Re: Moore on Skolem's Paradox

What's paradoxical about that, you say? Perhaps this. Common sense
suggests that

(*) The meanings of two terms are different iff their extensions are
demonstrably different.

For example, the word "pissed" in English means "drunk", in American
English it means "angry". When Tom Wolfe (an American writer) has an
English character say "Steiner just came over and asked me where you
were. Not out of curiousity either. He acted extremely pissed". It
reads oddly to an English reader, who imagines Steiner was acting
drunk, whereas (if you understand the context) it means he was acting
angry. But the difference, though confusing, can be demonstrated in
some way. For example, point to a happily drunk person and say (in
English) "he is pissed". There are similarly amusing differences
between the English meaning of "pants" (the objects worn under
trousers) and the American meaning (the trousers themselves). B.t.w.
what do Americans call the objects worn under trousers?

Even supposing the difference is not normally observable - imagine
the substance we call "water" has a slightly different chemical
composition in America than in England - such a difference could be
demonstrated in some way, by chemical reduction or by electron
microscopy, perhaps.

If by contrast the difference in extension is not in principle
demonstrable, then the two terms (surely) have the same meaning. Yet
this is what Skolem's theorem seems to deny. We are to suppose that
certain terms in two apparently identical languages (like American and
English) have different extensions (i.e. they apply to different
mathematical objects), but that the difference is not demonstrable in
any way, because demonstration in mathematics is a method of proof, and
all methods of proof will give the same result in both languages.

That is too much to swallow. It is not a paradox, rather it is a
conflict (rather like the notion of a set having the same number as its
proper subset) between our ordinary intuitions, and what mathematics
says.

Actually, you can see why Wittgenstein hated mathematical logic so
much. It is essential to his relativism that identity of usage =
identity of meaning. That's the entire point of whole sections of the
Philosophical Investigations.

.

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