TY - GEN
T1 - A Semantics for a Failed Axiomatization of K
AU - Omori, Hitoshi
AU - Skurt, Daniel
N1 - Funding Information:
1 The work reported in this paper was initially supported by JSPS KAKENHI Grant Number JP18K12183, and later by a Sofja Kovalevskaja Award of the Alexander von Humboldt-Foundation, funded by the German Ministry for Education and Research. Some part of the results were presented at Oberseminar Logik und Sprachtheorie in Tübingen in May 2019. We would also like to thank the three referees for their careful reading and helpful comments, suggestions and corrections that improved the paper. Email: [email protected] 2 Some of the observations of this article were presented at Trends in Logic XVIII in Milan in September 2018, at the Kolloquium Philosophie & Linguistik in Göttingen in November 2018, at the Philosophisches Kolloquium in Leipzig in December 2018. DS would like to thank the audience for their helpful comments and encouragements. Email: [email protected]
Publisher Copyright:
© 2020 College Publications. All rights reserved.
PY - 2020
Y1 - 2020
N2 - In "Yet another "choice of primitives" warning: Normal modal logics", Lloyd Humberstone discussed a failed axiomatization for the normal modal logic K with ⋄ as the only primitive modal operator. More specifically, Humberstone observed that a simple translation of the standard axiomatization for K, where all occurrences of the necessity operator are replaced by -⋄-, will not be a complete axiomatization, since ⋄p → ⋄ - p is not derivable. As a result, the emerging proof system resists the standard Kripke semantics. However, to the best of the authors' knowledge, no semantics for the failed axiomatization of K is known in the literature. The aim of this article is to offer the first sound and complete semantics for the failed axiomatization of K by making use of a semantical framework suggested by John Kearns. In short, Kearns' semantics is a combination of non-deterministic semantics together with an additional hierarchy of valuations. We will also discuss a small question left open by Humberstone in the same paper. In view of the results presented in this article, we hope to establish part of the versatility of Kearns' semantics.
AB - In "Yet another "choice of primitives" warning: Normal modal logics", Lloyd Humberstone discussed a failed axiomatization for the normal modal logic K with ⋄ as the only primitive modal operator. More specifically, Humberstone observed that a simple translation of the standard axiomatization for K, where all occurrences of the necessity operator are replaced by -⋄-, will not be a complete axiomatization, since ⋄p → ⋄ - p is not derivable. As a result, the emerging proof system resists the standard Kripke semantics. However, to the best of the authors' knowledge, no semantics for the failed axiomatization of K is known in the literature. The aim of this article is to offer the first sound and complete semantics for the failed axiomatization of K by making use of a semantical framework suggested by John Kearns. In short, Kearns' semantics is a combination of non-deterministic semantics together with an additional hierarchy of valuations. We will also discuss a small question left open by Humberstone in the same paper. In view of the results presented in this article, we hope to establish part of the versatility of Kearns' semantics.
KW - Non-deterministic Semantics
KW - Normal Modal Logics
KW - Primitive Connectives
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M3 - Conference contribution
AN - SCOPUS:85108726799
SN - 1904987206
T3 - Advances in Modal Logic
SP - 481
EP - 501
BT - Advances in Modal Logic, AiML 2020
A2 - Olivetti, Nicola
A2 - Verbrugge, Rineke
A2 - Negri, Sara
A2 - Negri, Sara
A2 - Sandu, Gabriel
PB - College Publications
T2 - 13th Conference on Advances in Modal Logic, AiML 2020
Y2 - 24 August 2020 through 28 August 2020
ER -