Organon F
Volume 26, August 2019, Issue 3, Pages 505–536
ISSN 2585-7150 (online) ISSN 1335-0668 (print)
Research Article | Special issue on Reflecting on the Legacy of C.I. Lewis: Contemporary and Historical Perspectives on Modal Logic
Semantic Tableau Versions of Some Normal Modal Systems with Propositional Quantifiers
Daniel Rönnedal
In Symbolic Logic (1932), C. I. Lewis developed five modal systems S1 − S5. S4 and S5 are so-called normal modal systems. Since Lewis and Langford’s pioneering work many other systems of this kind have been investigated, among them the 32 systems that can be generated by the five axioms T, D, B, 4 and 5. Lewis also discusses how his systems can be augmented by propositional quantifiers and how these augmented logics allow us to express some interesting ideas that cannot be expressed in the corresponding quantifier-free logics. In this paper, I will develop 64 normal modal semantic tableau systems that can be extended by propositional quantifiers yielding 64 extended systems. All in all, we will investigate 128 different systems. I will show how these systems can be used to prove some interesting theorems and I will discuss Lewis’s so-called existence postulate and some of its consequences. Finally, I will prove that all normal modal systems are sound and complete and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete.
C.I. Lewis, modal logic, propositional quantifiers, semantic tableaux.
Author
Daniel Rönnedal
Affiliation
Stockholm University
Address
Department of Philosophy, University of Stockholm, Universitetsvägen 10 D, 106 91 Stockholm, Sweden
daniel.ronnedal@philosophy.su.se
Received
12 July 2018
Accepted
2 January 2019
Published online
24 August 2019
Publishers
Institute of Philosophy of the Slovak Academy of Sciences
Institute of Philosophy of the Czech Academy of Sciences
APA
Rönnedal, D. (2019). Semantic Tableau Versions of Some Normal Modal Systems with Propositional Quantifiers. Organon F, 26(3), 505–536. https://doi.org/10.31577/orgf.2019.26309
Chicago
Rönnedal, Daniel. 2019. "Semantic Tableau Versions of Some Normal Modal Systems with Propositional Quantifiers." Organon F 26 (3): 505–536. https://doi.org/10.31577/orgf.2019.26309
Harvard
Rönnedal, D. (2019). Semantic Tableau Versions of Some Normal Modal Systems with Propositional Quantifiers. Organon F, 26(3), pp. 505–536. https://doi.org/10.31577/orgf.2019.26309
© Daniel Rönnedal
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