Abstract
In their book Entailment, Anderson and Belnap investigate the consequences of defining ‘Lp’ (it is necessary that p) in system E as (p→p)→p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions {⊢ E Lp→p, ⊢EL(p→q)→(Lp→Lq), ⊢E p→Lp} if and only if its has the form C 1→.C2→ ...→. Cn→p, where each C iis equivalent in E to either p→p or ((p→p)→p)→p.
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References
A. Anderson and N. Belnap, Entailment: The Logic of Relevance and Necessity, Princeton University Press, 1975.
R. K. Meyer, R i-the bounds of finitude, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 16, pp. 385–387.
D. Henry and M. Byrd, Sugihara's criterion and some structural parallels between E → and S3 →, forthcoming in Zeitschrift für Mathematische Logik and Grundlagen der Mathematik.
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Henry, D., Byrd, M. On defining necessity in terms of entailment. Stud Logica 38, 95–104 (1979). https://doi.org/10.1007/BF00370435
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DOI: https://doi.org/10.1007/BF00370435