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Symposi .urn: ON TH;; ONTOLOGICAL SIGNIFICANCE OF THE LoWENHEIM-SKOLEM THEOREM I share with the previous speaker the conviction that the Lowenheim-Skolem theorem has no direct philosophical implica- tions. This phrase should be clarified. What is implied is a propo- sition and to say there are philosophical implications implies that there are philosophical propositions. This runs counter to the idea that philosophy is an activity rather than a doctrine, an idea to which with reservations I subscribe. However part if not all of this activity consists in the assertion of propositions, which are not however philosophical propositions in themselves, but become philosophical in virtue of being asserted in the course of philosoph- ical activity. Hence no proposition has philosophical implications in the strict sense, but perhaps every proposition may with pro- priety be asserted in the course of philosophical activity. Almost any proposition may I suppose initiate philosophical activity, and I take the invitation to contribute the present paper as a request to perform a philosophical activity initiated (after those introduc- tory remarks) by the assertion of the Lowenheim-Skolem theorem. The assertions made by me subsequently to this assertion I shall call indirect implications of the Lowenheim-Skolem theorem, using the word 'implication' in its colloquial rather than its technical sense. My initial remark that the theorem has no direct philosophi- cal implications is therefore a direct consequence of my view that philosophy is an activity rather than a doctrine. I do not maintain that philosophy is wholly or primarily an activity of clarification. In particular I cannot see that clarifica- tion is the principal goal of ethics, though it might be an important 57
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58 SIGNIFICANCE OF LijWENHEIM-SKOLEM THEOREM instrument in achieving that goal. None the less clarification is part of philosophy, or at least the clarification of certain issues is. Much of the activity which I will perform in this paper will be clarificatory, that is, it will be devoted to stating in non-technical terms what the Lowenheim-Skolem theorem is. Why is this a philosophical activity? Would a clarification of say the binomial theorem be philosophical? Clearly not; more exactly, it seems highly dubious that the assertion of the binomial theorem could profitably initiate a philosophical discourse, except perhaps by way of illustration of some general aspect of mathematics for which purpose a good many other theorems would have served equally well. The reason why the menheim-Skolem theorem seems a fruitful proposition with which to begin a philosophical discourse, while the binomial theorem does not, is that we are inclined to ask "What does the Liiwenheim-Skolem theorem really mean?" while we are not inclined to ask "What does the binomial theorem really mean?" I take such questions seriously. A question is an expression of intellectual anxiety and an answer is an attempt at resolution of that anxiety.
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