Taking Peirce’s Graphs Seriously

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Notes

¹ See, for instance, the review by Frederik Stjernfelt, Transactions of the Charles S. Peirce Society, Vol. 57:1 (Winter 2021), pp. 114–127.

² That Peirce insists that his graphs are scribed on a sheet of assertion already raises fundamental, and largely hitherto unaddressed, questions in philosophical logic and the philosophy of language about the primacy of that speech act above others, at least for the purpose of logical investigation. Why, for instance, couldn’t logical diagrams be inscribed instead upon a sheet of denial? In the aforementioned second Logical Tract of 1903 (selection 30 of LOF, Vol 2:1), Peirce connects this question to the naturalness of representing conjunction by juxtaposition on a logical diagram (see LOF, 2:1, pp. 137ff). Here Peirce makes much of the idea that scribing two diagrams together on a logical sheet should have the same effect as scribing them both independently. Thus the reason why he thinks his ‘phemic sheet’ should be associated with the pragmatic force of assertion rests on the fact that it is assertion (and not denial) that distributes into conjunction.

³ The ovals can also interact with the lines alone to form graphs with meaningful content. For instance, wholly enclosing a line of identity with an oval signifies the denial of any existing thing, while drawing an oval across a segment of a line of identity tells us that there are at least two non-identical existences. One interesting distinction between the graph notation and conventional algebraic notation is that the graphical notation is token-referential rather than type-referential. That is, reference to one and the same thing is secured through attachment to the same particular line of identity, rather than being indexed to distinct occurrences of the same term-type.

⁴ See LOF, Vol 1, p. 16.

⁵ In his commentary on the materials in Volume 1, Pietarinen tells a fascinating story about the emergence of the scroll as device to signify a conditional. Peirce’s own story about the best way to graph the conditional is included in The Logical Tracts (No. 2) [LOF 2, pp. 140ff].

⁶ The bilateralism that one can discern in the writings of Frank Ramsey might well have been due to Peirce, of whose work Ramsey was well aware. See Beisecker (2019).

⁷ As it was, in the entry on ‘Symbolic Logic’ in the Baldwin Dictionary of Philosophy and Psychology (Vol. 2, pp. 645–650, reprinted in CP 4.372–4393) – one of the few places where Peirce was able to publish his theory of the graphs – Peirce (and Christine Ladd-Franklin) were forced to adopt a bastardized single-line or ‘linear’ notation for the graphs, in which the ovals were replaced by matching parentheses.

⁸ Alternately, one might also think of this modality as the one Sporting Life put to song in Porgy and Bess: It ain’t necessarily so.

⁹ See, for instance, Francesco Berto and Greg Restall, ‘Negation on the Australian Plan,’ Journal of Philosophical Logic (2019) 48::1119–1144. Like Peirce, Berto and Restall argue that negation is primarily used to express exclusions or incompatibilities. Ironically, then, the so-called ‘Australian plan for negation’ (which is often contrasted with the ‘American plan’) actually has roots in America, specifically in Arisbe, near the Delaware Water Gap.