemplified by an object called the round square (and, really, any object)—e.g. the property of not being a computer, and the property of not being a pyramid. Note that this strategy has forced "is" to abandon its predicative use, and now functions abstractly.

When one now analyzes the round square cupola using the MOTdc, one will find that it now avoids the three common paradoxes: (1) The violation of the law of noncontradiction, (2) The paradox of claiming the property of existence without actually existing, and (3) producing counterintuitive consequences. Firstly, the MOTdc shows that the round square does not exemplify the property of being round, but the property of being round and square. Thus, there is no subsequent contradiction. Secondly, it avoids the conflict of existence/non-existence by claiming non-physical existence: by the MOTdc, it can only be said that the round square simply does not exemplify the property of occupying a region in space. Finally, the MOTdc avoids counterintuitive consequences (like a 'thing' having the property of nonexistence) by stressing that the round square cupola can be said merely to encode the property of being round and square, not actually exemplifying it. Thus, logically, it does not belong to any set or class.

In the end, what the MOTdc really does is create a kind of object: a nonexistent object that is very different from the objects we might normally think of. Occasionally, references to this notion, while obscure, may be called "Meinongian objects."

The dual property strategy

Making use of the notion of "non-physically existent" objects is controversial in philosophy, and created the buzz for many articles and books on the subject during the first half of the 20th century. There are other strategies for avoiding the problems of Meinong's theories, but they suffer from serious problems as well.

First is the dual property strategy,[3] also known as the nuclear–extranuclear strategy.[3]

Mally introduced the dual property strategy,[28][29] but did not endorse it.[27] The dual property strategy was eventually adopted by Meinong.[27] Other proponents of this approach include: Terence Parsons and Richard Routley.[4]

According to Meinong, it is possible to distinguish the natural (nuclear) properties of an object, from its external (extranuclear) properties. Parsons identifies four types of extranuclear properties: ontological, modal, intentional, technical—however, philosophers dispute Parson's claims in number and kind. Additionally, Meinong states that nuclear properties are either constitutive or consecutive, meaning properties that are either explicitly contained or implied/included in a description of the object. Essentially the strategy denies the possibility for objects to have only one property, and instead they may have only one nuclear property. Meinong himself, however, found this solution to be inadequate in several ways and its inclusion only served to muddle the definition of an object.

The other worlds strategy

There is also the other worlds strategy.[3] Similar to the ideas explained with possible worlds theory, this strategy employs the view that logical principles and the law of contradiction have limits, but without assuming that everything is true. Enumerated and championed by Graham Priest, who was heavily influenced by Routley, this strategy forms the notion of "noneism". In short, assuming there exist infinite possible and impossible worlds, objects are freed from necessarily existing in all worlds, but instead may exist in impossible worlds (where the law of contradiction does not apply, for example) and not in the actual world. Unfortunately, accepting this strategy entails accepting the host of problems that come with it, such as the ontological status of impossible worlds.

See also

Notes

  1. ^ Zalta, Edward N. (2004). "The Theory of Abstract Objects". The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University. Retrieved July 18, 2020.
  2. ^ "An Introduction to a Theory of Abstract Objects (1981)". ScholarWorks@UMass Amherst. 2009. Retrieved July 21, 2020.
  3. ^ a b c d e f Reicher, Maria (2014). "Nonexistent Objects". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  4. ^ a b c Dale Jacquette, Meinongian Logic: The Semantics of Existence and Nonexistence, Walter de Gruyter, 1996, p. 17. Cite error: The named reference "Jacquette" was defined multiple times with different content (see the help page).
  5. ^ Alexius Meinong, "Über Gegenstandstheorie" ("The Theory of Objects"), in Alexius Meinong, ed. (1904). Untersuchungen zur Gegenstandstheorie und Psychologie (Investigations in Theory of Objects and Psychology), Leipzig: Barth, pp. 1–51.
  6. ^ a b Zalta (1983:xi).
  7. ^ Ernst Mally (1912), Gegenstandstheoretische Grundlagen der Logik und Logistik (Object-theoretic Foundations for Logics and Logistics), Leipzig: Barth, §§33 and 39.
  8. ^ Zalta (1983:33).
  9. ^ Zalta (1983:36).
  10. ^ Zalta (1983:35).
  11. ^ Romane Clark, "Not Every Object of Thought Has Being: A Paradox in Naive Predication Theory", Noûs 12(2) (1978), pp. 181–188.
  12. ^ William J. Rapaport, "Meinongian Theories and a Russellian Paradox", Noûs 12(2) (1978), pp. 153–80.
  13. ^ Adriano Palma, ed. (2014). Castañeda and His Guises: Essays on the Work of Hector-Neri Castañeda. Boston/Berlin: Walter de Gruyter, pp. 67–82, esp. 72.
  14. ^ Alan McMichael and Edward N. Zalta, "An Alternative Theory of Nonexistent Objects", Journal of Philosophical Logic 9 (1980): 297–313, esp. 313 n. 15.
  15. ^ Daniel Kirchner, "Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL", Archive of Formal Proofs, 2020[2017].
  16. ^ Zalta (2022:239): "Some non-core λ-expressions, such as those leading to the Clark/Boolos, McMichael/Boolos, and Kirchner paradoxes, will be provably empty."
  17. ^ Zalta (1983:158).
  18. ^ Edward N. Zalta and Branden Fitelson, "Steps Toward a Computational Metaphysics", Journal of Philosophical Logic 36(2) (April 2007): 227–247.
  19. ^ Jesse Alama, Paul E. Oppenheimer, Edward N. Zalta, "Automating Leibniz's Theory of Concepts", in A. Felty and A. Middeldorp (eds.), Automated Deduction – CADE 25: Proceedings of the 25th International Conference on Automated Deduction (Lecture Notes in Artificial Intelligence: Volume 9195), Berlin: Springer, 2015, pp. 73–97.
  20. ^ https://rintintin.colorado.edu/~vancecd/phil375/Quine.pdf/
  21. ^ Alexius Meinong, "Über Gegenstandstheorie" ("The Theory of Objects"), in Alexius Meinong, ed. (1904). Untersuchungen zur Gegenstandstheorie und Psychologie (Investigations in Theory of Objects and Psychology), Leipzig: Barth, pp. 1–51.
  22. ^ Bertrand Russell, "On Denoting," Mind, New Series, Vol. 14, No. 56. (Oct. 1905), pp. 479–493. online text, doi:10.1093/mind/XIV.4.479, JSTOR text.
  23. ^ Zalta 1983, p. 5.
  24. ^ Gottlob Frege, The Foundations of Arithmetic, Northwestern University Press, 1980[1884], p. 87.
  25. ^ Jacek Paśniczek, The Logic of Intentional Objects: A Meinongian Version of Classical Logic, Springer, 1997, p. 125.
  26. ^ Mally, Ernst, Gegenstandstheoretische Grundlagen der Logik und Logistik, Leipzig: Barth, 1912, §33.
  27. ^ a b c Ernst Mally – The Metaphysics Research Lab
  28. ^ Mally, Ernst. 1909. "Gegenstandstheorie und Mathematik", Bericht Über den III. Internationalen Kongress für Philosophie zu Heidelberg (Report of the Third International Congress of Philosophy, Heidelberg), 1–5 September 1908; ed. Professor Dr. Theodor Elsenhans, 881–886. Heidelberg: Carl Winter’s Universitätsbuchhandlung. Verlag-Nummer 850. Translation: Ernst Mally, "Object Theory and Mathematics", in: Jacquette, D., Alexius Meinong, The Shepherd of Non-Being (Berlin/Heidelberg: Springer, 2015), pp. 396–404, esp. 397.
  29. ^ Dale Jacquette, Meinongian Logic: The Semantics of Existence and Nonexistence, Walter de Gruyter, 1996, p. 16.

References

Further reading

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