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140Mathematical RelativismHistory and Philosophy of Logic 10 (1): 53-65. 1989.We set out a doctrine about truth for the statements of mathematics—a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics—and argue that this doctrine, which we shall call ‘mathematical relativism’, withstands objections better than do other non-realist accounts.
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198Does the law of excluded middle require bivalence?Erkenntnis 31 (1). 1989.Determining whether the law of excluded middle requires bivalence depends upon whether we are talking about sentences or propositions. If we are talking about sentences, neither side has a decisive case. If we are talking of propositions, there is a strong argument on the side of those who say the excluded middle does require bivalence. I argue that all challenges to this argument can be met.
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262Expressions and TokensAnalysis 41 (4): 181-187. 1981.The purpose of this paper is to uncover and correct several confusions about expressions, tokens and the relations between them that crop up in even highly sophisticated writing about language and logic.
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157Williams’ 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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103What is an infinite expression?Philosophia 16 (1): 45-60. 1986.The following syllogism is considered: a string is not an expression unless it is tokenable; no one can utter, write, or in anyway token an infinite string; so no infinite string is an expression. The second premise is rejected. But the tokenability of an infinite sentence is not sufficient for it being an infinite expression. A further condition is that no finite sentence expresses that sentence’s truth-conditions. So it is an open question whether English contains infinite expressions.
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Chapter 8: Thesis OnePoznan Studies in the Philosophy of the Sciences and the Humanities 90 215-240. 2006.
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164Thompson Clarke and the problem of other mindsInternational Journal of Philosophical Studies 13 (1): 1-14. 2005.The force of sceptical inquiries into out knowledge of other people is a paradigm of the force that philosophical views can have. Sceptical views arise out of philosophical inquiries that are identical in all major respects with inquiries that we employ in ordinary cases. These inquiries employ perfectly mundane methods of making and assessing claims to know. This paper tries to show that these inquiries are conducted in cases that lack certain contextual ingredients found in ordinary cases. The…Read more
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257There Is A Problem with Substitutional QuantificationTheoria 68 (1): 4-12. 2002.Whereas arithmetical quantification is substitutional in the sense that a some-quantification is true only if some instance of it is true, it does not follow (and, in fact, is not true) that an account of the truth-conditions of the sentences of the language of arithmetic can be given by a substitutional semantics. A substitutional semantics fails in a most fundamental fashion: it fails to articulate the truth-conditions of the quantifications with which it is concerned. This is what is defended…Read more
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408A Wittgensteinian Philosophy of MathematicsLogic and Logical Philosophy 15 (2): 55-69. 2005.Three theses are gleaned from Wittgenstein’s writing. First, extra-mathematical uses of mathematical expressions are not referential uses. Second, the senses of the expressions of pure mathematics are to be found in their uses outside of mathematics. Third, mathematical truth is fixed by mathematical proof. These theses are defended. The philosophy of mathematics defined by the three theses is compared with realism, nominalism, and formalism.
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Chapter 6: Arithmetic and NecessityPoznan Studies in the Philosophy of the Sciences and the Humanities 90 159-182. 2006.
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63Roman Suzuko on Situational IdentitySorites 15 42-49. 2004.This paper gives a semantical account for the (i)ordinary propositional calculus, enriched with quantifiers binding variables standing for sentences, and with an identity-function with sentences as arguments; (ii)the ordinary theory of quantification applied to the special quantifiers; and (iii)ordinary laws of identity applied to the special function. The account includes some thoughts of Roman Suszko as well as some thoughts of Wittgenstein's Tractatus.
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93Absurdity and spanningPhilosophia 2 (3): 227-238. 1972.On the basis of observations J. J. C. Smart once made concerning the absurdity of sentences like 'The seat of the bed is hard', a plausible case can be made that there is little point to developing a theory of types, particularly one of the sort envisaged by Fred Sommers. The authors defend such theories against this objection by a partial elucidation of the distinctions between the concepts of spanning and predicability and between category mistakenness and absurdity in general. The argument su…Read more
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113Nagel, Internalism, and RelativismJournal of Philosophical Research 1990 310-319. 1990.In this paper we give (1) a new interpretation to Nagel’s THE POSSIBILITY OF ALTRUISM and (2) use that interpretation to show that internalism and anti-realism are compatible, despite appearances to the contrary.
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146Indenumerability and substitutional quantificationNotre Dame Journal of Formal Logic 23 (4): 358-366. 1982.We here establish two theorems which refute a pair of what we believe to be plausible assumptions about differences between objectual and substitutional quantification. The assumptions (roughly stated) are as follows: (1) there is at least one set d and denumerable first order language L such that d is the domain set of no interpretation of L in which objectual and substitutional quantification coincide. (2) There exist interpreted, denumerable, first order languages K with indenumerable domains…Read more
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192Is heaven a possible world?International Journal for Philosophy of Religion 12 (1). 1981.The goal of theodicy is to show how God could create our world with all its evil. This paper argues that the theodicist can achieve her goal only if she gives up one of these three propositions: (1) evil does not exist in heaven; (2) heaven is better than the present world; (3) heaven is a possible world. Second, it is argued that the theodicist can reject (3) without giving up her belief that heaven exists, so that (3) is her best alternative.
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1308Has Nozick Justified the State?Pacific Philosophical Quarterly 62 (4): 411-415. 1981.In ANARCY, STATE AND UTOPIA Robert Nozick says that the fundamental question of political philosophy, one that precedes questions about how the state should be organized, is whether there should be any state at all. In the first part of his book he attempts to justify the state. We argue that he is not successful.
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138Is English inconsistent?Erkenntnis 15 (3). 1980.The significance of the semantical paradoxes for natural languages is examined. If Tarski’s reflections on the issue are correct, English is inconsistent. Paul Ziff responds to Tarskian reflections by arguing to the conclusion that no natural language is or can be inconsistent. The authors reject Ziff’s argument, but they defend something similar to its conclusion: no language, natural or otherwise, is or can be inconsistent in the way that Tarski holds languages capable of formulating the Epime…Read more
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88Eternal sentencesAustralasian Journal of Philosophy 54 (1). 1976.The paper argues that two apparently attractive conceptions of an eternal sentence are defective. An alternative conception is presented which the authors think allows greater insight into the nature of semantic concepts.
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78A semantical account of the vicious circle principleNotre Dame Journal of Formal Logic 20 (3): 595-598. 1979.Here we give a semantical account of propositional quantification that is intended to formally represent Russell’s view that one cannot express a proposition about "all" propositions. According to the account the authors give, Russell’s view bears an interesting relation to the view that there are no sets which are members of themselves.
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758Pragmatics 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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53Dialogues Concerning Natural NumbersPeter Lang. 2009.Two philosophical theories, mathematical Platonism and nominalism, are the background of six dialogues in this book. There are five characters in these dialogues: three are nominalists; the fourth is a Platonist; the main character is somewhat skeptical on most issues in the philosophy of mathematics, and is particularly skeptical regarding the two background theories.
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57Can there be a Proof that Some Unprovable Arithmetic Sentence Is True?Dialectica 43 (3): 289-292. 1989.
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175The Tree Theory and IsomorphismAnalysis 41 (1): 6-11. 1980.A main thesis of Fred Sommers' type theory, is that an isomorphism exists between any natural language and the categories discriminated by that language. Here the author gives an explanation of what this claim comes to. And then it is argued that, so understood, the claim is incompatible with Zermelo-Fraenkel set theory. Finally, it is argued against trying to salvage the isomorphism thesis by appealing to some other set theory.
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130The 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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Chapter 2: Notes to GrundlagenPoznan Studies in the Philosophy of the Sciences and the Humanities 90 45-72. 2006.
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499Steiner 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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317Tarski 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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University of Nebraska, LincolnRetired faculty
Lincoln, Nebraska, United States of America