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Abstract object theory
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<h2 id="overview">Overview</h2> <p class="note">See also: <a href="/facts/Dual_copula_strategy/HfJtTD6S">Dual copula strategy</a></p> <p><i>Abstract Objects: An Introduction to Axiomatic Metaphysics</i> (1983) is the title of a publication by Edward Zalta that outlines abstract object theory. </p><p>AOT is a <a href="/facts/Dual_predication_approach/HfJtTD6S">dual predication approach</a> (also known as "dual copula strategy") to abstract objects<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> influenced by the contributions of <a href="/facts/Alexius_Meinong/AIrFc5vh">Alexius Meinong</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and his student <a href="/facts/Ernst_Mally/fdubLx8q">Ernst Mally</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> On Zalta's account, there are two modes of <a href="/facts/Predicate_(mathematical_logic)/5WazAvKF">predication</a>: some objects (the ordinary <a href="/facts/Abstract_and_concrete/uNshak6S">concrete</a> ones around us, like tables and chairs) <i>exemplify</i> properties, while others (abstract objects like numbers, and what others would call "<a href="/facts/Nonexistent_object/6gkyd8ND">nonexistent objects</a>", like the <a href="/facts/Round_square_copula/HfJtTD6S">round square</a> and the mountain made entirely of gold) merely <i>encode</i> them.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> 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.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> For every set of properties, there is exactly one object that encodes exactly that set of properties and no others.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> This allows for a <a href="/facts/Formal_system/KaeDAjK7">formalized</a> <a href="/facts/Ontology/QJKTsEIJ">ontology</a>. </p><p>A notable feature of AOT is that several notable paradoxes in naive predication theory (namely <a href="/facts/Romane_Clark/171oLcaM">Romane Clark</a>'s paradox undermining the earliest version of <a href="/facts/H%C3%A9ctor-Neri_Casta%C3%B1eda/ry9sDLsF">Héctor-Neri Castañeda</a>'s <a href="/facts/Guise_theory/ry9sDLsF">guise theory</a>,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Alan McMichael's paradox,<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> and Daniel Kirchner's paradox)<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> do not arise within it.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> AOT employs <a href="/facts/Range_of_quantification/bgmvwxoo">restricted</a> <a href="/facts/Abstraction/p7l9eHGd">abstraction</a> <a href="/facts/Axiom_schema/vKNTz5IT">schemata</a> to avoid such paradoxes.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p><p>In 2007, Zalta and <a href="/facts/Branden_Fitelson/orbE0JdR">Branden Fitelson</a> introduced the term computational metaphysics to describe the implementation and investigation of formal, axiomatic metaphysics in an <a href="/facts/Automated_reasoning/x5EdkKoU">automated reasoning</a> environment.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Abstract_and_concrete/uNshak6S">Abstract and concrete</a></li> <li><a href="/facts/Abstractionism_(philosophy_of_mathematics)/7srXNWGa">Abstractionism (philosophy of mathematics)</a></li> <li><a href="/facts/Algebra_of_concepts/cTc2er83">Algebra of concepts</a></li> <li><a href="/facts/Mathematical_universe_hypothesis/b7JIz2kF">Mathematical universe hypothesis</a></li> <li><a href="/facts/Modal_Meinongianism/hyJ5YHbQ">Modal Meinongianism</a></li> <li><a href="/facts/Modal_neo-logicism/YvHs6X3E">Modal neo-logicism</a></li> <li><a href="/facts/Object_of_the_mind/6gkyd8ND">Object of the mind</a></li> <li><a href="/facts/Objective_precision/rY38FB2v">Objective precision</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Zalta, Edward N. (1983). <a href="https://mally.stanford.edu/abstract-objects.pdf"><i>Abstract Objects: An Introduction to Axiomatic Metaphysics</i></a> (PDF). Dordrecht: D. Reidel.</li> <li>Zalta, Edward N. (1988). <a href="https://mally.stanford.edu/intensional-logic.pdf"><i>Intensional Logic and the Metaphysics of Intentionality</i></a> (PDF). Cambridge, MA: The MIT Press/Bradford Books.</li> <li>Zalta, Edward N. (February 10, 1999). <a href="http://doors.stanford.edu/principia-1999-02-10.pdf"><i>Principia Metaphysica</i></a> (PDF). Center for the Study of Language and Information, Stanford University.</li> <li>Kirchner, Daniel; Benzmüller, Christoph; Zalta, Edward N. (March 2020). <a href="https://mally.stanford.edu/Papers/mechanizing-principia.pdf">"Mechanizing <i>Principia Logico-Metaphysica</i> in Functional Type Theory"</a> (PDF). <i>Review of Symbolic Logic</i>. 13 (1): 206–218.</li> <li>Zalta, Edward N. (May 22, 2024). <a href="https://mally.stanford.edu/principia.pdf"><i>Principia Logico-Metaphysica</i></a> (PDF). Center for the Study of Language and Information, Stanford University.</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Kirchner, Daniel (2021). <a href="https://d-nb.info/1262308674/34"><i>Computer-Verified Foundations of Metaphysics and an Ontology of Natural Numbers in Isabelle/HOL</i></a> (PhD thesis). Free University of Berlin.</li> <li>Zalta, Edward N. (May 2020). <a href="https://mally.stanford.edu/Papers/typed-object-theory.pdf">"Typed object theory"</a> (PDF). In Falguera López, José Luis; Martínez-Vidal, Concha (eds.). <a href="https://books.google.com/books?id=2pbiDwAAQBAJ"><i>Abstract objects: For and against</i></a>. Synthese library: Studies in epistemology, logic, methodology, and philosophy of science. Vol. 422. Cham, Switzerland: <a href="/facts/Springer_Nature/TpTgApAh">Springer Nature</a>. pp. 59–88. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-030-38242-1_4">10.1007/978-3-030-38242-1_4</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-030-38241-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1129207159">1129207159</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="http://mally.stanford.edu/theory.html" target="_blank">http://mally.stanford.edu/theory.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Zalta, Edward N. (1981). An Introduction to a Theory of Abstract Objects (Thesis). UMass Amherst. doi:10.7275/f32y-fm90. hdl:20.500.14394/12282. <a href="https://scholarworks.umass.edu/bitstreams/d2a3ed8c-6f57-43b0-a55e-ba91ae0dc839/download" target="_blank">https://scholarworks.umass.edu/bitstreams/d2a3ed8c-6f57-43b0-a55e-ba91ae0dc839/download</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Dale Jacquette, Meinongian Logic: The Semantics of Existence and Nonexistence, Walter de Gruyter, 1996, p. 17. <a href="/wiki/Dale_Jacquette" target="_blank">/wiki/Dale_Jacquette</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="/wiki/Alexius_Meinong" target="_blank">/wiki/Alexius_Meinong</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Zalta 1983, p. xi. - Zalta, Edward N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics (PDF). Dordrecht: D. Reidel. <a href="https://mally.stanford.edu/abstract-objects.pdf" target="_blank">https://mally.stanford.edu/abstract-objects.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Mally, Ernst (1912). Gegenstandstheoretische Grundlagen der Logik und Logistik [Object-theoretic Foundations for Logics and Logistics] (PDF) (in German). Leipzig: Barth. §§33 and 39. <a href="https://mally.stanford.edu/mally-book/ObjectTheoreticFoundationsOfLogic2.pdf" target="_blank">https://mally.stanford.edu/mally-book/ObjectTheoreticFoundationsOfLogic2.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Zalta 1983, p. xi. - Zalta, Edward N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics (PDF). Dordrecht: D. Reidel. <a href="https://mally.stanford.edu/abstract-objects.pdf" target="_blank">https://mally.stanford.edu/abstract-objects.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Zalta 1983, p. 33. - Zalta, Edward N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics (PDF). Dordrecht: D. Reidel. <a href="https://mally.stanford.edu/abstract-objects.pdf" target="_blank">https://mally.stanford.edu/abstract-objects.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Zalta 1983, p. 36. - Zalta, Edward N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics (PDF). Dordrecht: D. Reidel. <a href="https://mally.stanford.edu/abstract-objects.pdf" target="_blank">https://mally.stanford.edu/abstract-objects.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Zalta 1983, p. 35. - Zalta, Edward N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics (PDF). Dordrecht: D. Reidel. <a href="https://mally.stanford.edu/abstract-objects.pdf" target="_blank">https://mally.stanford.edu/abstract-objects.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Clark, Romane (1978). "Not Every Object of Thought Has Being: A Paradox in Naive Predication Theory". Noûs. 12 (2): 181–188. JSTOR 2214691. <a href="/wiki/Romane_Clark" target="_blank">/wiki/Romane_Clark</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Rapaport, William J. (1978). "Meinongian Theories and a Russellian Paradox". Noûs. 12 (2): 153–180. <a href="/wiki/William_J._Rapaport" target="_blank">/wiki/William_J._Rapaport</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>* Palma, Adriano, ed. (2014). Castañeda and his guises: Essays on the work of Hector-Neri Castañeda. Philosophische Analyse / Philosophical Analysis (in Breton). Boston/Berlin: De Gruyter. pp. 67–82, esp. 72. ISBN 978-1-61451-663-7. <a href="978-1-61451-663-7" target="_blank">978-1-61451-663-7</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>McMichael, Alan; Zalta, Edward N. (1980). "An alternative theory of nonexistent objects". Journal of Philosophical Logic. 9 (3): 297–313, esp. p. 313 n. 15. doi:10.1007/BF00248396. ISSN 0022-3611. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Daniel Kirchner, "Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL", Archive of Formal Proofs, 2017. <a href="http://isa-afp.org/entries/PLM.html" target="_blank">http://isa-afp.org/entries/PLM.html</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Zalta 2024, p. 253: "Some non-core λ-expressions, such as those leading to the Clark/Boolos, McMichael/Boolos, and Kirchner paradoxes, will be provably empty." - Zalta, Edward N. (May 22, 2024). Principia Logico-Metaphysica (PDF). Center for the Study of Language and Information, Stanford University. <a href="https://mally.stanford.edu/principia.pdf" target="_blank">https://mally.stanford.edu/principia.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Zalta 1983, p. 158. - Zalta, Edward N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics (PDF). Dordrecht: D. Reidel. <a href="https://mally.stanford.edu/abstract-objects.pdf" target="_blank">https://mally.stanford.edu/abstract-objects.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Fitelson, Branden; Zalta, Edward N. (March 14, 2007). "Steps toward a computational metaphysics" (PDF). Journal of Philosophical Logic. 36 (2): 227–247. doi:10.1007/s10992-006-9038-7. ISSN 0022-3611. <a href="https://mally.stanford.edu/Papers/computation.pdf" target="_blank">https://mally.stanford.edu/Papers/computation.pdf</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>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. <a href="/wiki/Edward_N._Zalta" target="_blank">/wiki/Edward_N._Zalta</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> </ol>