FOM: Peano

[Dean Buckner]

I'm interested in the history of the singular term, and how it got into
logic.  This is an ancient idea, but was never part of syllogistic.
Aristotle never discusses the singular term at all (in his logic at least).
It is treated as a universal by all logicians from Ockham onwards (see e.g.
Mill's system of logic II. ii. 1).  I had thought the whole idea originated
with Frege.

But then of course it was Peano (apparently) who introduced the symbol for
"is a member of", and who distinguished it from the sign for "is a subset
of".  And this distinction makes no sense without singular terms.  If
propositions are only of the form "all /some men are mortal", we can make do
with just the sign
for "is a subset of".  Likewise if "Socrates" is a general word satisfied by
one individual, so that "Socrates is mortal" says that the singleton class
containing whichever individual this is, is a subset of (NOT member of) the
class of mortal things.

We only need "is a member of" if we need singular terms.  Do we credit Peano
with this innovation?  What was the context?

With this also comes the idea of a set as something over and above the
individuals it contains, and the introduction of a singular term say
"{Aristotle, Plato, Socrates}" to denote something different from the
individuals Aristotle, Plato, and Socrates.

Also there comes the nasty idea of "direct reference", the strange semantic
connection between a singular term and the object it "refers to", and the
distinction between "satisfaction" (as in Aristotle satisfies "philosopher")
and "reference" (as in Aristotle is the reference of "Aristotle").  and the
nastier problem of explaining "Pegasus does not exist" which Frege wisely
evades, leaving it to his descendants to deal with!  (See e.g. Gareth Evans
tortuous and painful discussion in 10.4 of *Varieties*).

Do we blame Peano for this rather than Frege?  My hunch is that all these
ideas came into logic at the same time because of mathematicians who had for
a  long time (I assume) been used to distinguish between variables and
constants, and arguments and functions.  This comes out clearly in Frege's
earlier writing.

Would be interested in views, and particularly anyone who has access to the
Peano paper where this idea first appeared, and which I cannot locate (I
believe 1888?)

As late as 1895 you find Frege heaping abuse upon Schroder for failing to
grasp the distinction (see Geach and Black pp 86-106).


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