csp-frege-revolu

HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? 30
Returning to van Heijenoort’s list of properties, it should be clear from the evidence which we have presented
that, under this interpretation, Peirce indeed had both a calculus ratiocinator, or sentential calculus with derivation,
defined in terms of illation (property 1) and a characterica universalis, or quantification theory and notation for quantification
theory (property 3), and that these are clearly present in a single unified package in “On the Algebra of Logic:
A Contribution to the Philosophy of Notation” [Peirce 1885].
The other aspect of this universality is that, as a language, it is not restricted to a specific universe of discourse,
but that it operates on the universal domain, what Frege called the Universum. Thus, the universe of discourse for
Frege and Russell is the universal domain, or the universe. It is in virtue of the Begriffsschrift’s and the Principia system’s
universe of discourse being the universe, that enables these logical systems to say (to put it in colloquial terms)
everything about everything in the universe. One might go even further, and with van Heijenoort understand that, ultimately,
Frege was able to claim that there are only two objects in the Universum: the True and the False, and that
every proposition in his system assigns the Bedeutung of a proposition to one or the other.
Johannes Lenhard [2005] reformulates van Heijenoort’s distinction between logic as calculus and logic as language
in ontological term, by suggesting that the concept of logic as a language upholds a model carrying an ontological
commitment, and arguing in turn that Hilbert’s formalism, expressed in his indifference to whether our axioms
apply to points, lines, and planes or to tables, chairs, and beer mugs, bespeaks a model theory which is free of any ontology.
It is precisely in this sense that Lenhard [2005, 99] cites Hintikka [1997, 109] as conceiving of Hilbert as opening
the path to model theory in the twentieth century. This view of an ontologically challenged conception was already
articulated by Hilbert’s student Paul Isaac Bernays (1888–1977), who defined mathematical “existence” in terms of
constructibility and non-contradictoriness within an axiom system and in his [1950] “Mathematische Existenz und
Widerspruchsfreiheit”.
What makes the logic of the Begriffsschrift (and of the Principia) a language preeminently, as well as a calculus,
rather than a “mere” calculus, was that it is a logica docens, and it is absolute. The absoluteness guarantees that the
language of the Begriffsschrift is a language, and in fact a universal language, and fulfills the Leibniz programme of
establishing it as a mathesis universa, which is both a language and a calculus. In answer to the question of what Frege
means when he says that his logical system, the Begriffsschrift, is like the language Leibniz sketched, a lingua
characteristica, and not merely a logical calculus, [Korte 2010, 183] says that: “According to the nineteenth century
studies, Leibniz’s lingua characteristica was supposed to be a language with which the truths of science and the constitution
of its concepts could be accurately expressed.” [Korte 2010, 183] argues that “this is exactly what the
Begriffsschrift is: it is a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic
language, since the meaning of its complex expressions depend only on the meanings of their constituents and on
the way they are put together.” Korte argues that, contrary to Frege’s claims, and those by van Heijenoort and Sluga in
agreement with Frege, the Begriffsschrift is, indeed, a language, but not a calculus. 34
Because of this universality, there is, van Heijenoort argues, nothing “outside” of the Universum. (This should
perhaps set us in mind of Wittgenstein, and in particular of his proposition 5.5571 of the Tractatus logicophilosophicus
[Wittgenstein 1922], that “The limits of my language are the limits of my world”—“Die Grenzen
meiner Sprache bedeuten die Grenzen meiner Welt.”) If van Heijenoort had cared to do so, he would presumably have
quoted Prop. 7 from the Tractatus, that, by virtue of the universality of the logica docens and its universal universe of
discourse, anything that can be said must be said within and in terms of the logica docens (whether Frege’s variant or
Whitehead-Russell’s), and any attempt to say anything about the system is “wovon man nicht sprechen kann.” In van
Heijenoort’s terminology, then, given the universality of the universal universe of discourse, one cannot get outside of
the system, and the system/metasystem distinction becomes meaningless, because there is, consequently, nothing outside
of the system. It is in this respect, then, that van Heijenoort argued that Frege and Russell were unable to pose, let
alone answer, metalogical questions about their logic. Or, as Wittgenstein stated it in his Philosophische Grammatik
[Wittgenstein 1973, 296]: “Es gibt keine Metamathematik,” explaining that “Der Kalkül kann uns nicht prinzipielle
Aufschlüssen über die Mathematik geben,” and adding that “Es kann darum auch keine “führenden Probleme” der
mathematischen Logik geben, denn das wären solche….”
It was, as van Heijenoort [1967b; 1977; 1986b; 1987], Goldfarb [1979], and Gregory H. Moore [1987; 1988]
established, the model-theoretic turn, enabled by the work of Löwenheim, Skolem, and Herbrand, in turn based upon
the classical Boole-Peirce-Schröder calculus, and opened the way to asking and treating metasystematic questions
about the logical systems of Frege and Russell, and, as Moore [1987; 1988] also showed, helped establish the firstorder
functional calculus of Frege and Russell as the epitome of “mathematical” logic.
Turning then specifically to Peirce, we can readily associate his concept of a logica docens as a general theory
of semiotics with van Heijenoort’s conception of Frege’s Begriffsschrift and Whitehead-Russell’s Principia as instanc-