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.
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16 May 2020
This erratum is to correct in the paper of Daniele Chiffi and Ahti-Veikko Pietarinen, On the Logical Philosophy of Assertive Graphs.
Notes
See e.g. Bellucci and Pietarinen (2016), Pietarinen (2006), Roberts (1973), Shin (2002) and Zeman (1964) for these systems and the explanations of what the qualification ‘roughly’ means. Peirce also initiated the development of the gamma part of the theory of logical graphs, in which we find graphical modal (propositional and quantificational) logic, graphical epistemic logic and graphical higher-order logic, among others (see Ma and Pietarinen 2017a, b, c, d).
The cross bar is a design feature added (i) in order to respect the history of the development of relevant notation for disjunctive assertions, and (ii) in order to not to confuse the connection with quantificational lines that may occur in relevant extensions of AGs.
The term “blot” is derived from Peirce’s originals (R S-30, 1906). We should imagine it to refer to the blackening of the entire area on which that blot occurs, thus acquiring the meaning that there is no room left for any assertion on that area—that is, everything is false. For convenience, we represent such a blackening as a heavy black dot so that we need not represent very large (and possible infinite) areas of the sheet as black.
The sign
is Peirce’s original and favourite design for logical consequence relation.
That is, they are sound and complete, which can be shown by a Lindenbaum–Tarski construction as the underlying algebraic theory (Heyting algebra) is a variety and defines a congruence relation. A similar (though by no means identical) graphical intuitionistic system is Ma and Pietarinen (2018), with more on e.g. admissible rules. The set of rules for AGs differs from graphical intuitionistic system in order to compensate for the lack of polarities—admittedly additional rules have to be introduced to do that and also because there are more logical primitives in AGs.
An exception is when the consequent of the cornering whose antecedent we are to insert is occupied by the blot (and the antecedent J to be inserted is not the blot \(\CIRCLE \)).
This restriction to the applicability of the rule (InsA) is important, as the rule would otherwise permit derivation of invalid principles.
For additional comparisons between the notations employed in AGs, intuitionistic logic of graphs and classical AGs, and especially in relation to how conditionals are expressed in them, see Pietarinen and Chiffi (2018).
We have preliminarily discussed, in another context, some of them in Chiffi and Pietarinen (2018).
This is not to claim that AGs would be the first graphical deep-inference system, as systems whose notation is explicitly diagrammatic may also have that property, among them not only the logic of existential graphs but also for example spider diagrams (Howse et al. 2005), of which there is now a considerable and growing body of research available (Chapman et al. 2018).
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Acknowledgements
This work has been presented at The First Int. Logic Day, Tallinn University of Technology, Tallinn, Estonia; Assertion and Belief, University of Padua, Italy; Department of Computer Science Seminar, University of Tübingen, Germany; 9th Interational Conference on the Theory and Applications of Diagrams, Edinburgh, Scotland; Around Peirce, 6th World Congress and School on Universal Logic, Vichy, France; Grupo de Investigación en Lógica, Lenguaje e Información (GILLIUS), University of Sevilla, Spain; Philosophy Seminar, Nazarbayev University. We thank the audience of these events for valuable suggestions, among them F. Bellucci, M. Capraru, M. Carrara, J.-M. Chevalier, C. De Florio, M. Fontaine, C. Barés Gómez, V. Morato. Á. Nepomuceno, M. Plebani, P. Schroeder-Heister, G. Sundholm, L. Tranchini, T. Uustalu. Above all, our thanks go to the two reviewers of the present journal for helpful comments that led to a number of improvements.
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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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Chiffi, D., Pietarinen, AV. On the Logical Philosophy of Assertive Graphs. J of Log Lang and Inf 29, 375–397 (2020). https://doi.org/10.1007/s10849-020-09315-6
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DOI: https://doi.org/10.1007/s10849-020-09315-6