Re: Aristotle's horse in De Morgan's harness.
From: Owen (oorionus_at_yahoo.com)
Date: 11/20/04
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Date: Sat, 20 Nov 2004 14:32:30 -0500
"Kenneth Doyle" <nobody@notmail.com> wrote in message
news:Xns95A6F3D3DDE46nobodynotmailcom@61.9.191.5...
> http://www.math.psu.edu/simpson/papers/philmath/node9.html
>
> "... the 19th century logician Augustus DeMorgan noted that the
inference
> all horses are animals,
> therefore, the head of a horse is the head of an animal
>
> is beyond the reach of Aristotelean logic. Yet this same inference may
be
> paraphrased as 'if all horses are animals, then for all x, if x is the
head
> of some horse then x is the head of some animal.'"
"Kenneth Doyle" <nobody@notmail.com> wrote in message
news:Xns95A6F3D3DDE46nobodynotmailcom@61.9.191.5...
> http://www.math.psu.edu/simpson/papers/philmath/node9.html
>
> "... the 19th century logician Augustus DeMorgan noted that the
inference
> all horses are animals,
> therefore, the head of a horse is the head of an animal
>
> is beyond the reach of Aristotelean logic. Yet this same inference may
be
> paraphrased as 'if all horses are animals, then for all x, if x is the
head
> of some horse then x is the head of some animal.'"
" .. and this corresponds to a logically valid formula
of the predicate calculus. Here H, A, R denote ``is a horse'', ``is an
animal'', ``is the head of'', respectively. Thus DeMorgan's conclusion is
indeed a logical consequence of his premise."
-----------------------------------------------------------------------
Apparently Bertrand Russell did not agree.
See: Principia Mathematica, page 291.
>... the 19th century logician Augustus DeMorgan noted that the
inference
> all horses are animals,
> therefore, the head of a horse is the head of an animal
> is beyond the reach of Aristotelean logic.
"It must be confessed that this was a merit in Aristotle's logic, since
the proposed inference is fallacious without the added premiss "E!(the head
of the horse in question)". E.g. it does not hold for an oyster or a hydra,
But with the addition E!R'y , the above gives an important and common type
of asyllogistic inference."
Owen
> Lacking an academic grounding in aristotlean logic, I'm guessing that in
> this context, "beyond the reach" in the above quote is merely a matter
of
> form. Certainly, the inference seems valid but the aristotlean
syllogism
> has no power to express it(?). If someone out there happens to have
done a
> recent essay on this, I'd be interested to read it.
>
> I'm struggling at the moment, to aquire the discipline to ignore what
seems
> intuitively obvious in order to learn the formal limitations of logical
> systems; it's an interesting exercise.
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