FOM: "Consistent":Three Questions.
Robert Tragesser
RTragesser at compuserve.com
Tue Dec 1 09:55:59 EST 1998
[Thanks to Davis, Tait, Poston, et al.; here I think is a
slightly better framing of my concerns.]
Three questions, one technical, two historical-philosophical,
about "consistency".
[1] TECHNICAL QUESTION.
Is there an example of, or can one prove the existence of, a
theory framed in a higher order classical logic [or at least a logic so
understood that the class of semantical logical consequences and the
class of syntactic consequences do not coincide; but hopefully the
logic and the theory would be nontrivial wrt my question] that is
(a) inconsistent,
(b) a contradiction is not derivable.
[2] HISTORICAL QUESTION 1.
In German one has, _konsistent_ and _widerspruchslos_. I'm
curious about the history of these terms, in the context of logic and/or
mathematical demonstration. In GA III, Hilbert uses the second. (Well,
I find _Widerspruchsfreiheit_ and _Widerspruchslosigkeit_; is there a
difference in connotation?) I don't find 'konsistent'.
Cantor speaks of "(in)konsistente Vielheit". In, e.g., _Cantor
an Dedekind 28 Juli 1899_, Cantor has Omega' as "keine konsistente
Vielheit" because it supports "ein Widerspruch". Is Cantor's use of
'konsistent' rather than 'widerspruchslos/frei' stylistic only, a
matter of diction?
In English one says "consistent" rather than
"contradiction-free" because of the awkwardness of the latter; but one
wonders if this doesn't sometimes conceal some important distinctions.
[3] HISTORICAL QUESTION 2.
This goes to the heart of my concern: a _reductio ad absurdu_m
argument in the primary sense shows a proposition to be false by showing
that it implies something "absurd". The latter meaning not only "false"
but also "not true to" or even "dissonant with" (etc.). It does seem
that "negation" in intuitionistic reasoning is most resonantly
understood in these terms (reducible to something intuitionistically
absurd; "false" for "absurd" won't do, of course). Here one then has a
use for "consistent"--true to intuitionistic reasoning -- which only by
slight of hand is explicable in terms of "contradiction-free".
With classical logic, one is perhaps even required to restrict
"absurdities" to contradictions; then of course one can use RAA
arguments as proofs rather than as disproofs only (it is only by a kind
of ruse that in intuitionistic reasoning what is essentially a disproof
of p is made to appear to be a proof of "-p".).
I am still wondering if for example supplying the upper
half-plane model for Bolyai-Lobach. geometry does not _fundamentally_
(from the point of view of the mathematical achievement or effect) show
B-L geometry to be "consistent" more in the sense of "consistent with
mathematics", that is, it does bring B-L geometry fully into the fold
of mathematics. Hanging out here (if I may speak colloquially) is the
thought that in a fundamental sense, consistency proofs ought to lend
mathematical signficance to a subject whose mathematical significance is
in doubt. (Whereas one can worry over the significance of just
supplying indirectly -- as in Go"del's completeness proof -- a model in
the natural numbers that has no striking mathematical significance.)
Of course, what I driving me is the great discomfort I've always
felt with mathematics that is too logic driven, perhaps because I place
a great value on understanding in some full or perhaps melodramatic
sense. (The line about depriving a mathematician of the law of the
excluded middle is like depriving a boxer of their fists has struck me
as issuing from the sort of person who goes to boxing matches in the
hopes of seeing a smashing knock-out punch, or listens to symphonies
impatient for "the good parts".)
Robert Tragesser
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