, Volume 195, Issue 8, pp 3621–3650 | Cite as

Gamma graph calculi for modal logics

  • Minghui MaEmail author
  • Ahti-Veikko Pietarinen


We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic.


Gamma graphs Peirce Existential graphs Broken-cut operator Modal logic Epistemic logic 



We thank the four anonymous reviewers of the present journal for comments. Earlier versions of the paper were presented by the second author at the Workshop on Existential Graphs held in Helsinki in August 2016, at the Charles S. Peirce International Centennial Congress held at the University of Massachusetts Lowell in July 2014, and at the Modalities and Modal Logic Conference held at the University of Copenhagen in May 2012. This paper is dedicated to Jay J. Zeman, who passed away just at the time of completion of the present paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Institute of Logic and CognitionSun Yat-Sen UniversityGuangzhouChina
  2. 2.Chair of PhilosophyTallinn University of TechnologyTallinnEstonia

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