FOM: Re: : Universal Generalization (aka UI)
Dean.Buckner at btopenworld.com
Tue Feb 26 14:46:54 EST 2002
There's a background to my previous arguments on UG/UI.
Aristotelian logic does not anywhere contain the idea of a singular
proposition. Aristotle only mentions them at the
beginning, (De Interpretatione Chapter 7) and thereafter ignores them. His
scheme allows for only four kinds of proposition: universal affirmative
("All men are mortal"), particular affirmative ("Some men are
philosophers"), universal negative ("No philosophers are rich"),
particular negative ("Some men are not philosophers").
Thus it would have been absurd to prove a universal conclusion using an
intermediate step invoving particulars. The inference from "An A is a B"
and "a B is a C" proceeds directly: A is a logical part of B, which is a
logical part of C, so an A is a C. To be a mathematician is to be a man, to
be a man is to be mortal, so ...
See An. Post. 2.11 where he says the premisses of an argument are the
"cause" of the conclusion.
The syllogism "all men are mortal-Socrates is a man- So Socrates is
mortal" was not Aristotelian at all, but is from Sextus Empiricus. Sextus'
that if "all men are mortal" asserts that Socrates is mortal, then
conclusion contained in first premiss. If not, then we may want to
question the premiss. A development of this argument discussed in detail by
Mill in A System of Logic. The argument in my earlier posting was
different, but in the same spirit. My point was we cannot reason from
arbitrary objects or "singulars" to a general conclusion. We must
implicitly assume the
universal proposition that we conclude. If not, always open to reason that
the singular cases covered, are not ALL the cases.
I could make same objection to the "angel" argument. The angel examines
each object in the "domain" in turn. But how do we know s/he's covered
_every_ object? How do we know the subproof applies to ALL these things.
On the "re-usable template" argument, how do we know it is re-usable? How
do we know the template applies to all "objects" in the "domain"?
The ideas of Fitch and Thomason i think are old and date back to origin of
"natural deduction". I'm no expert, but see Jeff Pelletier's totally
excellent history of NDS on his website. Fitch's method he says is a
streamlined version of Jaskowski's, and Thomason's in turn a derivative of
( Pelletier, FJ , 1998, "A Brief History of Natural Deduction" in
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