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168Automating 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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1Science provides us with representations of atoms, elementary particles, polymers, populations, genetic trees, economies, rational decisions, aeroplanes, earthquakes, forest fires, irrigation systems, and the world’s climate. It's through these representations that we learn about the world. This entry explores various different accounts of scientific representation, with a particular focus on how scientific models represent their target systems. As philosophers of science are increasingly acknow…Read more
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60Frege's theorem and foundations for arithmeticIn Ed Zalta (ed.), Stanford Encyclopedia of Philosophy, Stanford Encyclopedia of Philosophy. 2012.The principal goal of this entry is to present Frege's Theorem (i.e., the proof that the Dedekind-Peano axioms for number theory can be derived in second-order logic supplemented only by Hume's Principle) in the most logically perspicuous manner. We strive to present Frege's Theorem by representing the ideas and claims involved in the proof in clear and well-established modern logical notation. This prepares one to better prepared to understand Frege's own notation and derivations, and read Fre…Read more
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210In Defence of the Law of Non-ContradictionIn Graham Priest, Jc Beall & Bradley P. Armour-Garb (eds.), The law of non-contradiction : new philosophical essays, Oxford University Press. 2004.The arguments of the dialetheists for the rejection of the traditional law of noncontradiction are not yet conclusive. The reason is that the arguments that they have developed against this law uniformly fail to consider the logic of encoding as an analytic method that can resolve apparent contradictions. In this paper, we use Priest [1995] and [1987] as sample texts to illustrate this claim. In [1995], Priest examines certain crucial problems in the history of philosophy from the point of vi…Read more
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212In defense of the law of noncontradictionIn Graham Priest, Jc Beall & Bradley P. Armour-Garb (eds.), The law of non-contradiction : new philosophical essays, Oxford University Press. 2004.The arguments of the dialetheists for the rejection of the traditional law of noncontradiction are not yet conclusive. The reason is that the arguments that they have developed against this law uniformly fail to consider the logic of encoding as an analytic method that can resolve apparent contradictions. In this paper, we use Priest [1995] and [1987] as sample texts to illustrate this claim. In [1995], Priest examines certain crucial problems in the history of philosophy from the point of view …Read more
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1In Defence of the Law of Non-ContradictionIn Graham Priest, Jc Beall & Bradley P. Armour-Garb (eds.), The law of non-contradiction : new philosophical essays, Oxford University Press. 2004.
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29Lambert, Mally and the Principle of IndependenceGrazer Philosophische Studien 25 (1): 447-459. 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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71Lambert, mally, and the principle of independenceGrazer Philosophische Studien 25 (1): 447-459. 1985.In this paper, the author analyzes critically some of the ideas found in Karel Lambert's recent book, Meinong and the Principle of Independence (Cambridge: Cambridge University Press, 1983). Lambert attempts to forge a link between the ideas of Meinong and the free logicians. The link comes in the form of a principle which, Lambert says, these philosophers adopt, namely, Mally's Principle of Independence, which Mally himself later abandoned. Instead of following Mally and attempting to formulate…Read more
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2Models in scienceIn Ed Zalta (ed.), Stanford Encyclopedia of Philosophy, Stanford Encyclopedia of Philosophy. 2012.
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1The ergodic hierarchyIn Ed Zalta (ed.), Stanford Encyclopedia of Philosophy, Stanford Encyclopedia of Philosophy. 2012.The so-called ergodic hierarchy (EH) is a central part of ergodic theory. It is a hierarchy of properties that dynamical systems can possess. Its five levels are egrodicity, weak mixing, strong mixing, Kolomogorov, and Bernoulli. Although EH is a mathematical theory, its concepts have been widely used in the foundations of statistical physics, accounts of randomness, and discussions about the nature of chaos. We introduce EH and discuss its applications in these fields.
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10Metaphysics of Routley StarAustralasian Journal of Logic 21 (4): 141-176. 2024.This paper investigates two forms of the Routley star operation, one in Routley & Routley 1972 and the other in Leitgeb 2019. We use the background of object theory to define both forms of the Routley star operation and show how the basic principles governing both forms become derivable and need not be stipulated. Since no mathematics is assumed by our background formalism, the existence of the Routley star image s* of a situation s is therefore guaranteed not by set theory but by a theory of ab…Read more
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45Number Theory and Infinity Without MathematicsJournal of Philosophical Logic 53 (5): 1161-1197. 2024.We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of ‘ordinary’ mathematics? (2) How should Frege’s theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won’t give rise to different numbers at different worlds? (3) Can one reconstruct Frege’s theory of numbers in a non-modal setting without mathematical primitives suc…Read more
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880Stanford Encyclopedia of Philosophy (edited book)Stanford Encyclopedia of Philosophy. 2012.The Stanford Encyclopedia of Philosophy organizes scholars from around the world in philosophy and related disciplines to create and maintain an up-to-date reference work. Co-Principal Editors: Edward N. Zalta and Uri Nodelman
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12Frege's logic, theorem, and foundations for arithmeticIn Ed Zalta (ed.), Stanford Encyclopedia of Philosophy, Stanford Encyclopedia of Philosophy. 2012.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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4Reply 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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179Bennett 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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272A 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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568What 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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309Naturalized 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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162Is 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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593In 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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267In 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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190Steps 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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82Mathematical 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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74On 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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22Lambert, 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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258Mechanizing 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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122Unifying 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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Metaphysics and Epistemology |
Philosophy of Mathematics |
Formal Philosophy |
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