nach oben

Journal of Logic, Language and Information

Erschienen in:

20.04.2020

On the Logical Philosophy of Assertive Graphs

verfasst von: Daniele Chiffi, Ahti-Veikko Pietarinen

Erschienen in: Journal of Logic, Language and Information | Ausgabe 4/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Nächster Artikel Correction to: On the Logical Philosophy of Assertive Graphs

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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

https://static-content.springer.com/image/art%3A10.1007%2Fs10849-020-09315-6/MediaObjects/10849_2020_9315_Figaa_HTML.gif

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).

Artemov, S., & Iemhoff, R. (2007). The basic intuitionistic logic of proofs. Journal of Symbol Logic, 72, 439–451.CrossRef

Bellucci, F., & Pietarinen, A.-V. (2016). Existential graphs as an instrument of logical analysis: Part I. Alpha. The Review of Symbolic Logic, 9, 209–237.CrossRef

Bellucci, F., & Pietarinen, A.-V. (2017a). Assertion and denial: A contribution from logical notation. Journal of Applied Logic, 24, 1–22.

Bellucci, F., & Pietarinen, A.-V. (2017b). From Mitchell to Carus: 14 years of logical graphs in the making. Transactions of the Charles S. Peirce Society, 52(4), 539–575.CrossRef

Bellucci, F., Moktefi, A., & Pietarinen, A.-V. (2017a). Simplex sigillum veri: Peano, Frege and Peirce on the primitives of logic. History and Philosophy of Logic, 39(1), 80–95.CrossRef

Bellucci, F., Chiffi, D., & Pietarinen, A.-V. (2017b). Assertive graphs. Journal of Applied Non-Classical Logics, 28(1), 72–91.CrossRef

Boyd, K. (2016). Peirce on assertion, speech acts, and taking responsibility. Transactions of the Charles S. Peirce Society, 52(1), 21–46.CrossRef

Brünnler, K. (2003). Deep inference and symmetry in classical proof. Ph.D. thesis, Technische Universität Dresden.

Bruscoli, P., & Guglielmi, A. (2009). On the proof complexity of deep inference. ACM Transactions on Computational Logic, 10(2), 1–34.CrossRef

Bruscoli P., & Straßburger L. (2017). On the length of medial-switch-mix derivations. In: J. Kennedy, & R. de Queiroz (Eds.), Logic, language, information, and computation. WoLLIC 2017. Lecture notes in computer science (Vol. 10388). Berlin: Springer. https://doi.org/10.1007/978-3-662-55386-2

Carrara, M., Chiffi, D., & De Florio, C. (2017). Assertions and hypotheses: A logical framework for their opposition relations. Logic Journal of the IGPL, 25(2), 131–144.

Chiffi, D. & Pietarinen, A.-V. (2018). Assertive and existential graphs: A comparison. In: P. Chapman, G. Stapleton, A. Moktefi, S. Perez-Kriz & F. Bellucci (Eds.), Diagrammatic representation and inference. Diagrams 2018. Lecture notes in computer science (Vol. 10871). Springer.

Chapman P., Stapleton G., Moktefi A., Perez-Kriz S., Bellucci F. (eds.) (2018). Diagrammatic representation and inference. Diagrams 2018. Lecture notes in computer science (Vol. 10871). Springer.

Došen, K. (1994). Logical constants as punctuation marks. In D. M. Gabbay (Ed.), What Is a logical system? (pp. 273–296). Oxford: Clarendon Press.

Dummett, M. (1975/1983). The philosophical basis of intuitionistic logic. In: P. Benacerraf, & H. Putnam (eds.). Philosophy of mathematics: Selected readings (2nd Ed., pp. 97–129). Cambridge: Cambridge University Press.

Dummett, M. (1981). Frege. Philosophy of language (2nd ed.). London: Duckworth.

Guglielmi, A. (2010). Deep inference. http://alessio.guglielmi.name/res/cos/

Hacking, I. (1979). What is logic? Journal of Philosophy, 76, 285–319.CrossRef

Heyting, A. (1930). Sur la logique intuitionniste, Académie Royale de Belgique. Bulletin, 16, 975–963.

Heyting, A. (1956). Intuitionism. An introduction. Studies in logic and the foundations of mathematics. Amsterdam: North-Holland.

Howse, J., Stapleton, G., & Taylor, H. (2005). Spider diagrams. Journal of Computational Mathematics, 8, 145–194.

Iemhoff, R., & Metcalfe, G. (2009). Proof theory for admissible rules. Annals of Pure and Applied Logic, 159, 171–186.CrossRef

Ma, M., & Pietarinen, A.-V. (2017a). Peirce’s sequent proofs of distributivity. Lecture Notes in Computer Science, 10119(168–182), 13. https://doi.org/10.1007/978-3-662-54069-5.CrossRef

Ma, M., & Pietarinen, A.-V. (2017b). Graphical sequent calculi for modal logics. Electronic Proceedings in Theoretical Computer Science, 243, 91–103. https://doi.org/10.4204/EPTCS.243.7.CrossRef

Ma, M., & Pietarinen, A.-V. (2017c). Proof analysis of Peirce’s alpha system of graphs. Studia Logica, 105(3), 625–647.CrossRef

Ma, M., & Pietarinen, A.-V. (2017d). Gamma graph calculi for modal logics. Synthese. https://doi.org/10.1007/s11229-017-1390-3.

Ma, M., & Pietarinen, A.-V. (2018). A graphical deep inference system for intuitionistic logic. Logique & Analyse, 245, 73–114.

Peirce, C. S. (1967). Manuscripts in the Houghton Library of Harvard University. Identified by Richard Robin, Annotated Catalogue of the Papers of Charles S. Peirce, Amherst: University of Massachusetts Press, 1967, and The Peirce Papers: A supplementary catalogue. Transactions of the C. S. Peirce Society, 7(1971), 37–57.

Pietarinen, A.-V. (2006). Signs of logic: Peircean themes on the philosophy of language, games, and communication. Dordrecht: Springer.

Pietarinen, A.-V., & Chiffi, D. (2018). Assertive and existential graphs: A comparison. In: P. Chapman, G. Stapleton, A. Moktefi, S. Perez-Kriz, & F. Bellucci (Eds.), Diagrammatic representation and inference. Diagrams 2018. Lecture notes in computer science (Vol. 10871). Springer.

Prior, A. (1960). The runabout inference-ticket. Analysis, 21, 38–39.CrossRef

Roberts, D. D. (1973). The existential graphs of Charles S. Peirce. The Hague: Mouton.CrossRef

Shin, S.-J. (2002). The iconic logic of Peirce’s graphs. Cambridge, MA: MIT Press.CrossRef

Sundholm, B. G. (1997). Implicit epistemic aspects of constructive logic. Journal of Logic, Language, and Information, 6, 191–212.CrossRef

Troelstra, A. S., & van Dalen, D. (1988). Constructivism in mathematics (Vol. I). Amsterdam: North Holland.

Zeman, J. (1964). The graphical logic of Charles S. Peirce. Ph.D. thesis, University of Chicago.

Titel: On the Logical Philosophy of Assertive Graphs
verfasst von: Daniele Chiffi
Ahti-Veikko Pietarinen
Publikationsdatum: 20.04.2020
Verlag: Springer Netherlands
Erschienen in: Journal of Logic, Language and Information / Ausgabe 4/2020
Print ISSN: 0925-8531
Elektronische ISSN: 1572-9583
DOI: https://doi.org/10.1007/s10849-020-09315-6

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 4/2020

Natural Density and the Quantifier “Most”

Correction to: On the Logical Philosophy of Assertive Graphs

Iterated Admissibility Through Forcing in Strategic Belief Models

Towards an Algebraic Semantics for Implicatives

Formalizing GDPR Provisions in Reified I/O Logic: The DAPRECO Knowledge Base

A Four-Valued Dynamic Epistemic Logic

Premium Partner