Paul Oppenheimer | Stanford University (original) (raw)
Papers by Paul Oppenheimer
Physical review, Nov 1, 1992
Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard mo... more Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard model on a 4 x 4 square lattice at U/t = 0.5, 4, and 10, and even number of electrons with Filling factors n ranging from a quarter filling up to half filling. We compare the ground-state energy and density of states at U/t = 0.5 and 4 with second-order perturbation theory in U/t in the paramagnetic phase, and find that while the agreement is reasonable at U/t = 0.5, it becomes worse as the perturbatively determined (i.e., using Stoner s criterion) boundary of the paramagnetic to spin-density-wave instability is approached. In the strong coupling regime (U/t = 10), we find reasonable agreement between the density of states of the Hubbard and the t-J model especially for low doping fractions. In general, we find that at half filling the filled states are separated from the empty states by a gap. At U/t = 10, the density of states shows two bands clearly separated by a Mott-Hubbard gap of order U.
Australasian Journal of Philosophy, Jun 1, 2011
The authors investigate the ontological argument computationally. The premises and conclusion of ... more The authors investigate the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine prover9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. prover9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the argument into better focus. Also, the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a 'diagonal argument' and, moreover, one used to establish a positive conclusion rather than a paradox.
Grazer Philosophische Studien, Aug 13, 1980
Studia Neoaristotelica, 2007
Philosophical Perspectives, 1991
Saint Anselm of Canterbury offered several arguments for the existence of God. We examine the fam... more Saint Anselm of Canterbury offered several arguments for the existence of God. We examine the famous ontological argument in Proslogium ii. Many recent authors have interpreted this argument as a modal one. 1 But we believe that Jonathan Barnes has argued persuasively that Anselm's argument is not modal. 2 Even if one were to construe the word 'can' in the definite description 'that than which none greater can be conceived' in terms of metaphysical possibility, the logic of the ontological argument itself doesn't include inferences based on this modality. In this paper, we develop a reading of Anselm's Proslogium that contains no modal inferences. Rather, the argument turns on the difference between saying that there is such a thing as x and saying that x has the property of existence. We formally represent the claim that there is such a thing as x * Originally published in Philosophical Perspectives 5: The Philosophy of Religion,
Journal of Logic and Computation, Jun 18, 2010
Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing ... more Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an oversimplification: one can't assimilate predication to functional application. * This paper is forthcoming in the Journal of Logic and Computation. The authors would like to thank Uri Nodelman for his observations on the first draft of this paper. We'd also like to thank Bernard Linsky for observations on the second draft, which led us to reconceptualize the significance of our results within a more historical context. We'd also like to acknowledge one of the referees of this journal, whose comments led us to clarify and better document the claims in the paper.
Springer eBooks, 2015
Our computational metaphysics group describes its use of automated reasoning tools to study Leibn... more 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 of Leibniz's theory can be represented for investigation by means of automated theorem provers and finite model builders. The fundamental theorem of Leibniz's theory is derived using these tools.
History of Philosophy & Logical Analysis, 2021
Formulations of Anselm’s ontological argument have been the subject of a number of recent studies... more 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 description, to a representation that analyzes it as an arbitrary name.
Australasian Philosophical Review, 2020
Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of... more Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of Anselm's argument for the existence of God. We showed that there is a valid argument for God's existence in Proslogion II. However, in that paper, we deliberately decided not to include a discussion and analysis of the soundness of the argument. In these afterthoughts, we shall take up this question. We plan to argue for the following:
It is commonly believed that it makes no difference whether one starts with relational types or f... more It is commonly believed that it makes no difference whether one starts with relational types or functional types in formulating type theory, since one can either start with relations as primitive and represent functions as relations or start with functions as primitive and represent relations as functions. It is also commonly believed that the formula-based logic of relational type theory is equivalent to the term-based logic of functional type theory. However, in this paper, the authors argue that there are systems with logics that can be ...
Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of... more Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of Anselm’s argument for the existence of God. We showed that there is a valid argument for God’s existence in Proslogion II. However, in that paper, we deliberately decided not to include a discussion and analysis of the soundness of the argument. In these afterthoughts, we shall take up this question. We plan to argue for the following: 1. Anselm’s argument for Premise 1 is not valid. This casts doubt on the truth of Premise 1 of the ontological argument. 2. If Premise 1 is revised so as to be clearly true, and the rest of the ontological argument is modified so as to preserve validity with the revised Premise 1, then the resulting argument is sound but doesn’t have the conclusion that Anselm wishes to establish. Our analysis in what follows appeals to the theory of abstract objects (Zalta 1983) and to the distinction between exemplifying and encoding a property fundamental to that theor...
Journal of Logic and Computation
In this paper, we present a formalism for handling polysemy in spatial expressions based on super... more In this paper, we present a formalism for handling polysemy in spatial expressions based on supervaluation semantics called standpoint semantics for polysemy (SSP). The goal of this formalism is, given a prepositional phrase, to define its possible spatial interpretations. For this, we propose to characterize spatial prepositions by means of a triplet langle\langle langleimage schema, semantic feature, spatial axis$\rangle $. The core of SSP is predicate grounding theories, which are formulas of a first-order language that define a spatial preposition through the semantic features of its trajector and landmark. Precisifications are also established, which are a set of formulae of a qualitative spatial reasoning formalism that aims to provide the spatial characterization of the trajector with respect to the landmark. In addition to the theoretical model, we also present results of a computational implementation of SSP for the preposition ‘in’.
Though Frege was interested primarily in reducing mathemat- ics to logic, he succeeded in reducin... more Though Frege was interested primarily in reducing mathemat- ics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By con- trast, Whitehead & Russell reduced an important part of mathe- matics to logic by defining functions in terms of relations (using the definite description operator). We argue that there
Lecture Notes in Computer Science, 2015
Our computational metaphysics group describes its use of automated reasoning tools to study Leibn... more 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 of Leibniz's theory can be represented for investigation by means of automated theorem provers and finite model builders. The fundamental theorem of Leibniz's theory is derived using these tools.
The Review of Metaphysics, Jul 22, 2011
PDC Homepage Home » Products » Purchase. LOGIN; PRODUCTS: All Products; Online Resources; Journal... more PDC Homepage Home » Products » Purchase. LOGIN; PRODUCTS: All Products; Online Resources; Journals & Series; Digital Media; Books & Reference Works. MEMBERSHIPS: Societies & Associations; Conference Registrations. E-COLLECTION: About; Alphabetically; By Category; By Type; Price Lists; Terms and Conditions. SERVICES: Conference Exhibits; Conference Registrations; Electronic Publishing; Journal Advertising; Mailing Lists; Marketing ...
Grazer Philosophische Studien, 1980
Philosophical Perspectives, 1991
Physical review, Nov 1, 1992
Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard mo... more Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard model on a 4 x 4 square lattice at U/t = 0.5, 4, and 10, and even number of electrons with Filling factors n ranging from a quarter filling up to half filling. We compare the ground-state energy and density of states at U/t = 0.5 and 4 with second-order perturbation theory in U/t in the paramagnetic phase, and find that while the agreement is reasonable at U/t = 0.5, it becomes worse as the perturbatively determined (i.e., using Stoner s criterion) boundary of the paramagnetic to spin-density-wave instability is approached. In the strong coupling regime (U/t = 10), we find reasonable agreement between the density of states of the Hubbard and the t-J model especially for low doping fractions. In general, we find that at half filling the filled states are separated from the empty states by a gap. At U/t = 10, the density of states shows two bands clearly separated by a Mott-Hubbard gap of order U.
Australasian Journal of Philosophy, Jun 1, 2011
The authors investigate the ontological argument computationally. The premises and conclusion of ... more The authors investigate the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine prover9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. prover9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the argument into better focus. Also, the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a 'diagonal argument' and, moreover, one used to establish a positive conclusion rather than a paradox.
Grazer Philosophische Studien, Aug 13, 1980
Studia Neoaristotelica, 2007
Philosophical Perspectives, 1991
Saint Anselm of Canterbury offered several arguments for the existence of God. We examine the fam... more Saint Anselm of Canterbury offered several arguments for the existence of God. We examine the famous ontological argument in Proslogium ii. Many recent authors have interpreted this argument as a modal one. 1 But we believe that Jonathan Barnes has argued persuasively that Anselm's argument is not modal. 2 Even if one were to construe the word 'can' in the definite description 'that than which none greater can be conceived' in terms of metaphysical possibility, the logic of the ontological argument itself doesn't include inferences based on this modality. In this paper, we develop a reading of Anselm's Proslogium that contains no modal inferences. Rather, the argument turns on the difference between saying that there is such a thing as x and saying that x has the property of existence. We formally represent the claim that there is such a thing as x * Originally published in Philosophical Perspectives 5: The Philosophy of Religion,
Journal of Logic and Computation, Jun 18, 2010
Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing ... more Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an oversimplification: one can't assimilate predication to functional application. * This paper is forthcoming in the Journal of Logic and Computation. The authors would like to thank Uri Nodelman for his observations on the first draft of this paper. We'd also like to thank Bernard Linsky for observations on the second draft, which led us to reconceptualize the significance of our results within a more historical context. We'd also like to acknowledge one of the referees of this journal, whose comments led us to clarify and better document the claims in the paper.
Springer eBooks, 2015
Our computational metaphysics group describes its use of automated reasoning tools to study Leibn... more 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 of Leibniz's theory can be represented for investigation by means of automated theorem provers and finite model builders. The fundamental theorem of Leibniz's theory is derived using these tools.
History of Philosophy & Logical Analysis, 2021
Formulations of Anselm’s ontological argument have been the subject of a number of recent studies... more 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 description, to a representation that analyzes it as an arbitrary name.
Australasian Philosophical Review, 2020
Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of... more Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of Anselm's argument for the existence of God. We showed that there is a valid argument for God's existence in Proslogion II. However, in that paper, we deliberately decided not to include a discussion and analysis of the soundness of the argument. In these afterthoughts, we shall take up this question. We plan to argue for the following:
It is commonly believed that it makes no difference whether one starts with relational types or f... more It is commonly believed that it makes no difference whether one starts with relational types or functional types in formulating type theory, since one can either start with relations as primitive and represent functions as relations or start with functions as primitive and represent relations as functions. It is also commonly believed that the formula-based logic of relational type theory is equivalent to the term-based logic of functional type theory. However, in this paper, the authors argue that there are systems with logics that can be ...
Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of... more Our 1991 paper on the logic of the ontological argument contained an analysis of the structure of Anselm’s argument for the existence of God. We showed that there is a valid argument for God’s existence in Proslogion II. However, in that paper, we deliberately decided not to include a discussion and analysis of the soundness of the argument. In these afterthoughts, we shall take up this question. We plan to argue for the following: 1. Anselm’s argument for Premise 1 is not valid. This casts doubt on the truth of Premise 1 of the ontological argument. 2. If Premise 1 is revised so as to be clearly true, and the rest of the ontological argument is modified so as to preserve validity with the revised Premise 1, then the resulting argument is sound but doesn’t have the conclusion that Anselm wishes to establish. Our analysis in what follows appeals to the theory of abstract objects (Zalta 1983) and to the distinction between exemplifying and encoding a property fundamental to that theor...
Journal of Logic and Computation
In this paper, we present a formalism for handling polysemy in spatial expressions based on super... more In this paper, we present a formalism for handling polysemy in spatial expressions based on supervaluation semantics called standpoint semantics for polysemy (SSP). The goal of this formalism is, given a prepositional phrase, to define its possible spatial interpretations. For this, we propose to characterize spatial prepositions by means of a triplet langle\langle langleimage schema, semantic feature, spatial axis$\rangle $. The core of SSP is predicate grounding theories, which are formulas of a first-order language that define a spatial preposition through the semantic features of its trajector and landmark. Precisifications are also established, which are a set of formulae of a qualitative spatial reasoning formalism that aims to provide the spatial characterization of the trajector with respect to the landmark. In addition to the theoretical model, we also present results of a computational implementation of SSP for the preposition ‘in’.
Though Frege was interested primarily in reducing mathemat- ics to logic, he succeeded in reducin... more Though Frege was interested primarily in reducing mathemat- ics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By con- trast, Whitehead & Russell reduced an important part of mathe- matics to logic by defining functions in terms of relations (using the definite description operator). We argue that there
Lecture Notes in Computer Science, 2015
Our computational metaphysics group describes its use of automated reasoning tools to study Leibn... more 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 of Leibniz's theory can be represented for investigation by means of automated theorem provers and finite model builders. The fundamental theorem of Leibniz's theory is derived using these tools.
The Review of Metaphysics, Jul 22, 2011
PDC Homepage Home » Products » Purchase. LOGIN; PRODUCTS: All Products; Online Resources; Journal... more PDC Homepage Home » Products » Purchase. LOGIN; PRODUCTS: All Products; Online Resources; Journals & Series; Digital Media; Books & Reference Works. MEMBERSHIPS: Societies & Associations; Conference Registrations. E-COLLECTION: About; Alphabetically; By Category; By Type; Price Lists; Terms and Conditions. SERVICES: Conference Exhibits; Conference Registrations; Electronic Publishing; Journal Advertising; Mailing Lists; Marketing ...
Grazer Philosophische Studien, 1980
Philosophical Perspectives, 1991