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).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
References
Bellucci, F., Pietarinen, A.-V.: Existential graphs as an instrument of logical analysis: part I. Alpha. Rev. Symb. Logic 9, 209–237 (2016)
Bellucci, F., Pietarinen, A.-V.: From Mitchell to Carus: 14 years of logical graphs in the making. Trans. Charles S. Peirce Soc. 52(4), 539–575 (2017a)
Bellucci, F., Pietarinen, A.-V.: Assertion and denial: a contribution from logical notations. J. Appl. Logics 25, 1–22 (2017b)
Bellucci, F., Chiffi, D., Pietarinen, A.-V.: Assertive graphs. J. Appl. Non-Class. Logics (2018). https://doi.org/10.1080/11663081.2017.1418101
Carrara, M., Chiffi, D., De Florio, C.: Assertions and hypotheses: a logical framework for their opposition relations. Logic J. IGPL 25(2), 131–144 (2017)
Chiffi, D., Pietarinen, A-V.: On the Logical Philosophy of Assertive Graphs, preprint
Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Louis Nebert, Halle a. S. (1879)
Guglielmi, A.: Deep Inference. http://alessio.guglielmi.name/res/cos/
Heyting, A.: Intuitionism: An Introduction. North Holland, Amsterdam (1956)
Ma, M., Pietarinen, A.-V.: Proof analysis of Peirce’s alpha system of graphs. Stud. Logica. 105(3), 625–647 (2017)
Ma, M., Pietarinen, A.-V.: A graphical deep inference system for intuitionistic logic. Log. Anal., in press
Oostra, A.: Los gráficos Alfa de Peirce aplicados a la lógica intuicionista. Cuad. Sist. Peir. 2, 25–60 (2010)
Peirce, C.S.: On the algebra of logic. Am. J. Math. 3(1), 15–57 (1880)
Peirce, C.S.: On the algebra of logic: a contribution to the philosophy of notation. Am. J. Math. 7(2), 180–196 (1885)
Peirce, C.S.: Manuscripts in the Houghton Library of Harvard University. Cited as R or L followed by manuscript or letter number (1967)
Roberts, D.D.: The Existential Graphs of C.S. Peirce. Mouton, The Hague (1973)
Searle, J.R., Vanderveken, D.: Foundations of Illocutionary Logic. Cambridge University Press, Cambridge (1985)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-91376-6_51
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91375-9
Online ISBN: 978-3-319-91376-6
eBook Packages: Computer ScienceComputer Science (R0)