The Genealogy of Landon D. C. Elkind Richard Zach November 23, - - PowerPoint PPT Presentation

The Genealogy of ∨

Landon D. C. Elkind Richard Zach November 23, 2020 JHAP/BRRC Seminar

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Outline

The mystery of ∨ From Leibniz to Peano Russell The adoption of ∨ Conclusions

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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?

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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.

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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)

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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).

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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.

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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)

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Russell

Timeline

The principal data points in this timeline are:

• 1902 “On Likeness”
• 1903 “Classes”
• 1905/1906 “The Theory of Implication”

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∨ for union

In his spring 1902 manuscript “On Likeness,” (Russell, 1902, p. 440), Russell uses ‘∨’ for the first time (although not for disjunction): ∗ 1⋅1 (푅)퐿(푅′) = 푅, 푅′휀Rel E 1 → 1 푆휀(휎 = 휌 ∨ ̆ 휌 푅′ = ̆ 푆푅푆) Df

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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). =∶ ( E 푓) {(푥) 휙푥 ≡ 퐹푥} ∨ ( E 푓) {(푥) 휙푥 ≡ 퐹푥} Df

Figure 1: Use of ∨ for disjunction in Russell, 1903

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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.

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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. ∗ 4⋅1. 푝 ∨ 푞 = ∼푝 ⊃ 푞 Df

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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.

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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.

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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).

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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).

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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.

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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.

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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)

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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.

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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.

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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.

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References iii

Jevons, William Stanley (1883). The Principles of Science: A Treatise on Logic and Scientific Method. London: Macmillan and Co. archive.org: principlesofsci00jevo/. Kennedy, Hubert (1973). Selected Works of Giuseppe Peano. Toronto: University of Toronto Press. Kneale, William and Martha Kneale (1962). The Development of Logic. Oxford: Oxford University Press. Ladd, Christine (1883). “On the algebra of logic”. In: Studies in logic, by members of the Johns Hopkins University. Ed. by Charles Sanders Peirce. Boston, MA: Little, Brown, and Co., p. 17–71. google books: V7oIAAAAQAAJ.

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References iv

Leibniz, Gottfried Wilhelm (1679). “Matheseos universalis pars prior: De terminis incomplexis”. Gottfried Wilhelm Leibniz Bibliothek, Leibniz Handschriften LH 35, 1,

• 30. url: http://digitale-sammlungen.gwlb.de/sammlungen/sammlungsliste/

werksansicht/?no_cache=1&tx_dlf[id]=1656&tx_dlf[page]=18. – (1863). Leibnizens mathematische Schriften. Zweite Abtheilung: die mathematischen Abhandlungen Leibnizens enthaltend. Ed. by Carl Immanuel Gerhardt. Vol. 3. Leibnizens Gesammelte Werke aus den Handschriften der königlichen Bibliothek zu Hannover, Dritte Folge: Mathematik 7. Halle: Schmidt. google books: xDXtVCOjOHMC. Löwenheim, Leopold (1915). “Über Möglichkeiten im Relativkalkül”. In: Mathematische Annalen 76(4):447–470. doi: 10.1007/BF01458217.

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References v

McColl, Hugh (1877). “The calculus of equivalent statements and integration limits”. In: Proceedings of the London Mathematical Society s1-9(1):9–20. doi: 10.1112/plms/s1-9.1.9. Peano, Giuseppe (1888). Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva. Turin: Fratelli Bocca. “The Geometrical Calculus According to the Ausdehnungslehre of H. Grassman, Preceded by the Operations of Deductive Logic”. In: Selected works of Giuseppe

• Peano. Ed. and trans. by Hubert C. Kennedy. Toronto: University of Toronto Press,

1973, p. 75–100.

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References vi

Peano, Giuseppe (1889). Arithmetices principia nova methodo exposita. Rome: Fratelli

• Bocca. url: http://archive.org/details/arithmeticespri00peangoog. “The

Principles of Arithmetic, Presented by a New Method”. In: Selected works of Giuseppe Peano. Ed. and trans. by Hubert C. Kennedy. Toronto: University of Toronto Press, 1973, p. 100–134. – (1891). “Principii di logica matematica”. In: Rivista di matematica 1:1–10. “The Principles of Mathematical Logic”. In: Selected works of Giuseppe Peano. Ed. and

• trans. by Hubert C. Kennedy. Toronto: University of Toronto Press, 1973, p. 153–161.

– (1894). Introduction au formulaire mathématique. Turin: Fratelli Bocca. archive.org: formulairedemat03peangoog. – (1897). Formulaire mathématique. Vol. 2. Turin: Fratelli Bocca. archive.org: formulairedemat02peangoog.

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References vii

Peirce, Charles Sanders (1870). Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic. Cambridge: Welch, Bigelow, and Co. google books: fFnWmf5oLaoC. Quine, Willard Van Orman (1934). A System of Logistic. Cambridge, MA: Harvard University Press. archive.org: systemoflogistic0000quin. – (1940). Mathematical Logic. 1st ed. New York: W. W. Norton & Co. archive.org: mathematicallogi00quin. – (1950). Methods of Logic. 1st ed. New York: Henry Holt and Co. – (1965). Elementary Logic. revised. New York: Harper and Row. Ramsey, Frank P. (1926). “The foundations of mathematics”. In: Proceedings of the London Mathematical Society, Second Series 25(1):338–384. doi: 10.1112/plms/s2-25.1.338.

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References viii

Russell, Bertrand (1902). “On Likeness”. “On Likeness”. In: Toward the ‘Principles of Mathematics’ 1900–02. Ed. by Gregory H. Moore. The Collected Papers of Bertrand Russell (3). London and New York: Routledge, 1994, p. 437–451. – (1903). “Classes”. “Classes”. In: Toward the ‘Principles of Mathematics’ 1900–02.

• Ed. by Gregory H. Moore. The Collected Papers of Bertrand Russell (4). Routledge,

1994, p. 3–37. Schröder, Ernst (1877). Der Operationskreis des Logikkalkuls. Leipzig: Teubner. archive.org: deroperationskr00schrgoog. – (1890). Vorlesungen über die Algebra der Logik (exakte Logik). Vol. 1. Leipzig: Teubner.

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References ix

Skolem, Thoralf (1920). “Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen”. In: Videnskasselskapets skrifter, I. Matematisk-naturvidenskabelig klasse 1920(4):1–36. Tarski, Alfred (1935). “Einige methodologische Untersuchungen über die Definierbarkeit der Begriffe”. In: Erkenntnis 5:80–100. doi: 10.1007/BF00172286. Watson, Richard A. (1993). “Shadow history in philosophy”. In: Journal of the History

• f Philosophy 31:95–109. doi: 10.1353/hph.1993.0018.

Whitehead, Alfred North (1898). A Treatise on Universal Algebra, with Applications. Cambridge: Cambridge University Press. Wittgenstein, Ludwig (1921). “Logisch-philosophische Abhandlung”. In: Annalen für Naturphilosophie 14:198–262.

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