Abstract
This paper focuses on two important notational devices that were embedded in Husserl’s and Peirce’s notations, the sign of inclusion and the sign of assertion. Husserl first criticizes, then follows Schröder in taking inclusion to be a simpler concept than equality, and endows his logical notation with a sign of inclusion. This was due to Peirce’s notational innovations and arguments, by which Husserl is indirectly influenced through of Schröder. Further, like Frege Husserl endows his notation with a sign of assertion, claiming that without some such sign it would be impossible to distinguish the assertion of a propositional content from its occurrence unasserted in some larger propositional context. Against this idea of Frege and Husserl, Peirce had a powerful argument—the compositional structure of a formula is de facto the sign of its assertion—, and Peirce’s algebraical and graphical notations can be seen as notational realizations of this principle.
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Notes
- 1.
See Centrone (2010, pp. 128–147).
- 2.
- 3.
The expression “philosophy of notation” was coined by Peirce to indicate the philosophical investigation of the properties of logical notations; cf. Peirce (1885).
- 4.
Boole’s approach was based on equation. In The Laws of Thought he divided propositions into Primary, which relate to things, and Secondary, which relate to propositions, a distinction “nearly but not quite” co-extensive with that between categorical and hypothetical propositions. The propositions “The sun shines” and “The earth is warmed” are primary, the proposition “If the sun shines, the earth is warmed” is secondary. Both primary and secondary propositions can be symbolized in the same notation and subjected to the same laws. The difference between the two cases is only a difference in interpretation (as logic of classes or logic of propositions). Both kinds of proposition are expressed by an equation, symbolized in the notation by the sign “=” (Boole 1854, IV, 1, 53–54). Jevons had followed Boole in considering equation the “fundamental form of reasoning” (Jevons 1869, 14).
- 5.
Cf. Brady (2000, p. 145).
- 6.
“Die Formel sagt nämlich als disjunktives Urteil aus: wenn A ⊆ B ist, so ist entweder A untergeordnet B; oder aber […] es ist A gleich B” (1891, 109). In formulas: A =(= B = {A ⊂ B} + {A = B}.
- 7.
“Wenn a ⋹ b und zugleich b ⋹ a ist, so werde gesagt, es sei: a = b (gelesen a gleich b).”
- 8.
- 9.
Since Leibniz and Kant, the rule that “the greater the extension, the less the comprehension” was daily bread of logicians; we find it in Hamilton, Jevons, and many others. But for Peirce, the law only holds in a definite state of information; whenever we come to any new piece of knowledge of whatever kind, the state of information is altered, and the law of inverse proportionality does not hold anymore. See W1, 275 and Bellucci (2015b).
- 10.
The properties of the fundamental relative of quantity are “first, that it is transitive; second, that everything in the system is q of itself, and, third, that nothing is both q of and q’d by anything except itself. The objects of a system having a fundamental relation of quantity are called quantities, and the system is called a system of quantity” (W4, 299–300). See also “The Logic of Quantity,” MSS 13–17.
- 11.
On Peirce’s 1881 axiomatization of arithmetic and its centrality in his later thought see Shields (2012).
- 12.
Cf. MS 408, 121, 142 (1894); MS 594, 62–63 (1894); MS 441 (1898); see also Bellucci (2016).
- 13.
Cf. Dipert (1981).
- 14.
“Die Subsumption soll die ‘einfachere,’ ‘ursprünglichere’ Beziehung sein; sie drücke nur einen Gedanken aus, jene anderen aber deren zwei. a =(= b drücke nämlich aus: Alle a sind b. Dagegen z.B. a = b: Alle a sind b und alle b sind a usw. Aber all dies ist offenbar unrichtig. Schröder verwechselt beständig die sprachlichen Ausdrücke der Klassenurteile mit denen ihrer äquivalenten Inhaltsurteile. Halt man sich nur exakt an den wirklichen Inhalt der Zeichen, dann ist von jeder Behauptung des Verf. genau das Gegenteil wahr. a = b besagt z.B., die Klasse a ist gleich der Klasse b, a =(= b hingegen, die Klasse a ist untergeordnet oder gleich der Klasse b. Die Subsumptionsbeziehung ist also, wie das Zeichen es exakt ausdrückt, die aus Gleichheit und Subordination zusammengesetzte, und diese daher die ‚ursprünglichere.’ Lage hierin wirklich ein Argument, so müssten wir vielmehr diese Beziehungen für den Aufbau des Kalküls bevorzugen.”
- 15.
“Je zwei Sätze können, was immer ihr sonstiger Gehalt sei, zu einem neuen Satz konjunktiv verbunden werden, z.B. ‘Gott ist gerecht, und die Bösen werden bestraft’. Diese Verbindung wollen wir durch Nebeneinanderschreiben der Buchstaben bezeichnen. AB bedeutet also, wenn wir die Großbuchstaben als Zeichen irgendwelcher einzelner Sätze nehmen, die konjunktive Verknüpfung des Satzes A und des Satzes B; zu lesen: A und B. Ebenso bezeichnen wir durch A + B die disjunktive Verknüpfung; das Zeichen ist also zu lesen als: A oder B” (1896, 253). Cf. Husserl (1902, 239).
- 16.
“Eine dritte elementare Weise, aus zwei Sätzen einen zu bilden, ist die hypothetische: Wenn A, so B. Aber diese Verbindungsweise repräsentiert zugleich die Grundform der Beziehungzwischen Sätzen. Die Gültigkeit von A zieht die von B nach sich, und damit sind beide Sätze in ein Verhältnis gebracht, aus dem für jeden eigentümlich relative Beschaffenheiten entspringen: Grund zu sein und Folge zu sein. Dieses fundamentale Verhältnis bezeichnen wir so: €. Den wichtigen Fall, wo A € B und zugleich B € A, bezeichnen wir durch A = B” (1896, 253).
- 17.
- 18.
See Baker and Hacker (1984, 99).
- 19.
See again Backer and Hacker (1984, 85).
- 20.
Cf. “the assertoric force of a sentence is thus shown by its not being enclosed in the context of a longer sentence” (Geach 1965, 456); “[t]he correct employment of these symbols makes clear, for example, that the content of the antecedent of a conditional is not asserted, even though the content of the conditional as a whole is” (Baker & Hacker 1984, 84).
- 21.
- 22.
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Bellucci, F. (2019). Philosophy of Notation in the 19th Century. Peirce, Husserl, and All the Others on Inclusion and Assertion. In: Shafiei, M., Pietarinen, AV. (eds) Peirce and Husserl: Mutual Insights on Logic, Mathematics and Cognition. Logic, Epistemology, and the Unity of Science, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-030-25800-9_4
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