Gamma graph calculi for modal logics

Minghui Ma*, Ahti Veikko Pietarinen

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

23 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)3621-3650
Number of pages30
Issue number8
Early online date7 Apr 2017
Publication statusPublished - Aug 2018

Scopus Subject Areas

  • Philosophy
  • Social Sciences(all)

User-Defined Keywords

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


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