Lord Russell is desired by Mr. Bertrand Russell to convey the following message to Mr. Broad: — “I have read him in this month’s Mind with much pleasure and approval, but I don’t altogether like shrieks3 upside-down. Tell him he should get into communication with Whitehead as to notation.”
[document] The letter was edited from a photocopy of the unsigned, typed letter is in Broad’s papers in Trinity College Library, Cambridge. BR’s message is not in the extant letters to his brother, and probably was communicated to Frank during a visit to Brixton.
C.I. Broad Frank Russell got the second initial wrong.
in this month’s Mind … shrieks Exclamation marks — known as “shrieks” — serve a number of purposes in the symbolism of Principia Mathematica, e.g. to indicate a predicative function or that a descriptive function is uniquely satisfied. In “A General Notation for the Logic of Relations”, Mind 27 (1918): 284–303, C.D. Broad gave them additional roles, e.g. to distinguish relational propositions from relational complexes. The relevant one in this context is “R(!, ‘‘β)”, the class of things which have the relation R to some member of β. (The Principia notation for this is R‘‘β [PM, *37.01].) Similarly, in Broad’s notation, “R(‘‘α, !)” represents the class of things to which some member of α has the relation R. But also to be considered are the class of things which have the relation R to every member of β and the class of things to which every member of α has the relation R. For these Broad turned the shriek upside-down: “R(¡, ‘‘β)” and “R(‘‘α, ¡)”, respectively. It’s not obvious why BR didn’t like the upside-down shriek notation, but perhaps it was because it might be confused with the other Principia notation. For example, “¡” is hard to distinguish in writing from “İ”, which, by the usual PM principles, would represent the product of two relations (except that in PM BR and Whitehead omitted the over-printed dot). Broad may, in fact, have had this in mind, for the situation he used “¡” to represent is a kind of product (albeit more properly of classes than relations), while that symbolized by “!” is a kind of sum. Alternatively, BR may have thought that using an upside-down shriek would be hard to distinguish in writing from an over-dotted “1”, which is needed in relation-arithmetic (cf. PM, *181).
Rather surprisingly, the formalism of Principia allows only for two-term (or dyadic) relations. In fact the class of relations, Rel, is defined for dyadic relations only. Broad’s main purpose in the paper was to extend the notation to allow for relations of greater adicity. This, he said, would be of especial importance in dealing with geometry. Indeed, he demonstrated the power of his symbolism by translating Hilbert’s four axioms of order for Euclidean geometry into it. (Cf. David Hilbert, Foundations of Geometry, trans. E.J. Townsend [Chicago: Open Court, 1902].) This, presumably, was why BR recommended that he get in touch with Whitehead, who was working (on his own) on the never-finished, fourth volume of Principia, which would have dealt with geometry.
Though the purpose of Broad’s paper was simply to propose a symbolism, and he explicitly avoided taking a stand on logical and philosophical issues, the fact that BR approved of the paper indicates the extent to which he was moving away from some of the doctrines in Principia. In particular, Broad’s notational distinction between relational propositions and relational complexes, which would be a very useful one in a more ordinary sort of system than Principia, has no role in Principia, where there are no propositions.
All that said, Broad’s notation never did catch on.