### Abstract

Original language | English |
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Journal | Synthese |

Publication status | Published - 2017 |

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**Gamma Graph Calculi for Modal Logics.** / Pietarinen, Ahti-Veikko; Ma, Minghui.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Gamma Graph Calculi for Modal Logics

AU - Pietarinen, Ahti-Veikko

AU - Ma, Minghui

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

M3 - Article

JO - Synthese

JF - Synthese

SN - 0039-7857

ER -