[FOM] Understanding universal quantification
Steve Newberry
stevnewb at ix.netcom.com
Sat Feb 22 21:51:26 EST 2003
Peter,
Actually, the way quantification got *started* was as a purely *syntactical*
operation: The prefix generalization of the hitherto dyadic infix operators
of sum and product, disjunction and conjunction, to arbitrarily many operands,
was introduced by William Stanley Jevons in his "Principles of Science" (1874)
which also redefined "logical sum" from the 'exclusive-or' of Boole to the
'inclusive-or' and its attendant dual-symmetries, which survive to this
day. The
term "quantifier", and the use of (capital) sigma and pi for existential and
universal quantifiers was initiated in the works of Charles Saunders Peirce.
The *semantic* and ontological analyses of quantification appear to be of more
recent occurrence.
W.r.t. the ontological entailments of universal/existential quantification, I
can offer one simple notion which has helped me to resolve the apparent
dichotomy
between "actual infinity" and "potential infinity": namely that the
intuition of
potential infinity seems to be well captured by the notion of a
sequence S of
arbitrarily great extent, as e.g., <1,2,3, ... >, whereas the intuition of
actual
infinity seems to be adequately captured by the notion of the union (in the
sense
of Cantor) of the sequence, U(S). If we cook up a sort of inverse-union
operation,
say "disunion", symbolized by prefix ' * ' [asterisk] then we can "ping-pong"
back and forth between the two concepts in a manner which clearly indicates
that
the only difference between the two concepts is psychological, rather than
fundamentally ontological.
S =df= <1,2,3, ...> which is potential, incomplete; U(S) =df= { 1,2,3, ...
} which
is actual (completed); and *(U(S)) is once again <1,2,3, ...> .
That said, I'm confident that there will be some who will wish to argue the
point.
SN
At 04:45 PM 2/22/2003 -0800, you wrote:
>GR:
>
>It seems thus that universal
>quantification would not commit one to the assertion of the ontological
>existence of any totality (omega, power set).
>
>PA:
>Conceded for the sake of argument.
>
>GR:
>
>It is sometimes claimed that universal quantification is meaningful
>even without reference to the domain of quantification of the variable,
>the intended meaning of the quantification being conveyed by one's
>understanding of the introduction and elimination rules of a (natural
>deduction, sequents) logical system.
>
>PA:
>
>Well, no quantification without identity (Frege,Quine) and no identity
>without indiscernibility (Leibniz, Cantor, Frege), so quantification at
>least presupposes a prior quantization of the domain of discourse (as the
>old saying goes, why do you think they call it quantification . . .).
>However this nuance escapes formal notice (but not Geach's) so we (i.e.
>proof theorists) live in a world indiscernible from a world in which the
>claim you report is true.
>
>Quantification is fundamentally a semantic notion. All Kaplan aside, accept
>no syntactic surrogates my friend, for all theory is gray, but the tree of
>life is green.
>
>I hope that clears things up :-)
>
>
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