Simplex sigillum veri: Peano, Frege, and Peirce on the Primitives of Logic

Abstract

We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis.

Acknowledgements

Our thanks go to the anonymous referees of the journal.

Notes

1 In the Enciclopedia entry, Padoa (1930, p. 14) states the principles of the ethics of notation as follows: ‘The meaning of a formula (either simple or complex) draws its origin from tradition or from the author who first used it or who first meant to use in a quite different way than the traditional one’. This comes surprisingly close to Peirce's principle of the ethics of notation: ‘The person who introduces a conception into science has both the right and the duty of prescribing a terminology and a notation for it; and his terminology and notation should be followed except so far as it may prove positively and seriously disadvantageous to the progress of science. If a slight modification is sufficient to remove the objection, a much greater one should be avoided’ (Peirce 1904b, 1, emphasis in the original). (Cf. Peirce 1904a and Peirce 1905).

2 De Int. 16a3-8.

3 Padoa’s 1900 method gives a sufficient but not a necessary condition for logical constants to be undefinable from the other primitives of a theory: there are terms of a theory of which the method fails to show their undefinability. A sequence of developments to prove and improve the method emerged sometime later, beginning with Tarski and Lindenbaum in 1926 and followed by McKinsey, Beth, Craig and Robinson, among others.

4 In addition, Padoa considers that ‘[b]esides logical symbols, variables and punctuation marks also belong to the logical vocabulary’ (Padoa 1930, p. 12).

5 This recognition is limited to the sense of scope concerning the binding of variables. The other function of the notion of scope is that of denoting logical priority, namely the dependence between quantifiers, and Frege seems to have been much less keen to appreciate the latter than the former sense. See Hintikka 1997.

6 Cf. Schlimm 2017.

7 It may be added that Ernst Schröder followed Peirce in this respect. Although Schröder introduced a sign for inclusion in 1873, in the 1877 Operationskreis he adopted Boole's equational approach (judgments are expressed as equations between classes). Later, in the Vorlesungen, Schröder adopts an inclusional approach with judgments expressed as the subsumption relation between classes (see Peckhaus 1991, p. 181). Other logicians who used inclusion or implication signs in the 1870s include Robert Grassmann, Hugh MacColl, Charles L. Dodgson and, of course, Frege (see Durand-Richard and Moktefi 2014).

8 This is how negation is defined in intuitionist logic: A is an abbreviation of (A → ⊥). Peirce was not far from an invention of intuitionistic logic, given that his 1885 algebra of logic, which corresponds to Boolean algebras, has an intuitionistic kernel in that its most important rule (or its leading principle, known as Peirce's Rule), and which is needed to prove the full distributivity laws, says that the material implication is a right residuation of conjunction. This is the same residuation law by which intuitionistic implication is algebraically introduced. Removing the law of excluded middle from his 1880 propositional calculus would yield a calculus for Heyting algebras and thus straight at intuitionistic logic. And remove the law he did, thought only much later, in the triadic logic of 1909 and in the logic of vagueness (see Lane 1999).

9 Cf. Peirce 1885 = Peirce 1993, pp, 174–75; Peirce 1896b, p. 9.

10 Cf. also Peirce 1906. The details of the derivation, which involves the introduction of the pseudo-graph, are in Zalamea 2012 and in Bellucci and Pietarinen 2016a.

11 We incidentally note how closely this idea resembles the later discovery of one-constant operators in quantified logic (Schönfinkel 1924; Bimbò 2010). Schönfinkel took the ‘“eternal” essence of propositions’ to consist of the phenomenon that argument places and operators ‘belong together’. Peirce's logical graphs are built upon such a concrete notation that can also expose to view how they belong together. Pursuing this line of thought further we would realize that the Beta graphs show how quantifiers and identity likewise belong together. After Schönfinkel, Curry and Feys 1958 proposed the operator that is a combination of a universal quantifier and the conditional as a useful primitive that gave rise to combinatory logic; in Peirce's terms the Curry-Feys operator is the ‘scroll that binds’.

Additional information

Funding

Research supported by the Estonian Research Council (PUTJD 72, The Philosophy of Notation, 2015–2017; PUT 1305, Abduction in the Age of Fundamental Uncertainty, 2016–2018).