The Genealogy of ∨ Landon D. C. Elkind Richard Zach November 23, 2020 JHAP/BRRC Seminar 1
Outline The mystery of ∨ From Leibniz to Peano Russell The adoption of ∨ Conclusions 2
The mystery of ∨
The mystery of ∨ • Most logic symbols not universally accepted. • However, ∨ is. • Textbook story: ∨ is an abbreviation of Latin vel . • Is that really so? Who started this and when? 3
Shadow history • The usual answers are something of a “shadow history” Watson, 1993. • W. Kneale and M. Kneale (1962, p. 520): “the system [which includes ∨ ] is that introduced by Peano in his Notations de logique mathematique of 1894, developed in the successive editions of his Formulaire de mathematiques, and then perfected by Whitehead and Russell in their Principia Mathematica of 1910.” • Cajori (1929, p. 307) says that the earliest use occurs in Whitehead and Russell’s 1910 Principia . 4
From Leibniz to Peano
Leibniz As + is a conjunctive sign, i.e., a sign of juxtaposition, and corresponds to and, so that a + b is a and b simultaneously, there is a disjunctive sign as well, i.e., a sign that means an alternative, corresponding to or [vel]. Thus, for me, a “ v b means a or b. Matheseos universalis pars prior: De terminis incomplexis (Leibniz, 1679, p. 9v, Leibniz, 1863, p. 57) 5
Algebra of logic • Leibniz’s use of v remained unknown. • Boole used + for (exclusive) or (Boole, 1847, p. 51). • Jevons: ⋅ | ⋅ and + (Jevons, 1883, pp. 67–68). • Peirce: + (exclusive), + , (inclusive) (Peirce, 1870, p. 9), ⌣ ∣ . • Others using + : Schröder (1877) and Schröder (1890), McColl (1877), Ladd (1883), Whitehead (1898). 6
Peano • Uses ∪ (and ∩ ) in (Peano, 1888) and after. • These symbols adopted from Grassmann (1844). • Consciously avoids + as confusing logical and arithmetical operations. • Still uses ∪ for both union and disjunction. • First to mention Leibniz’s use of v and connection to vel in print. 7
Peano and Leibniz Instead of 푎 ✂✁ 푏 Leibniz has 푎 u 푏 (where u is the initial of uel). . . (Peano, 1891, p. 9) The sign ◦ corresponds to the Latin aut; the sign ✂✁ to vel. (Peano, 1894, p. 10) One may consider the sign ✂✁ as a deformation of v, the initial letter of “vel,” used by Leibniz as well for the same purpose. However, in the “Arithmetices principia” (Peano, 1889), which contains the first theory rendered in symbols, I have chosen the shape of logical signs in such a way as to avoid any confusion. (Peano, 1897, No. 3, 16) 8
Russell
Timeline The principal data points in this timeline are: • 1902 “On Likeness” • 1903 “Classes” • 1905/1906 “The Theory of Implication” 9
∨ for union In his spring 1902 manuscript “On Likeness,” (Russell, 1902, p. 440), Russell uses ‘ ∨ ’ for the first time (although not for disjunction): 휌 푅 ′ = ̆ ∗ 1 ⋅ 1 ( 푅 ) 퐿 ( 푅 ′ ) = 푅, 푅 ′ 휀 Rel Df E 1 → 1 푆휀 ( 휎 = 휌 ∨ ̆ 푆푅푆 ) 10
The birthday of ∨ In his 1903 “Classes,” another manuscript (some of which is lost), we find Russell’s first known use of ‘ ∨ ’ for propositional disjunction rather than for class union (Russell, 1903, p. 9): ∗ 12 ⋅ 58 Quad ( 휙 ) . =∶ ( Df E E 푓 ) {( 푥 ) 휙푥 ≡ 퐹푥 } ∨ ( 푓 ) {( 푥 ) 휙푥 ≡ 퐹푥 } Figure 1: Use of ∨ for disjunction in Russell, 1903 11
The birthday of ∨ Comments on the subsequent definition for ✂✁ show that Russell in this 1903 piece uses ‘ ∨ ’ only for disjunction. Thus, ‘ ∨ ’ (or at least its bolder ancestor ‘ ∨ ’) for disjunction (alone) was (re-)born in 1903. 12
First published ∨ In summer 1905, Russell wrote “The Theory of Implication,” which was published in the American Journal of Mathematics in April 1906. This piece is the first published use (rather than mention) of ∨ for disjunction since Leibniz’s one, i.e. in over 225 years. Df ∗ 4 ⋅ 1 . = ∼ 푝 ⊃ 푞 푝 ∨ 푞 13
Why ∨ ? What spurred Russell’s choice of ‘ ∨ ’ for disjunction? He does not explicitly say. Given the textual evidence, we think that Russell was ultimately inspired by 1. Peano’s raising of the possible choice of ‘ ∨ ’ for disjunction 2. Russell’s interest in the analogy of ‘ ∨ ’ and ‘ ✂✁ ’ 3. Russell’s interest in separating the symbols for class union and propositional disjunction in a way that Peano (deliberately) did not. 14
The adoption of ∨
Logic after Principia • Principia system adopted by philosophers (Wittgenstein, 1921; Ramsey, 1926; Carnap, 1929; Quine, 1934; Tarski, 1935). • Lvov-Warsaw school used Polish notation (usually 퐴푝푞 for the alternation of 푝 and 푞 ) • Algebraists like Löwenheim (1915), Skolem (1920), and Huntington (1933), continued to use the Boole-Peirce-Schröder notation. 15
The Hilbert school • Hilbert at al. worked intensively on logic in the 1920s • First relied on Schröder (1890). • Behmann (1918) introduced Principia . • Hilbert & Bernays built modern FOL 1917–1922. • First notation: + and × for and and or. • From 1921: &, v (and → ). • Widely adopted after textbook Principles of Theoretical Logic (Hilbert and Ackermann, 1928). 16
Quine In mathematical logic the ambiguity of ordinary usage is resolved by adopting a special symbol ‘ ∨ ’, suggestive of ‘vel’, to take the place of ‘or’ in the inclusive sense. Mathematical logic (Quine, 1940, pp. 12–13) In modern logic it is customary to write ‘ ∨ ’, reminiscent of vel, for ‘or’ in the nonexclusive sense: ‘ 푝 ∨ 푞 ’. Methods of logic (Quine, 1950, p. 5) For ‘ 푝 or 푞 ’ the notation is ‘ 푝 ∨ 푞 ’; here ‘’ ∨ ’ stands for the Latin vel, which means ‘or’ in the inclusive sense. Elementary logic (rev. ed) (Quine, 1965, p. 52). 17
Conclusions
The appealing short story told that our ‘ ∨ ’ for disjunction comes from the Latin vel and is due to Leibniz is not well-supported by the textual record. It was Russell who first systematically used ‘ ∨ ’ for disjunction, and it likely was not Latin-inspired. Simply put, the short story overlooks the messier and longer story of our near-uniform use of ‘ ∨ ’ for disjunction. 18
Why care about ∨ ? • Notations (and definitions), not just results, are important—and so is their history. • Sheds light on philosophical issues. . . • E.g. distinctions: algebraic/logical (Peano), class/propositions (Russell) • Use of notation indicates influences, commitments. 19
Tips • Practically any obscurity can make a decent paper topic. • Rabbit holes are deep (the ‘answers’ raise more questions). • Detective work is hard. • Publications don’t tell the whole story. • Looking into minor figures can be fruitful. • Editors are very important (e.g. Morley and Gregory H. Moore). • The internet is your friend in times of closed libraries. • Google Books • Archive.org • Hathi Trust • Library & archive websites (eg Leibniz archive) • Colleagues (by email or social media) 20
The triumphant declarations among some philosophers inspired and encouraged by the new logic, e.g. in the Carnap piece, was hard-won. Symbolism itself and the notions they are designed to express were (and still are) active topics of research. Symbols are theory-laden, too, and can be wrought under a faulty theory. This is a point Frege was keen to stress in justifying conceptual notation for scientific ends, and it was and is an active thread of the analytical tradition. 21
References i References Behmann, Heinrich (1918). “Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead”. Dissertation. Universität Göttingen. Boole, George (1847). The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning . Cambridge: Macmillan, Barclay, and Macmillan. google books: zv4YAQAAIAAJ . Cajori, Florian (1929). A History of Mathematical Notations, Volume 2: Notations Mainly in Higher Mathematics . La Salle, IL: Open Court. google books: wUkkzQEACAAJ . 22
References ii Carnap, Rudolf (1929). Abriss der Logistik . Vienna: Springer. Grassmann, Hermann (1844). Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik . Leipzig: O. Wigand. archive.org: dieausdehnungsl00grasgoog . Hilbert, David and Wilhelm Ackermann (1928). Grundzüge der theoretischen Logik . 1st ed. Berlin: Springer. repr. “Grundzüge der theoretischen Logik”. In: ed. by William Bragg Ewald and Wilfried Sieg. Berlin and Heidelberg: Springer, 2013, p. 806–916. Huntington, Edward V. (1933). “New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia Mathematica”. In: Transactions of the American Mathematical Society 35(1):274–304. doi: 10.2307/1989325 . JSTOR: 1989325 . 23
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