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On the Logical Philosophy of Assertive Graphs Daniele Chiffi1 · Ahti-Veikko Pietarinen2,3,4

© Springer Nature B.V. 2020

Abstract The logic of assertive graphs (AGs) is a modification of Peirce’s logic of existential graphs (EGs), which is intuitionistic and which takes assertions as its explicit object of study. In this paper we extend AGs into a classical graphical logic of assertions (ClAG) whose internal logic is classical. The characteristic feature is that both AGs and ClAG retain deep-inference rules of transformation. Unlike classical EGs, both AGs and ClAG can do so without explicitly introducing polarities of areas in their language. We then compare advantages of these two graphical approaches to the logic of assertions with a reference to a number of topics in philosophy of logic and to their deep-inferential nature of proofs. Keywords Assertion · Assertive graphs · Existential graphs · Peirce · Classical vs. non-classical logical graphs · Deep inference · Inferentialism

1 Introduction The notion of assertion plays an essential role in logic. It is a key ingredient in most logical systems, either implicitly or explicitly. Frege’s ideographical language of the

The work of the first author is supported by the Portuguese Foundation for Science and Technology, project (PTDC/MHC-FIL/0521/2014), and the Estonian Research Council PUT1305 Abduction in the Age of Fundamental Uncertainty. The work of the second author is partly supported by the Russian Academic Excellence Grant “5-100”, Formal Philosophy and the Estonian Research Council PUT1305 Abduction in the Age of Fundamental Uncertainty.

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Ahti-Veikko Pietarinen [email protected] Daniele Chiffi [email protected]

1

Politecnico di Milano, Milan, Italy

2

Nazarbayev University, Nur-Sultan, Kazakhstan

3

HSE University, Moscow, Russian Federation

4

Tallinn University of Technology, Tallinn, Estonia

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D. Chiffi, A.-V. Pietarinen

Begriffsschrift introduced a specific sign designating assertion, ‘’. It expresses the acknowledgement of the truth of the content of the assertion (Bellucci et al. 2017a). In Peirce’s graphical logic of existential graphs (EGs) which he introduced in 1896 there is no explicit sign for assertion (Bellucci and Pietarinen 2017b). Yet the notion of assertion surfaces virtually everywhere across his logical writings. The reason is that making an assertion signals liability that the utterer of the logical statement bears on the truth of the proposition (Peirce 1967). Peirce incorporated assertion as an implicit sign embedded in the notation of the Sheet of Assertion (SA) (Bellucci and Pietarinen 2017a). It is an embedded sign, since SA represents both logical truth and the assertoric nature of graphical logical formulas scribed upon the sheet. In intuitionistic logic, an explicit notion of assertion became commonplace in analyzing inference and proofs (Artemov and Iemhoff 2007; Carrara et al. 2017; Dummett 1975; Iemhoff and Metcalfe 2009), and to explicate certain topics in the philosophy of logic such as the mean