Logic as Calculus and Logic as Language
Answering Schröder’s criticisms of Begriffsschrift, Frege states that, unlike Boole’s, his logic is not a calculus ratiocinator, or not merely a calculus ratiocinator, but a lingua characterica.1 If we come to understand what Frege means by this opposition, we shall gain a useful insight into the history of logic.
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- 1.Schröder’s criticisms are contained in his review of Begriffsschrift, published in Zeitschrift für Mathematik und Physik 25 (1880), Historisch-literarische Abtheilung, 81–94. Frege’s reply was an address to a learned society, delivered on 27 January 1882 and published in its proceedings, ‘Über den Zweck der Begriffsschrift’, Sitzungsberichte der Jenaischen Gesellschaft für Medicin und Naturwissenschaft für das Jahr 1882 (Jena 1883), pp. 1–10, reprinted in Gottlob Frege, Begriffsschrift und andere Aufsätze, Hildesheim 1964, pp. 97–106. On the origin of the expression ‘lingua characterica’ see Günther Patzig’s footnote 8, on p. 10 of Gottlob Frege, Logische Untersuchungen, Göttingen 1966.Google Scholar
- 2.See, for instance, Frege’s comments on Boole in ‘Über den Zweck der Begriffsschrift’ (mentioned in footnote 1), pp. 1–2.Google Scholar
- 3.In ‘Über die Begriffsschrift des Herrn Peano und meine eigene’, Berichte über die Verhandlungen der Königlichen Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-physische Classe 48 (1896), 361–378, Frege writes on p. 371: “Boole’s logic is a calculus ratiocinator, but no lingua characterica; Peano’s mathematical logic is in the main a lingua characterica and, subsidiarily, also a calculus ratiocinator, while my Begriffsschrift intends to be both with equal stress.” Here the terms are used with approximately the meanings given in the present paragraph: Boole has a propositional calculus but no quantification theory; Peano has a notation for quantification theory but only a very deficient technique of derivation; Frege has a notation for quantification theory and a technique of derivation.Google Scholar
- 4.Begriffsschrift, § 13.Google Scholar
- 5.Here the influence of Frege on Wittgenstein is obvious. — Frege’s refusal to entertain metasystematic questions explains perhaps why he was not too disturbed by the statement ‘The concept Horse is not a concept’. The paradox arises from the fact that, since concepts, being functions, are not objects, we cannot name them, hence we are unable to talk about them. Some statements that are (apparently) about concepts can easily be translated into the system; thus, ‘the concept Φ(ξ) is realized’ becomes (Ex) Φ (x)’. The statements that resist such a translation are, upon examination, metasystematic; for example, ‘there are functions’ cannot be translated into the system, but we see, once we have ‘caught on’, that there are function signs among the signs of the system, hence that there are functions.Google Scholar
- 6.Kurt Gödel, ‘Die Vollständigkeit der Axiome des logischen Funktionenkalküls’, Monatshefte für Mathematik und Physik 37, 349–360; English translation by Stefan Bauer-Mengelberg in J. van Heijenoort, From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967.Google Scholar
- 7.Here ‘axiomatic’ is used for a method of formal derivation based on axioms and rules of inference, and this use should not be confused with broader uses, as in ‘the axiomatic method in geometry’. — Let us remark that, unlike Frege, Russell never emphasized the formal aspect of logical proofs and that, in particular, the system of Principia Mathematica does not measure up to the standards that Frege set for a formal system. (On this point see Kurt Gödel, ‘Russell’s Mathematical Logic’, in The Philosophy of Bertrand Russell (ed. by Paul Arthur Schilpp), New York 1944, pp. 123–153, especially p. 126; see also W. V. Quine, ‘Whitehead and the Rise of Modern Logic’, in The Philosophy of Alfred North Whitehead (ed. by Paul Arthur Schilpp), New York 1941, pp. 125–163, especially p. 140.) The notion of formal system was again brought into the forefront by Hilbert, in the ‘twenties. That is perhaps why the (in our sense) axiomatic systems of logic are called Hilbert-type systems by Kleene (Introduction to Metamathematics, p. 441). If the historical priority is to be respected, they should rather be called Frege-type systems.Google Scholar
- 8.For the sake of simplicity I take the formulation of the theorem for quantification theory without identity.Google Scholar
- 9.Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen’, Videnskapssels- kapets skrifter, L Matematisk-naturvidenskabelig klasse, no. 4.Google Scholar