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2Frege's logic, theorem, and foundations for arithmeticIn The Stanford Encyclopedia of Philosophy, The Metaphysics Research Lab. 2014.In this entry, Frege’s logic is introduced and described in some detail. It is shown how the Dedekind-Peano axioms for number theory can be derived from a consistent fragment of Frege’s logic, with Hume’s Principle replacing Basic Law V
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3Reply to P. Ebert and M. Rossberg’s Friendly Letter of ComplaintIn Alexander Hieke & Hannes Leitgeb (eds.), Reduction, abstraction, analysis: proceedings of the 31th International Ludwig Wittgenstein-Symposium in Kirchberg, 2008, De Gruyter. pp. 311-320. 2009.
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159Bennett and “proxy actualism”Philosophical Studies 142 (2): 277-292. 2009.Karen Bennett has recently argued that the views articulated by Linsky and Zalta (Philos Perspect 8:431–458, 1994) and (Philos Stud 84:283–294, 1996) and Plantinga (The nature of necessity, 1974) are not consistent with the thesis of actualism, according to which everything is actual. We present and critique her arguments. We first investigate the conceptual framework she develops to interpret the target theories. As part of this effort, we question her definition of ‘proxy actualism’. We then d…Read more
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248A defense of contingent logical truthsPhilosophical Studies 157 (1): 153-162. 2012.A formula is a contingent logical truth when it is true in every model M but, for some model M , false at some world of M . We argue that there are such truths, given the logic of actuality. Our argument turns on defending Tarski’s definition of truth and logical truth, extended so as to apply to modal languages with an actuality operator. We argue that this extension is the philosophically proper account of validity. We counter recent arguments to the contrary presented in Hanson’s ‘Actuality, …Read more
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521What is neologicism?Bulletin of Symbolic Logic 12 (1): 60-99. 2006.In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic used in the reduction, (2) allow the addition of anal…Read more
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281Naturalized platonism versus platonized naturalismJournal of Philosophy 92 (10): 525-555. 1995.In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a prio…Read more
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140Is Lewis a meinongian?Australasian Journal of Philosophy 69 (4). 1991.The views of David Lewis and the Meinongians are both often met with an incredulous stare. This is not by accident. The stunned disbelief that usually accompanies the stare is a natural first reaction to a large ontology. Indeed, Lewis has been explicitly linked with Meinong, a charge that he has taken great pains to deny. However, the issue is not a simple one. "Meinongianism" is a complex set of distinctions and doctrines about existence and predication, in addition to the famously large o…Read more
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249In defense of the contingently nonconcretePhilosophical Studies 84 (2-3): 283-294. 1996.In "Actualism or Possibilism?" (Philosophical Studies, 84 (2-3), December 1996), James Tomberlin develops two challenges for actualism. The challenges are to account for the truth of certain sentences without appealing to merely possible objects. After canvassing the main actualist attempts to account for these phenomena, he then criticizes the new conception of actualism that we described in our paper "In Defense of the Simplest Quantified Modal Logic" (Philosophical Perspectives 8: Philosoph…Read more
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554In defense of the simplest quantified modal logicPhilosophical Perspectives 8 431-458. 1994.The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which …Read more
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171Steps Toward a Computational MetaphysicsJournal of Philosophical Logic 36 (2): 227-247. 2007.In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in PROVER9 (a first-order automated reasoning system which is the successor to OTTER). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in PRO…Read more
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45Mathematical PluralismNoûs 58 (2): 306-332. 2024.Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach t…Read more
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33On Anselm’s Ontological Argument in Proslogion IIHistory of Philosophy & Logical Analysis 25 (2): 327-351. 2021.Formulations of Anselm’s ontological argument have been the subject of a number of recent studies. We examine these studies in light of Anselm’s text and (a) respond to criticisms that have surfaced in reaction to our earlier representations of the argument, (b) identify and defend a more refined representation of Anselm’s argument on the basis of new research, and (c) compare our representation of the argument, which analyzes that than which none greater can be conceived as a definite descripti…Read more
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10Lambert, Mally, and the Principle of IndependenceGrazer Philosophische Studien 26 (1): 447-495. 1985.In a recent book, K. Lambert argues that philosophers should adopt Mally's Principle of Independence (the principle that an object can have properties even though it lacks being of any kind) by abandoning a constraint on true predications, namely, that all of the singular terms in a true predication denote objects which have being. The constraint may be abandoned either by supposing there is a true predication in which one of the terms denotes a beingless object (Meinong) or by supposing there i…Read more
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41Frege's Logic, Theorem, and Foundations for ArithmeticStanford Encyclopedia of Philosophy. 2010.This entry explains Frege's Theorem by using the modern notation of the predicate calculus. Frege's Theorem is that the Dedekind-Peano axioms for number theory are derivable from Hume's Principle, given the axioms and rules of second-order logic. Frege's methodology for defining the natural numbers and for the derivation of the Dedekind-Peano axioms are sketched in some detail.
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58A (Leibnizian) Theory of ConceptsHistory of Philosophy & Logical Analysis 3 (1): 137-183. 2000.Three different notions of concepts are outlined: one derives from Leibniz, while the other two derive from Frege. The Leibnizian notion is the subject of his "calculus of concepts" (which is really an algebra). One notion of concept from Frege is what we would call a "property", so that when Frege says "x falls under the concept F", we would say "x instantiates F" or "x exemplifies F". The other notion of concept from Frege is that of the notion of sense, which played various roles within Fre…Read more
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226Mechanizing principia logico-metaphysica in functional type-theoryReview of Symbolic Logic 13 (1): 206-218. 2018.Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects.This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply…Read more
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97Unifying Three Notions of ConceptsTheoria 87 (1): 13-30. 2019.In this presentation, I first outline three different notions of concepts: one derives from Leibniz, while the other two derive from Frege. The Leibnizian notion is the subject of his “calculus of concepts” (which is really an algebra). One notion of concept from Frege is what we would call a “property”, so that when Frege says “x falls under the concept F”, we would say “x instantiates F” or “x exemplifies F”. The other notion of concept from Frege is that of the notion of sense, which played v…Read more
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81Computer Science and Metaphysics: A Cross-FertilizationOpen Philosophy 2 (1): 230-251. 2019.Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the b…Read more
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33Mechanizing principia logico-metaphysica in functional type theoryReview of Symbolic Logic 1-13. 2019.Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects. This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deepl…Read more
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616Automating Leibniz’s Theory of ConceptsIn Felty Amy P. & Middeldorp Aart (eds.), Automated Deduction – CADE 25: Proceedings of the 25th International Conference on Automated Deduction (Lecture Notes in Artificial Intelligence: Volume 9195), Berlin: Springer, Springer. pp. 73-97. 2015.Our computational metaphysics group describes its use of automated reasoning tools to study Leibniz’s theory of concepts. We start with a reconstruction of Leibniz’s theory within the theory of abstract objects (henceforth ‘object theory’). Leibniz’s theory of concepts, under this reconstruction, has a non-modal algebra of concepts, a concept-containment theory of truth, and a modal metaphysics of complete individual concepts. We show how the object-theoretic reconstruction of these components o…Read more
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78Mathematical descriptionsPhilosophical Studies 176 (2): 473-481. 2019.In this paper, the authors briefly summarize how object theory uses definite descriptions to identify the denotations of the individual terms of theoretical mathematics and then further develop their object-theoretic philosophy of mathematics by showing how it has the resources to address some objections recently raised against the theory. Certain ‘canonical’ descriptions of object theory, which are guaranteed to denote, correctly identify mathematical objects for each mathematical theory T, ind…Read more
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83The Stanford Encyclopedia of Philosophy: A Developed Dynamic Reference WorkMetaphilosophy 33 (1‐2): 210-228. 2003.The present information explosion on the World Wide Web poses a problem for the general public and the members of an academic discipline alike, of how to find the most authoritative, comprehensive, and up-to-date information about an important topic. At the Stanford Encyclopedia of Philosophy (SEP), we have since 1995 been developing and implementing the concept of a dynamic reference work (DRW) to provide a solution to these problems, while maintaining free access for readers. A DRW is much mor…Read more
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167Object Theory and Modal MeinongianismAustralasian Journal of Philosophy 95 (4): 761-778. 2017.In this paper, we compare two theories, modal Meinongianism and object theory, with respect to several issues that have been discussed recently in the literature. In particular, we raise some objections for MM, undermine some of the objections that its defenders raise for OT, and we point out some virtues of the latter with respect to the former.
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33Referring to Fictional CharactersDialectica 57 (2): 243-254. 2003.In this paper, the author replies to a question raised about theories of nonexistent objects. The question concerns the way names of fictional characters, when analyzed as names which denote nonexistent objects, acquire their denotations. Since nonexistent objects cannot causally interact with existent objects, it is thought that we cannot appeal to a‘dubbing’or a‘baptism’. The question is, therefore, what is the starting point of the chain? The answer is that storytellings are to be thought of …Read more
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266A Nominalist's Dilemma and its SolutionPhilosophia Mathematica 13 (3): 294-307. 2005.Current versions of nominalism in the philosophy of mathematics have the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni's _Deflating Existential Consequence_ has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In this paper, we argue that, as it stands, Azzouni's proposal does not yet succeed. It faces a dilemma to the effect that either th…Read more
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21An Introduction to a Theory of Abstract ObjectsDissertation, University of Massachusetts Amherst. 1981.An axiomatic theory of abstract objects is developed and used to construct models of Plato's Forms, Leibniz's Monads, Possible Worlds, Frege's Senses, stories, and fictional characters. The theory takes six primitive metaphysical notions: object ; n-place relations ,G,...); x,...x exemplify F x...x); x exists ; it is necessary that "); and x encodes F "). Properties and propositions are one place and zero place relations, respectively.objects are objects which necessarily fail to exist E!x"). Th…Read more
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