FOM: Plural Quantification and Comprehension
Allen Hazen
a.hazen at philosophy.unimelb.edu.au
Thu Sep 14 03:48:34 EDT 2000
George Boolos's idea that (monadic) Second Order Logic can be
interpreted as NOT having ontological commitments to entities other than
those the First Order variables range over (so: no classes, properties,
concepts, or other "spooky" things) by thinking of Second Order variables
as plural terms has been mentioned. Three notes: VARIANT, HISTORY,
DISCLAIMER.
VARIANT: As Stephen Yablo points out, Boolos's idea gives only MONADIC
Second Order Logic, which is in general weaker than FULL S.O.L. (The
Monadic S.O. theory of the successor operation is decidable, the Full S.O.
theory of it is Second Order Arithmetic.) For some applications this
doesn't matter: if the domain the F.O. variables range over comes equipped
with a pairing operator (i.e. the non-logical vocabulary includes some
expression of pairing and projection operations), the distinction between
Monadic and Full S.O.L. collapses. Is there a general way of getting past
the restriction?
There would be if Boolos's interpretation could be extended to HIGHER
types, because then we could appeal to tricks like the Wiener-Kuratowski
analysis of the ordered pair. By work of Goodman and Quine (cf. "On
ordered pairs and relations" in Quine's "Selected Logic Papers," which
summarizes results published in the JSL in 1941,1945 & 1946), MONADIC Nth
Order Logic is equivalent to Full Nth Order Logic for N greater than or
equal to 4.
Here is a story. (I initially told it to a philosopher attracted to
Boolos's story as a reductio; to my dismay his response was "Great! The
full theory of types is ontologically free!") The plural ending in English
and many other terrestrial languages is not iterable, but in Old Classical
Martian it was: in addition to the simple plural, there were perplural
nouns, used to refer to multiple pluralities. To give a hint of the flavor
of Old Martian idiom, pretend that "people" is the plural, and "peoples"
the perplural, of "person." Then to say there are infinitely many people
we could say, Boolosianly, "There are some peoples such that for every
people among thems there are people among thems who include a person not
among the first people." Since it is hard to think of a principled reason
why Earth idiom should be a more trustworthy guide to metaphysics than
Martian, if Boolos's interpretation shows that S.O.L. is ontologically
noncommittal, Boolos-on-Mars shows that Third O.L. is ontologically
noncomittal. And so on up.
---
HISTORY: Boolos's suggestion was taken up with alacrity by David Lewis,
who uses it to argue for the philosophical legitimacy of his variant
axiomatization of set theory in his (FUN!! Read it! Teach Undergraduate
Students From It!) monograph "Parts of Classes." Footnotes to Lewis's book
indicate some of the pre-Boolos history of the the plural quantifier idea.
Let me add two more references. (i) Stanley Martens (sp?) used the idea in
his Cornell philosophy Ph.D. thesis in the 1970s to justify using S.O.L. in
the context of a "structuralist" approach to mathematics. (ii) Peter
Simons, "On understanding Lesniewski," in "History and Philosophy of
Logic," v. 3 (1982), pp. 165-191, argues that Lesniewski had something like
a plural interpretation of S.O.L., and even something like my "perplural"
interpretation of H.O.L.
---
DISCLAIMER: I myself think plural quantification is a lie and a cheat, an
attempt to disguise what are CLEARLY ontological commitments behind a
verbal smokescreen. I argued this in a paper in the "Australasian Journal
of Philosophy," v. 71 (1993), pp. 132-144... but apparently not everybody
was convinced.
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