[FOM] Understanding universal quantification

[Dean Buckner]

On Sat, 22 Feb 2003, Steve Newberry wrote:

>> The *semantic* and ontological analyses of quantification appear to be of
more
>> recent occurrence.

On Sun Feb 23, John Goodrick wrote:

>Not much more recent, though; Frege introduced a notation for
>quantification, to be interpreted objectually (not substitionally), in his
._Begriffschrift_ in 1879.
> I had thought, before reading your message, that Frege was the first to
> introduce quantifiers into formal logic.  I wonder if Frege was aware of,
> or influenced by, this work of Jevons?-John

Frege was certainly aware of Jevon's work, as he discusses it in the
Grundlagen. But it is absurd to credit Frege with the idea of "quantifying".
This is fundamental to traditional logic also, in the "quantity" of the
proposition.  The traditional scheme consists of the following components:

(1) two terms, say "man" mortal", one the subject, one the predicate (doesn'
t matter which, they are essentially symmetrical).
(2) an operation that "affirms" or "denies" the predicate of the subject
(3) a sign that indicates the "quantity" of the subject - universal (all
men) or "particular" (some men).

This gives only four kinds of proposition  universal affirmative ("All men
are mortal"), particular affirmative ("Some men are philosophers"),
universal negative  ("No philosophers are rich"), and particular negative
("Some men are not philosophers").

The obvious defect of this scheme is that it has no place for proper names,
the representation of which was a difficulty which had perplexed logicians
for centuries [1].  The traditional approach (which seems to have originated
with Ockham, but no one really knows) was to represent proper names as
implicitly quantified, thus: "every man is mortal, every Socrates is a man,
every Socrates is a mortal".

Frege's contribution to logic was not the quantifiers, but a different view
of the proposition, in particular, of the predicate.  Instead of predicate
being a general term like "man", which is joined to another predicates by
the verb "is", Frege swallows the copula "is" into the predicate, which
becomes a "gappy" sentence with the proper name (or names) removed, thus "-
is a mortal".  With a negation operator on the outside (rather than the
middle), we then get the four general propositions of Aristotelian logic,
plus the propositions we get when we substitute proper names into the gappy
bits (singular propositions).

I.e. Instead of two sorts of general proposition (universal and "particular"
i.e. existential) we now have general propositions, plus an entirely new
sort.  Frege's contribution was not the idea of quantification at all, but
rather the idea that there are exist propositions which are _not_
quantified.

As to where Frege got this idea, it appears very early in his philosophy
[2], and he has a reluctance to give his sources.  Some ideas

(1)  Ueberweg's book on logic mentions the idea of concepts being
"subordinated" under one another, an idea which is crucial to Frege's
scheme.  This was a standard German textbook of the time, and mentioned by
Cantor in connection with Frege's work.
(2)  It's also a crucial idea in Mill's System of Logic (general terms
having a "connotation")
(3)  Frege's father wrote a grammar book in which the article "the" is
mentioned as forming names for general concepts
(4)  The German logician R.H. Lotze mentions the same idea in his _Logic_
[3].  Lotze taught Frege for one year at Gottingen.
(5)  The idea of representing propositions in functional form,with proper
names as arguments, and what's left over (the gappy sentence) the function,
and a "truth value" as a value of the function, is one that would naturally
occur to a mathematician.

The natural result of this to give two separate signs for predication where
before there was only one.  Under the new scheme, we have to distinguish the
sign that connects an object to a concept (the set membership sign, epsilon,
which comes from the Greek word for the copula), from the sign for
"subordination" that connects to concepts (is a subset of).  Without this we
do not have the distinction between "first" and other orders of logic.


Dean


Bcc Richard Mendelsohn, who appears to be writing an interesting book on
Frege.
Richard, this appears in the Foundations of Mathematics discussion group,
here:

http://www.cs.nyu.edu/pipermail/fom/


[1] Aristotle's logic simply ignores proper names. The famous "All men are
mortal - Socrates is a man - Socrates is a mortal" was a later innovation.
In De Interpretatione  7, 17a 37 he strictly distinguishes the terms that
can go into a syllogism as being those which are "a nature as to be
predicated of many subjects" and those which are individual.  In Metaphysics
D 9, 1018 a4 he expressly repudiates the doctrine that we can say "every
Socrates" as we say "every man".


[2] Probably after 1874, when his dissertation "Methods of Calculation based
on an Extension of the Concept of Magnitude" was submitted to the University
of Jena in 1874, and and before the publication of Begriffschift in 1879,
when Frege was 31.

I found some new material on the web by Richard Mendelsohn, which discusses
the problems of tracing the origins of Frege's thought, here:

http://comet.lehman.cuny.edu/mendel/papers/Adobe%20Versions/AdobeFregeBiogra
phy(corrected).pdf

Mendelsohn confirms the view of Frege as an autodidact who relied for his
philosophical information on a popular anthology put together by Baumann.
"There appear to be immense holes in Frege's knowledge of the history of
philosophy".

He confirms the view that the connection with Lotze's thought is tenuous
(but see [3] below)

[3] Lotze says (_Logic_ I . I . ~3, p.14) that the first operation of
thought is its _objectification_.  This is best exemplified, he says, by
those languages which have the definite article.  It is a form of pointing,
he says.  However "The logical objectification ... which the creation of a
name implies, does not give an external reality to the matter named; the
common world, in which others are expected to recognise what we point to,
is, speaking generally, only the world of thought; what we do here is to
ascribe to it _the first trace of an existence of its own and an inward
order which is the same for all thinking beings and independent of them_ [my
emphasis]: it is quite indifferent whether certain parts of this world
indicate something which has besides an independent reality outside the
thinking minds, or whether all that it contains exists only in the thoughts
of those who think it, but with equal validity for them all." (I . I . ~3,
p.15-16)

This bears points of resemblance to Frege's ideas about the compositionality
of thought.