Tracing the symbol for disjunction

There was an interesting Twitter thread launched by Richard Zach’s remark:

Here we also get a curious sub-question: when did Russell switch from taking “⊃” as primitive (as he does in 1901—see Collected Papers Vol. 3, Chapter 11, page 381) to taking “∨” and “∼” as primitive (as he does in Principia)?

The short story is that the earliest use of “∨” for disjunction is early 1903 in Russell’s manuscript “Classes” (see the picture below) and the switch to taking “∨” and “∼” as primitive rather than “⊃” occurred around 21 August 1906. The longer story, most of which comes from Gregory Moore’s introductions in The Collected Papers of Bertrand Russell, Volume 3, is retold below.


Russell’s 1901 works are already deeply influenced by Peano, particularly in their choice of symbolism. However, even there, Russell shows some independent choices of notation. For example, in “The Logic of Relations,” Russell has already introduced the use of dot “.” for conjunction. This is in addition to his and Peano’s shared use of the same symbol as a scope marker. But as Dirk Schlimm pointed out to Richard Zach, Peano does not use “.” for conjunction. Still, in 1901 and 1902, Russell is working from (that is, working his way out of) the standpoint of treating certain logical symbols as having a dual use, namely, for symbolizing relations between classes and between propositions.

This dual usage was made possible partly by Russell’s logic of propositions around this time: on that theory, propositions stand in relations of implication, disjunction, conjunction, and so on, just as classes would stand in relations of subset and membership. This conception of propositional logic is very different from the modern one in which symbols like “∧,” “∨,” and “¬” connect well-formed formulas to form a new formula. Rather, on Russell’s view (and probably Peano’s also), such symbols connect propositions as terms to make a formula.

As Gregory Moore notes (ibid., Chapter 15, page 439), in his spring 1902 “On Likeness” (which was not published until Collected Papers Vol. 3), Russell departs further from Peano’s notation. Russell now distinguishes between “⊃” for the relations of propositional implication and relation inclusion and “⊂” for class inclusion. Similarly, he introduces “∼” for propositional negation rather than using “–” for this and for a class’ complement.

Here, however, his usage of “∧” and “∨” is the reverse of our modern usage: Russell introduces the new notation “∧” and “∨” for class intersection and class union, whereas he uses the old notation “∩” for propositional conjunction and relation intersection, and he used “∪” for propositional disjunction and relational union. This use of “∧” and “∨” can be seen in *1·1 below, and this use of “∪” can be seen in *2·1 below.

CPBRVol3OnLikenessPage440

Collected Papers Vol. 3, Chapter 15, page 440: Russell’s 1902 “On Likeness”

As Moore says, this usage gets reversed in Russell’s early 1903 “Classes” (Collected Papers Vol. 4, Chapter 1a), so that “∨” now symbolizes disjunction and “∪” symbolizes class union. This persists from Principles through Principia, and seems to be origin of our modern usage.

CPBRVol4GeneralTheoryOfClassesPage9

Collected Papers Vol. 4, Chapter 1a, page 9: the introduction of “∨” for disjunction in *12·58 of Russell’s 1903 “Classes”

All of this is a post-Peano influence with Russell’s considered modifications. There is a second wave of influence post-Frege’s deeper influence on Russell in early 1903. A deeper chronology is given by Moore in Collected Papers Vol. 3, Introduction, §V (that’s section five and not section or!).

Finally, as Moore again notes (Collected Papers Vol. 5, Introduction, §VII), in 1905 Russell switched to taking “∼” as primitive when he came to the conclusion, for philosophical reasons, that negation could not be defined. In 1906 Russell took “∨” as primitive because “two primitive propositions are made superfluous,” as he says in a 21 August 1906 letter to Louis Couturat (ibid., page xlv).


As a side remark, in the 1902 paper, Russell also introduces “≡” for propositional equivalence (it had previously been used for identity of individuals) rather than using “=” for both propositional equivalence and class identity. Against the backdrop of Russell’s propositional logic on which propositions can be terms of relations, there may be a curious anticipation of propositional equality in modern type theories treating propositions-as-types.


As another side remark, it appears Peano never changed his notation in response to Russell’s innovations. Curiously, a cursory look at Peano’s 1913 review of Principia in the Formulario is largely just Peano showing how the theorems of Principia could be symbolized in his preferred notation. This might be seen as a refusal to “get with the times” of modernized notation that avoids dual uses of symbols. But Peano’s concern was more with the abbreviation and crisp communication of mathematical knowledge.

Frege mockingly criticized Peano’s notation choices as making “the convenience of the typesetter” the “summum bonum” in his 1897 “On Mr. Peano’s Conceptual Notation and My Own.” However, in Peano’s defense, we see now a renewed concern with conveniently crisp presentations of appropriately abbreviated mathematics in modern proof-assistant libraries. In addition, the context in Peano’s works was supposed to clarify which dual use was meant (or else reading it on either use was not illicit and allowed for a briefer presentation of results), much as a modern typing context can clarify syntactic “ambiguities.”

I do not say this to defend dual uses of notation, but only to indicate that Peano’s values and goals, the ones that guided his notation choices, are deeply relevant to us today, just as avoiding syntactic ambiguities was and still is.

 

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