Symposi
.urn:
ON
TH;;
ONTOLOGICAL SIGNIFICANCE OF
THE LoWENHEIMSKOLEM THEOREM
I share with the previous speaker the conviction that the
LowenheimSkolem 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 LowenheimSkolem theorem.
The assertions made by me subsequently to this assertion
I shall
call indirect implications of the LowenheimSkolem 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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SIGNIFICANCE OF LijWENHEIMSKOLEM 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 nontechnical
terms what the LowenheimSkolem 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 menheimSkolem 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 LiiwenheimSkolem 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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 Spring '07
 FITELSON
 Logic, Meaning of life, Mathematical logic, Formal language, Model theory, order functional calculus

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