csp-frege-revolu

HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? 17
According to van Heijenoort [1967b, 325], Boole left his propositions unanalyzed. What he means is that propositions
in Boole are mere truth-values. They are not, and cannot be, analyzed, until quantifiers, functions (or predicate
letters), variables, and quantifiers are introduced. Even if we accept this interpretation in connection with Boole’s algebraic
logic, it does not apply to Peirce. We see this in the way that the Peirceans approached indexed “logical polynomials.
Peirce provides quantifiers, relations, which operate as functions do for Frege, as well as variables and constants,
the latter denoted by indexed terms. It is easier to understand the full implications when examined from the
perspective of quantification theory. But, as preliminary, we can consider Peirce’s logic of relations and how to interpret
these function-theoretically.
With respect to Boole, it is correct that he conceived of propositions as adhering to the subject-predicate form
and took the copula as an operator of class inclusion, differing from Aristotle only to the extent that the subject and
predicate terms represented classes that were bound by no existential import, and might be empty. De Morgan, however,
followed Leibniz in treating the copula as a relation rather than as representing a subsistence between an object and
a property. Peirce followed De Morgan in this respect, and expanded the role of relations significantly, not merely
defining the subsistence or nonsubsistence of a property in an object, but as a defined correlation between terms, such
as the relation “father of” or his apparent favorite, “lover of”. Boole, that is to say, followed Aristotle’s emphasis on
logic as a logic of classes or terms and their inclusion or noninclusion of elements of one class in another, with the
copula taken as an inherence property which entailed existential import, and treated syllogisms algebraically as equations
in a logic of terms. Aristotle recognized relations, but relegated them to obscurity, whereas De Morgan undertook
to treat the most general form of a syllogism as a sequence of relations and their combinations, and to do so algebraically.
De Morgan’s algebraic logic of relations is, thus, the counterpart of Boole’s algebra of classes. We may summarize
the crucial distinctions by describing the core of Aristotle’s formal logic as a syllogistic logic, or logic of terms
and the propositions and syllogisms of the logic having a subject-predicate syntax, entirely linguistic, the principle
connective for which, the copula is the copula of existence, which is metaphysically based and concerns the inherence
of a property, whose reference is the predicate, in a subject; Boole’s formal logic as a logic of classes, the terms of
which represent classes, and the copula being the copula of class inclusion, expressed algebraically; and De Morgan’s
formal logic being a logic of relations whose terms are relata, the copula for which is a relation, expressed algebraically.
It is possible to then say that Peirce in his development dealt with each of these logics, Aristotle’s Boole’s, and De
Morgan’s, in turn, and arrived at a formal logic which combined, and then went beyond, each of these, by allowing his
copula of illation to hold, depending upon context, for terms of syllogisms, classes, and propositions, expanding these
to develop (as we shall turn to in considering van Heijenoort’s third condition or characteristic of the “Fregean revolution”),
a quantification theory as well. Nevertheless, Gilbert Ryle (1900–1976) [1957, 9–10] although admittedly acknowledging
that the idea of relation and the resulting relational inferences were “made respectable” by De Morgan,
but he attributed to Russell their codification by in The Principles of Mathematics—rather than to Peirce—their codification
and to Russell—rather than to Peirce and Schröder—their acceptance, again by Russell in the Principles. Again,
Ryle [1957, 9–10] wrote: “The potentialities of the xRy relational pattern, as against the overworked s–p pattern, were
soon highly esteemed by philosophers, who hoped by means of it to order all sorts of recalitrances in the notions of
knowing, believing….”
It should be borne in mind, however, that Boole did not explicitly explain how to deal with equations in terms
of functions, in his Mathematical Analysis of Logic [Boole 1847], although he there [Boole 1847, 67] speaks of “elective
symbols” rather than what we would today term “Boolean functions”, 18 and doing so indirectly rather than explicitly.
In dealing with the properies of elective functions, Boole [1847, 60–69] entertains Prop. 5 [Boole 1847, 67]
which, Wilfrid Hodges [2010, 34] calls “Boole’s rule” and which, he says, is Boole’s study of the deep syntactic parsing
of elective symbols, and which allows us to construct an analytical tree of the structure of elective equations. Thus,
for example, where Boole explains that, on considering an equation having the general form a1t1 + a2t2 + … + artr = 0,
resolvable into as many equations of the form t = 0 as there are non-vanishing moduli, the most general transformation
of that equation is form ��(a1t1 + a2t2 + … + artr) = ��(0), provided is is taken to be of a “perfectly arbitrary character
and is permitted to involve new elective symbols of any possible relation to the original elective symbols. What this
entails, says Hodges [2010, 4] is that, given ��(x) is a Boolean function of one variable and s and t are Boolean terms,
then we can derive ��(s) = ��(s) from s = t, and, moreover, for a complex expression ��(x) = fghjk(x) are obtained by
composition, such that fghjk(x) is obtained by applying f to ghjk(x), g to hjk(x), …, j to k(x), in turn, the parsing of
which yields the tree