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99Did the greeks discover the irrationals?Philosophy 74 (2): 169-176. 1999.A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.
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128Steiner versus Wittgenstein: Remarks on Differing Views of Mathematical TruthTheoria 20 (3): 347-352. 2005.Mark Steiner criticizes some remarks Wittgenstein makes about Gödel. Steiner takes Wittgenstein to be disputing a mathematical result. The paper argues that Wittgenstein does no such thing. The contrast between the realist and the demonstrativist concerning mathematical truth is examined. Wittgenstein is held to side with neither camp. Rather, his point is that a realist argument is inconclusive
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94The Lessons of the LiarTheory and Decision 11 (1): 55-70. 1979.The paper argues that the liar paradox teaches us these lessons about English. First, the paradox-yielding sentence is a sentence of English that is neither true nor false in English. Second, there is no English name for any such thing as a set of all and only true sentences of English. Third, ‘is true in English’ does not satisfy the axiom of comprehension.
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72Remarks on Peano ArithmeticRussell: The Journal of Bertrand Russell Studies 20 (1): 27-32. 2000.Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is…Read more
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163Tarski and Proper ClassesAnalysis 40 (4): 6-11. 1980.In this paper the authors argue that if Tarski’s definition of truth for the calculus of classes is correct, then set theories which assert the existence of proper classes (classes which are not the member of anything) are incorrect.
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Chapter 10: Thesis ThreePoznan Studies in the Philosophy of the Sciences and the Humanities 90 254-283. 2006.
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10Notes and DiscussionsDialectica 57 (3): 315-322. 2003.This paper seeks to explain why the argument from analogy seems strong to an analogist such as Mill and weak to the skeptic. The inference from observed behavior to the existence of feelings, sensations, etc., in other subjects is justified, but its justification depends on taking observed behavior and feelings, sensations, and so on, to be not merely correlated, but connected. It is claimed that this is what Mill had in mind
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142Quantifying over the realsSynthese 101 (1). 1994.Peter Geach proposed a substitutional construal of quantification over thirty years ago. It is not standardly substitutional since it is not tied to those substitution instances currently available to us; rather, it is pegged to possible substitution instances. We argue that (i) quantification over the real numbers can be construed substitutionally following Geach's idea; (ii) a price to be paid, if it is that, is intuitionism; (iii) quantification, thus conceived, does not in itself relieve us …Read more
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4Nagel, Internalism, and RelativismJournal of Philosophical Research 16 309-319. 1991.In this paper we (1) give a new interpretation to Thomas Nagel’s The Possibility of Altruism, and (2) use that account to show how internalism and anti-relativism are compatible, despite appearances to the contrary.
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60Assertion and beliefPhilosophical Studies 17 (5). 1966.This paper is written in opposition of various antecedent discussions of Moore’s paradox. It concludes that one cannot make an honest and primary truth-claim by producing ‘p, but I believe not-p’.
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398Bound Variables and Schematic LettersLogique Et Analyse 95 (95): 425-429. 1981.The paper purports to show, against Quine, that one can construct a language , which results from the extension of the theory of truth functions by introducing sentence letter quantification. Next a semantics is provided for this language. It is argued that the quantification is neither substitutional nor requires one to consider the sentence letters as taking entities as values.
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28Is Any Economic System Unjust?Southwest Philosophy Review 5 (2): 17-23. 1989.The morality of an economic system characterized as an Adam Smith type system is compared with one characterized by central planning. A prima facie case is made that, while the latter has attributes that satisfy a necessary condition for having moral attributes, the former does not and, as a result, has no moral attributes. But then a deeper look at the situation reveals that the directed systems really do not satisfy the necessary condition either. Both the directed and undirected systems end u…Read more
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80Williams’ Definition of ‘X is true’Analysis 30 (3): 95-97. 1970.C. J. F, Williams proposed ‘for some p ___ states that p & p’ as a satisfactory analysis of ‘___ is true’. This paper takes issue with this claim.
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100Do we need quantification?Notre Dame Journal of Formal Logic 25 (4): 289-302. 1984.The standard response is illustrated by E, J. Lemmon's claim that if all objects in a given universe had names and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex proposition. It is because these two requirements are not always met that we need universal quantification. This paper is partly in agreement with Lemmon and partly in disagreement. From the point of view of syntax and semantics we can replace a universal prop…Read more
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88Philosophical Logic: An Introduction to Advanced Topicscontinuum. 2010.This title introduces students to non-classical logic, syllogistic, to quantificational and modal logic. The book includes exercises throughout and a glossary of terms and symbols. Taking students beyond classical mathematical logic, "Philosophical Logic" is a wide-ranging introduction to more advanced topics in the study of philosophical logic. Starting by contrasting familiar classical logic with constructivist or intuitionist logic, the book goes on to offer concise but easy-to-read introduct…Read more
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Intensionality and Truth: An Essay on the Philosophy of A. N. PriorStudia Logica 63 (2): 287-290. 1999.
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195True Propositions: A Reply to C.J.F. WilliamsAnalysis 32 (3): 101-106. 1972.This paper replies to points Williams makes to his reply to Sayward’s criticism of Williams’s proposal of ‘for some p ___ states that p & p’ as an analysis of ‘___ is true’.
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654Why Substitutional Quantification Does Not Express ExistenceTheory and Decision 50 67-75. 1987.Fundamental to Quine’s philosophy of logic is the thesis that substitutional quantification does not express existence. This paper considers the content of this claim and the reasons for thinking it is true.
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380Pragmatics and indexicalityPragmatics Microfiche 1 (4). 1975.A conception of pragmatics distinguishes pragmatics from semantics proper in terms of indexicality: semantics is conceived as the quest for a truth definition for languages without indexical expressions; pragmatics is conceived as a quest for a truth definition for languages with indexical expressions. I argue that indexicality is not a feature that can be used to capture anything like what Morris and Carnap had in mind.
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19Do Moral Explanations Matter?Philosophy Research Archives 14 137-142. 1988.In a recent paper Nicholas Sturgeon claims moral explanations constitute one area of disagreement between moral realists and noncognitivists. The correctness of such explanation is consistent with moral realism but not with noncognitivism. Does this difference characterize other anti-realist views? I argue that it does not. Moral relativism is a distinct anti-realist view. And the correctness of moral explanations is consistent with moral relativism.
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39Can There be a Proof that an Unprovable Sentence of Arithmetic is True?Dialectica 43 (43): 289-292. 1989.Various authors of logic texts are cited who either suggest or explicitly state that the Gödel incompleteness result shows that some unprovable sentence of arithmetic is true. Against this, the paper argues that the matter is one of philosophical controversy, that it is not a mathematical or logical issue.
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672The Internal/External QuestionGrazier Philosophishe Studien 47 31-41. 1994.For Rudolf Carnap the question ‘Do numbers exist?’ does not have just one sense. Asked from within mathematics, it has a trivial answer that could not possibly divide philosophers of mathematics. Asked from outside of mathematics, it lacks meaning. This paper discusses Carnap ’s distinction and defends much of what he has to say
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133Prior’s Theory of TruthAnalysis 47 (2): 83-87. 1987.This paper is a critical exposition of Prior’s theory of truth as expressed by the following truth locutions: (1) ‘it is true that’ prefixed to sentences; (2) ‘true proposition’; (3) true belief’, ‘true assertion’, ‘true statement’, etc.; (4) ‘true sentence’.
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97Relativism and ontologyPhilosophical Quarterly 37 (148): 278-290. 1987.This paper deals with the question of whether there is objectivist truth about set-theoretic matters. The dogmatist and skeptic agree that there is such truth. They disagree about whether this truth is knowable. In contrast, the relativist says there is no objective truth to be known. Two versions of relativism are distinguished in the paper. One of these versions is defended.
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97A conversation about numbersPhilosophia 29 (1-4): 191-209. 2002.This is a dialogue in which five characters are involved. Various issues in the philosophy of mathematics are discussed. Among those issues are these: numbers as abstract objects, our knowledge of numbers as abstract objects, a proof as showing a mathematical statement to be true as opposed to the statement being true in virtue of having a proof.
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University of Nebraska, LincolnRetired faculty
Lincoln, Nebraska, United States of America