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Putnam's ModelTheoretic Argument Reconstructed
 The Journal of Philosophy
, 1999
"... Among those addressed by Putnam's modeltheoretic argument it is common opinion that the argument is invalid because questionbegging. If the standard analysis of the argument is along the right lines, then what has been called the `just more theory move' is to be held responsible for this. In the p ..."
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Among those addressed by Putnam's modeltheoretic argument it is common opinion that the argument is invalid because questionbegging. If the standard analysis of the argument is along the right lines, then what has been called the `just more theory move' is to be held responsible for this. In the present paper, an alternative reading of Putnam's argument is o#ered that makes the `just more theory move' come out perfectly legitimate, and the argument as a whole valid. 1 Introduction Metaphysical realism has been the main subject of the critical part of Putnam's work for over twenty years now. At the heart of this doctrine, as presented by Putnam, is a thesis which we shall here call Methodological Fallibilism (MF). What it says is that even an epistemically ideal theory may fall short of truth, where the notion of truth involved is that of Correspondence Truth (CT). 1 Note that MF is not merely about epistemically ideal theories. The claim is that no empirical theory, no matter how...
Reflections on Skolem's Paradox
"... In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the Lö ..."
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In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the LöwenheimSkolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...
Models and recursivity
, 2002
"... It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term ..."
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It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number. ” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward LöwenheimSkolem theorem, most theorists agree that the number theoretic version does not have skeptical consequences about the reference of “natural number ” analogous to the ‘relativity ’ Skolem claimed pertains to notions such as “uncountable ” and “cardinal. ” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extramathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we
Mind 118 (October 2009): 1043–1059 Beth’s Theorem and Deflationism
"... In 1999, Jeffrey Ketland published a paper which posed a series of technical problems for deflationary theories of truth. Ketland argued that deflationism is incompatible with standard mathematical formalizations of truth and that alternate deflationary formalizations are unable to explain some cent ..."
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In 1999, Jeffrey Ketland published a paper which posed a series of technical problems for deflationary theories of truth. Ketland argued that deflationism is incompatible with standard mathematical formalizations of truth and that alternate deflationary formalizations are unable to explain some central uses of the truth predicate in mathematics. He also used Beth’s definability theorem to argue that, contrary to deflationists’ claims, the Tschema cannot provide an ‘implicit definition ’ of truth. In this paper, I want to challenge this final argument. Whatever other faults deflationism may have, the Tschema does provide an implicit definition of the truth predicate. Or so, at any rate, I shall argue. 1 1 Notation and preliminaries Let me start by setting out some context. Let L = {0, 1, +, ×, <} be the language of firstorder arithmetic and let PA be the theory based on the firstorder Peano axioms. 2 Our goal is to add a new truth predicate 1 The paper at issue here is Ketland 1999. For reasons of space, the present paper will focus fairly tightly on Ketland’s discussion of implicit definition. I should note, however, that some of Ketland’s other arguments have also sparked considerable discussion in the literature. The reader interested in this broader discussion is advised to start with the survey in Shapiro 2002. They should then turn to Field 1999, Tennant 2002, and Tennant 2005 for some responses to Ketland’s arguments, and to Ketland 2005 for Ketland’s replies.