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**1 - 4**of**4**### Non-Standard Models of Arithmetic: a Philosophical and Historical perspective

, 2010

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### The Philosophical Review 116.3 (July 2007): 401–425 The Problem with Charlie: Some Remarks on Putnam

"... In his new paper, “Eligibility and Inscrutability, ” J. R. G. Williams presents a surprising new challenge to David Lewis ’ theory of interpretation. Although Williams frames this challenge primarily as a response to Lewis ’ criticisms of Putnam’s model-theoretic argument, the challenge itself goes ..."

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In his new paper, “Eligibility and Inscrutability, ” J. R. G. Williams presents a surprising new challenge to David Lewis ’ theory of interpretation. Although Williams frames this challenge primarily as a response to Lewis ’ criticisms of Putnam’s model-theoretic argument, the challenge itself goes to the heart of Lewis’ own account of interpretation. Further, and leaving Lewis ’ project aside for a moment, Williams ’ argument highlights some important—and some fairly general—points concerning the relationship between model theory and semantic determinacy. In these remarks, I plan to do three things. First, I’ll provide a brief overview of the three arguments that I’ll be looking at: Putnam’s original model-theoretic argument, Lewis ’ response to this argument, and Williams ’ new model-theoretic argument. In giving this overview, I’m going to suppress almost all of the model-theoretic details, so that the arguments ’ philosophical structures come out more clearly. For similar reasons, I’ll feel free to shamelessly oversimplify the arguments. Where this oversimplification becomes too shameless, I’ll try to redeem my scholarly credentials in the footnotes. Second, I’ll lay out what I take to be the most obvious objection to Williams ’ new argument, and I’ll then look at several ways of modifying Williams ’ model theory so as to overcome this objection. Finally, I’ll step back and make a few remarks concerning the broader philosophical significance of model-theoretic

### In Dale Jacquette (ed), Philosophy of Logic: 485–518 The Mathematics of Skolem’s Paradox

"... presents a new proof of a model-theoretic 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 model-theoretic puzzle that has come to be known as “Skolem’s Paradox.” Over the yea ..."

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presents a new proof of a model-theoretic 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 model-theoretic 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 first-order 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 “solu-tions ” 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