Abstract
The blot is a sign in Peirce’s diagrammatic syntax of existential graphs that has hitherto been neglected in the literature on logical graphs. It is needed in order to trigger the cut-as-negation to come out from the scroll, namely from the implicational sign of a positive implicational (paradisiacal) logic. Since the cut-as-negation presupposes the blot and the scroll, what does the blot represent? On the one hand, it stands for constant absurdity, but on the other hand, Peirce takes it to be an affirmative sign. This paper explores the blot and its logical and conceptual properties from the multiple perspectives of notation, rules of transformation, icons, and scriptibility of graphs. It explains the apparent conflict in the blot’s meaning in its capacity of giving rise to the pseudo-graph that exploits positive character of absurdity. In effect, the blot is the mirror image of the sheet of assertion, not its complementation. On the sheet, it acts as a non-juxtaposable singularity.
A.-V. Pietarinen—The paper was prepared within the framework of the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ‘5-100’.
N. Haydon—Research supported by the ESF funded Estonian IT Academy research measure (project 2014-2020.4.05.19-0001).
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Notes
- 1.
The caption numberings in quotations preserve those in Peirce’s original writings.
- 2.
The consequent should be “…then every assertion is true”. The meaning of the “red blot” as “…then every assertion is false” comes from an earlier lecture draft (R 450), which Peirce soon in his next draft (R 455) corrects to the original meaning of the blot as in Convention 10.
- 3.
In Peirce’s hand, a similar sequence looked like this (R 455(s)):
Peirce intended this to show that “the impossibility [that exists within the inloop] destroys the cut and all it contains” (ibid.). By this, Peirce is preparing ground for his decidability operations for the Alpha system (Roberts 1997).
- 4.
See Oostra (2010) on Alpha System with the scroll that agrees with propositional intuitionistic logic. In this case, new graph for disjunction needs to be introduced as in intuitionistic logic, logical connectives are not interdefinable. How such modifications demonstrate the potential insights of Peirce’s EGs has been discussed in Shafiei (2019). Moreover, Ma and Pietarinen (2018) have offered an EGs version for intuitionistic logic analyzing the nature of deep inference.
- 5.
When things are unscriptible, it is even not clear whether deduction works as the right mode of reasoning in that dark realm (Peirce once talked about the mode of reasoning of “correction”, which is not “deduction” when all propositions are unscriptible (Ma and Pietarinen 2019).
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Pietarinen, AV., Bellucci, F., Bobrova, A., Haydon, N., Shafiei, M. (2020). The Blot. In: Pietarinen, AV., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., Linker, S. (eds) Diagrammatic Representation and Inference. Diagrams 2020. Lecture Notes in Computer Science(), vol 12169. Springer, Cham. https://doi.org/10.1007/978-3-030-54249-8_18
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