Your third question concerns my remarks, stating that Peano
had a good notation without having a use for it, whereas Frege
had an awful notation, but did use it to make his secular (if
not millenial) discovery - that of a calculus. You write
> I'd like to make a place in this story for Charles S. Peirce, who introduced
> several symbols, including the crucial "-<" for set inclusion, together with
> what Russell would call "the logic of relations" in his "Description of a
> Notation for the Logic of Relatives from an Amplification of the Conceptions
> of Boole's Calculus of Logic" published in Memoirs of the American Academy of
> Arts & Sciences (n.s. 9, 1873) and for Ernst Schr"oder, who seems to have
> borrowed Peirce's notation and improved on it, beginning in 1877 ("Der
> Operationskreis des Logikkalkuls,").
but while, what you say here about Peirce's and Schr"oder's
contributions, is certainly correct by itself, there is no
place for them in the above "story", i.e. in the development
of the mathematical theory of deduction. [And mind you that
it was geometry as a deductive science that our discussion
started from.]
Let me now develop this answer in more detail.
(1) Relations as tool for a phaenomenological semantics
Analysis of thought demands that thought, expressed through
language, be expressed clearly. Symbols and formulas are
useful for that purpose, and no doubt the history of
inventions made to this end is a long one; though I have
never cared to pursue it, I recall that Euler's and
Ploquet's diagrams (later called Venn's) would have to be
mentioned there. As Boolean algebra did not suffice to
describe even the situations met in the syllogisms,
attributes expressing properties had to be considered,
written as monadic or multivariable relations.
So there arose, during the second half of the 19th century,
the study of relations [now understood in today's sense],
culminating in Schr"oder's three fat volumes: from the
mathematical point of view, a new sort of algebra which, at
that time, was never separated from its intended model, the
actual relations on sets. [Only since the 1940ies, abstract
relational algebras began to be studied by Tarski, Lorenzen
and others, leading first to representation theorems and,
recently, to deep results about decidability etc.] From the
point of view of the intended application, the analysis of
language as an analysis of thought, a rather precise
description of [certain] language uses was achieved, and in
today's words we may say that a good part of the semantics
of first-order languages was obtained.
It is here then that C.S.Peirce has his place [and as the
article you mention, from 1873, is inaccessible to me, I
refer to his familiar "On the algebra of logic. A contribution
to the philosophy of notation" in American J. Math. 7 (1885)
180-202 ]: he isolated the 'quantifying' expressions "some"
and "for all" as linguistic operators acting upon expressions
[an idea, Peirce writes, originating with O.H.Mitchell from
Johns Hopkins ], emphasized the dependence on the quantified
variable in their usage, used the symbols capital-Sigma and
capital-Pi to denote these operators, and (saying that this
also was proposed by Mitchell) pointed out the connection
with the corresponding operators on actual relations (nowadays
called cylindrifications). In summary: an important contribution
to the semantical analysis of language. - I might add the
relation-theoretical treatment of semantics through
Schr"oder did produce one important result: L"owenheim's
theorem of 1916 .
(2) Deduction as a mechanics of character strings
Analysis of thought, analysis of language is a useful ability
to have at hand. In mathematical lectures, which require
hard physical work to write on the blackboard, lecturers
usually apply a kind of shorthand to denote complicated
relationships, and often they employ for that some symbols
reminiscent to those used in logic. Indeed, it does not need
a course in logic to understand that
(a) (Ae)(Ax)(Ay)(Ed) ( |x-y| < d -> |fx - fy| < e )
expresses the continuity of a function f in the domain where
x and y range, and that
(b) (Ae)(Ed)(Ax)(Ay) ( |x-y| < d -> |fx - fy| < e )
expresses uniform continuity in that same domain. Arguing
semantically, the mathematician knows that uniform continuity
implies continuity, and he may also be aware that this
derivation of (a) from (b) makes use of a special case of a
'law of thought' observed already by Peirce:
(c) (Ed)(Ax) a(x,d) -> (Ax)(Ed) a(x,d) .
But then to argue semantically means to use the uncontrolled
thoughts in the backs of our heads, steps of implicitly
performed induction rather than those of explicit deduction.
Should it not be possible to quite mechanically view the
character strings
1. (Ed)(Ay) b(x,y,d,e) -> (Ay)(Ed) b(x,y,d,e)
2. (Ax)(Ed)(Ay) b(x,y,d,e) -> (Ax)(Ay)(Ed) b(x,y,d,e)
3. (Ed)(Ax)(Ay) b(x,y,d,e) -> (Ax)(Ed)(Ay) b(x,y,d,e)
4. (Ed)(Ax)(Ay) b(x,y,d,e) -> (Ax)(Ay)(Ed) b(x,y,d,e)
5. (Ae)(Ed)(Ax)(Ay) b(x,y,e,d) -> (Ae)(Ax)(Ay)(Ed) b(x,y,d,e)
as 1. an instance of (c) ,
2. from 1. by 'prefixing' its parts with (Ax) ,
3. an instance of (c) ,
4. from 3. and 2. by a form of 'enchainment' ,
5. from 4. by 'prefixing' its parts with (Ae) .
This, indeed, might be read as a deduction from the law (c)
of the implication of (a) by (b): a deduction where now the
manipulations of 'enchainment' and 'prefixing' become
nothing but mechanical manipulations of character strings -
without references to their content and without other silent
thoughts in the backs of our heads.
Yet no one before Frege - neither Schr"oder, Peano nor Peirce
[as far as we can conclude from his published articles] - even
conceived the idea of such a calculus, called also syntactical
(as it refers only to the syntactical form, not to the content
or meaning of the expressions it treats). Frege in 1879
did present such a calculus. Of course, it was awkward to use,
and in the form most familiar today it was developed only by
Hilbert after 1920 ; considerably improved types of calculi
were invented by Gentzen 1934 . [In addition, I may notice that
most mathematicians, exercising every day the presentation and
discovery of mathematical proofs, do not have the practise
required to express their semantical reasoning in a calculus;
as a matter of fact, even some logicians among them will feel
hard pressed when attempting to do so.]
(3) Frege's achievement
That deductions can be expressed with help of a purely
syntactical mechanism may be interesting to know, and it will
be useful for the computer programmer wanting to write a
program which checks given proofs for possible gaps. But what
is it that made me call 'secular' the discovery of Frege -
more precisely: his discovery of syntactical rules treating the
use of quantifiers ?
Mathematics is a man-made art, technically difficult sometimes,
but without epistemological problems in traditional fields such
as combinatorics, algebra and basic geometry. Such problems,
however, arise in situations involving the infinite. Because
how shall it be possible to prove a 'universal' statement
for all x : F(x)
where x ranges over infinitely many objects - say all rational
numbers ? We may be able to produce a proof of F(r) for one
rational number r , we may be able to produce a proof also of
F(q) for a second rational number q , but we are not able to
produce, for all the infinitely many rational numbers r,q,p,... ,
the infinitely many proofs of the infinitely many statements
F(r) , F(q) , F(p) , ... :
not during our finite lifetime, not with the finite amounts of
paper or of magnetic storage space.
But already in school algebra we learn how to prove certain
universal statements: computing with 'letters', we establish
for instance that
(x+y)(x+y) = xx + 2xy + yy and (x+y)(x-y)= xx - yy ,
and as, for these letters, we could have substituted in our
computation any numbers n and m , we then are justified to say
that these identities have been established "for all x" and
"for all y" . So what we can 'generalize' are the results of
computations with letters, or variables; considering now
formulas with variables, Frege's rule to prove (Ax) F(x) is
nothing but an extension of this observation:
if F(y) has a proof from a collection of hypotheses _C_ , symbolically
(IA0) _C_ => F(y) ,
then this proof can be prolonged to give a proof of (Ax) F(x) :
(IA1) _C_ => (Ax) F(x) ,
provided the variable y is free in (IA0), i.e. does not already
occur in the hypotheses _C_ outside the scope of quantifiers (Qy).
[In Frege 1879 this is stated on p.21 in the paragraph
beginning with the phrase "Auch ist einleuchtend, dass man aus
... "; in Frege 1893 it is stated at the start of p.32 in
symbolical notation ("Wir schreiben einen solchen "Ubergang so
..." and in verbalized form at the end of p.33 ("Wir fassen
dies in folgende Regel ..."); in Russell 1908 it is rule (8)
of section VI , p.246 , and in Whitehead-Russell 1910 it is
principle 9.13 . Frege does not need provisions on free
variables because he uses different alphabets for free and for
bound variables. ]
[The proviso about the freeness of the variable y cannot be
avoided. For instance, let _D_ a set of identities defining
groups with an operation + . We then know that the only y with
y+y=y is the neutral, or zero element, and writing down this
argument formally, we obtain a proof from the hypotheses _D_
and y+y=y of y+u=u , i.e.
_D_ , y+y=y => y+u=u hence also
_D_ , y+y=y => (Az) y+z=z by the above rule .
It now needs the proviso of our rule to prevent us to proceed
to the semantically absurd
_D_ , y+y=y => (Ax)(Az) x+z=z . ]
So Frege's mathematical achievement is the discovery of a
syntactical (mechanical) calculus for deduction. The calculus
refers to objects which are abstractions (or formalizations) of
grammatical (linguistical) objects -- in Frege's case
2-dimensional proof figures, since Russell then sequences (or
tree-like figures) of strings of characters.
However, the example of the above rule (IA) shows that the
syntactical calculus is more than a plain translation of
deductive reasoning as presented in the mathematician's way of
arguing (whose proofs appear as protocols of his mental
constructions performed on semantical objects, e.g. numbers and
sets). That (IA0) is a free variable proof contains the
reference to the linguistic material of the calculus - to its
language and to the language's notational parts (in particular:
free variables). Yet (IA) is the only way to arrive at proofs
of universally quantified statements from unquantified ones;
hence the only way to prove statements "for all x : F(x)"
[where the range of x is assumed not to be finite] is to prove
F(y) with y as a schematic, free variable, not subjected to
restraining additional conditions (and it was this insight
which made Hilbert in 1923 call (IA) [together with the
corresponding rule (EE) to be mentioned below in the appendix]
a 'transfinite' rule). There is no other method to prove the
statement F(x) "for infinitely many instances" (as the
mathematician will express himself), but to prove F(y) for a
schematic, free variable y .
So Frege's methodological, or epistemic, achievement consists
in the observation of the dependence on a (particular)
linguistic description in the proofs of statements "for all x :
F(x)" . There can be no investigation of infinitely many
proofs/verifications, but there can, instead, be the study of
schematic free variable proofs. Mathematics of the infinite,
when to be captured deductively, requires attention to, and
appropriate use of, the language employed to speak about it.
W.F.