FOM: Axiom of Extensionality

[Dean Buckner]

Parson's (Fine's) three principles are not that difficult to accept.  After
all,

(1) If Clarendon was one of Godolphin and Hyde, then necessarily so, I
suppose.  And (2) if not, then necessarily not.  This is a simple extension
of modal ideas about identity: if Cicero = Tully, then necessarily so.  At
least, if we can make any sense of this talk about "necessity", I'm not so
sure we can.  And (3) if there were such people as Godolphin and Hyde, and
if Clarendon was one of them, then (clearly) there was such a person as
Clarendon.  Indeed, if Clarendon was a politician, or if he was an Anglican
.. or .. then there was such a person as Clarendon!

It depends how you interpret these bits of symbolism (symbols only have a
meaning if you can translate them into real language).   I translate
Parson's "e" is "is one of", and his signs for a "set" as a plural term
signifying a bunch of people.  This is what 19C writers called a "definite
set" or enumeration.

"The class definite is an enumeration of actual individuals, as the peers of
the realm, the oceans of the globe ... The class indefinite is unenumerated.
Such classes are stars, planets, men poets ... (Bain)

But it's in the latter (indefinite) sense that we must read Mill's "the
objects which compose any given class are perpetually fluctuating".  If the
class just was the objects, it couldn't fluctuate at all.  What would "it"
be, to fluctuate?  A favourite 19C example (used in the 20C by Russell in
the famous essay) was "the centre of the material universe", which is an
indefinite singleton, always changing.  Or how about "the population of
London"?

In the "indefinite" sense, I can't see how any of the above three
principles, quoted by Parsons, are true.

Is that important?  Only because, historically, the modern idea of a set
grew out of the idea of the indefinite class.

"I have assumed [classes] to be indefinite; because for the purposes of
Logic, definite classes, as such, are almost useless" (Mill)

"Only because classes are determined by the properties that objects in them
are to have, and because we use phrases like .. " the class of objects that
are b", only so does it become possible to express thoughts in general by
stating relations between classes; only so do we get a logic" (Frege)

Dean