csp-frege-revolu

HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? 22
that is:
If F is satisfiable in every domain, then F is �0-satisfiable;
If F is n-satisfiable, then F is (n + 1)-satisfiable,
and indeed, his proof was in all respects similar to that which appeared in Löwenheim’s 1915 paper, where, for any
�����, a product vanishes (i.e. is satisfiable), its � th term vanishes.
In its most modern and strictest form, the LST says that:
For a �-ary universe, a well-formed formula F is �0-valid if it is �-valid for every finite �, provided there is no
finite domain in which it is invalid.
Herbrand’s FT was developed in order to answer the question: what finite sense can generally be ascribed to the
truth property of a formula with quantifiers, particularly the existential quantifier, in an infinite universe? The modern
statement of FT is:
For some formula F of classical quantification theory, an infinite sequence of quantifier-free formulas F1, F2, ...
can be effectively generated, for F provable in (any standard) quantification theory, if and only if there exists a � such
that F is (sententially) valid; and a proof of F can be obtained from F�.
For both LST and FT, the elimination of quantifiers, carried out as expansion in terms of logical sums and
products, as defined by Mitchell-Peirce-Schröder, is an essential tool. Moreover, it was precisely the Mitchell-Peirce-
Schröder definition, in particular as articulated by Schröder in his Algebra der Logik, that provided this tool for
Löwenheim, Skolem, and Herbrand.
For the role that Peirce’s formulation of quantifier theory played in the work of Schröder and the algebraic logicians
who followed him, as well as the impact which it had more widely, not only on Löwenheim, Skolem, and
Herbrand, Hilary Putnam [1982, 297] therefore conceded at least that: “Frege did “discover” the quantifier in the sense
of having the rightful claim to priority. But Peirce and his students discovered it in the effective sense.”
The classical Peirce-Schröder calculus also, of course, played a significant role in furthering the developments
in the twentieth century of algebraic itself, led by Alfred Tarski (1903–1983) and his students. Tarski was fully cognizant
from the outset of the significance of the work of Peirce and Schröder, writing, for example in “On the Calculus
of Relations” [Tarski 1941, 73–74], that:
The title of creator of the theory of relations was reserved for C. S. Peirce. In several papers published between
1870 and 1882, he introduced and made precise all the fundamental concepts of the theory of relations and formulated
and established its fundamental laws. Thus Peirce laid the foundation for the theory of relations as a
deductive discipline; moreover he initiated the discussion of more profound problems in this domain. In particular,
his investigations made it clear that a large part of the theory of relations can be presented as a calculus
which is formally much like the calculus of classes developed by G. Boole and W. S. Jevons, but which greatly
exceeds it in richness of expression and is therefore incomparably more interesting from the deductive point of
view. Peirce’s work was continued and extended in a very thorough and systematic way by E. Schröder. The
latter’s Algebra und Logik der Relative, which appeared in 1895 as the third volume of his Vorlesungen über
die Algebra der Logik, is so far the only exhaustive account of the calculus of relations. At the same time, this
book contains a wealth of unsolved problems, and seems to indicate the direction for future investigations.
It is therefore rather amazing that Peirce and Schröder did not have many followers. It is true that A. N.
Whitehead and B. Russell, in Principia mathematica, included the theory of relations in the whole of logic,
made this theory a central part of their logical system, and introduced many new and important concepts connected
with the concept of relation. Most of these concepts do not belong, however, to the theory of relations
proper but rather establish relations between this theory and other parts of logic: Principia mathematica contributed
but slightly to the intrinsic development of the theory of relations as an independent deductive discipline.
In general, it must be said that—though the significance of the theory of relations is universally recognized
today—this theory, especially the calculus of relations, is now in practically the same stage of development
as that in which it was forty-five years ago.