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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 the ..."
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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...
In Dale Jacquette (ed), Philosophy of Logic: 485–518 The Mathematics of Skolem’s Paradox
"... presents a new proof of a modeltheoretic result originally due to Leopold Löwenheim and then discusses some philosophical implications of this result. In the course of this latter discussion, the paper introduces a modeltheoretic puzzle that has come to be known as “Skolem’s Paradox.” Over the yea ..."
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presents a new proof of a modeltheoretic result originally due to Leopold Löwenheim and then discusses some philosophical implications of this result. In the course of this latter discussion, the paper introduces a modeltheoretic puzzle that has come to be known as “Skolem’s Paradox.” Over the years, Skolem’s Paradox has generated a fairly steady stream of philosophical discussion; nonetheless, the overwhelming consensus among philosophers and logicians is that the paradox doesn’t constitute a mathematical problem (i.e., it doesn’t constitute a real contradiction). Further, there’s general agreement as to why the paradox doesn’t constitute a mathematical problem. By looking at the way firstorder structures interpret quantifiers—and, in particular, by looking at how this interpretation changes as we move from structure to structure—we can give a technically adequate “solution ” to Skolem’s Paradox. So, whatever the philosophical upshot of Skolem’s Paradox may be, the mathematical side of Skolem’s Paradox seems to be relatively straightforward. In this paper, I challenge this common wisdom concerning Skolem’s Paradox. While I don’t argue that Skolem’s Paradox constitutes a genuine mathematical problem (it doesn’t), I do argue that standard “solutions ” to the paradox are technically inadequate. Even on the mathematical side, Skolem’s Paradox is more complicated—and quite a bit more interesting—than it’s usually taken to be. Further, because philosophical
Generalized Logical Consequence: Making Room for Induction in the Logic of Science
"... We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of ..."
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We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures in contrast to the permissibility of classical firstorder logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented.