Abstract
Peirce’s graphical logic of Existential Graphs (egs) has no specific sign for assertion, although the notion is used virtually everywhere in Peirce’s logical theories. We outline the new system of Assertive Graphs (ags) that makes the embedded notion of assertions in egs explicit, and show how to inferentially transform ags to a classical graphical logic clag, without having to introduce polarities explicitly. We compare the philosophy of notation of ags to egs, where the latter has polarities both in its intuitionistic and classical cases. Our comparison is framed with respect to three different representations of implication, namely as cuts, boxes and scrolls. We also identify three fundamental differences in the meaning of the Sheet of Assertion and compare those with Peirce’s own proposed interpretation.
A.-V. Pietarinen—The work supported by the Estonian Research Council Personal Research Grant PUT 1305 (Abduction in the Age of Fundamental Uncertainty, PI Pietarinen), and Nazarbayev University Social Policy Grant 2018–2019.
D. Chiffi—The work supported by the Estonian Research Council Personal Research Grant PUT 1305 (Abduction in the Age of Fundamental Uncertainty, PI Pietarinen).
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Notes
- 1.
The reference R is to Peirce’s manuscripts by the Robin number [15].
- 2.
The blot is assumed to blacken the entire area within which it occurs. Since it is impractical to show this by actually blackening large blank areas, a heavy bullet is used instead.
- 3.
This reflects Peirce’s own turn of phrase from 1885: “But I cannot doubt that others, if they will take up the subject, will succeed in giving the notation a form in which it will be highly useful in mathematical work. I even hope that what I have done may prove a first step toward the resolution of one of the main problems of logic, that of producing a method for the discovery of methods in mathematics” (added emphasis).
- 4.
A historical tidbit is that the first to notice that Peirce’s conception of negation in logic might mean that the LEM would not hold was Gerrit Mannoury, Brouwer’s supervisor. Peirce was surely keen to limit the applicability of the LEM to propositions that are determinate.
- 5.
That is, from the outside-in direction, from the context of the conditional inside conditional forms, see R 292b, 293, 300, 515, 650, 669.
- 6.
Similar remarks hold on other logical constants of GrIn besides the scroll. In intuitionistic Beta graphs the phenomenon that the blank of the continuous sheet is that of the space of all transformations becomes even more pronounced, since the line of identity could be interpreted as signalling an identity of proofs.
- 7.
“[I]f we take a piece of blank paper, and form the resolve to write upon it some part of what we think about some real or imaginary condition of things, then ...the whole sheet having been devoted to that purpose exclusively, by the common understanding called of the graphist (as the person who makes assertions by “scribing”,—that is, by writing, drawing, or otherwise putting—on the sheet so devoted is to be called), and the interpreter (i.e. the person to whose understanding the graphist addresses the assertions that he scribes on the sheet), the graphist is at liberty to scribe any assertion on the sheet that he may be disposed to assert” (R 678).
- 8.
There is a continuum of intermediate logics and as logical graphs they have not yet been studied. One would expect changes in the meaning of the SA to be an important indicator of logical differences between intermediate logics.
- 9.
“[S]ince a scroll both of whose closes are empty asserts nothing, it is to be imagined that there is an abundant store of empty scrolls on a part of the sheet that is out of sight, whence one of them can be brought into view whenever desired” (R 669, 1910). Here Peirce speaks of the scrolls as double cuts. This convention implies the double-cut rule. By taking the “abundant store of empty scrolls” to refer to intuitionistic domains we get a different semantics for implication.
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Pietarinen, AV., Chiffi, D. (2018). Assertive and Existential Graphs: A Comparison. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_51
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