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On Defining Necessity in Terms of Entailment

Dennis Henry and Michael Byrd
Studia Logica: An International Journal for Symbolic Logic
Vol. 38, No. 2 (1979), pp. 95-104
Published by: Springer
Stable URL: http://www.jstor.org/stable/20014934
Page Count: 10
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On Defining Necessity in Terms of Entailment
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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 $\{\vdash _{E}Lp\rightarrow p,\vdash _{E}L(p\rightarrow q)\rightarrow (Lp\rightarrow Lq),\dashv _{E}p\rightarrow Lp\}$ if and only if its has the form $C_{1}\rightarrow .C_{2}\rightarrow...\rightarrow .C_{n}\rightarrow p$, where each $C_{i}$ is equivalent in E to either p → p or ((p → p) → p) → p.

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