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Rules in Relevant Logic. II: Formula Representation

Ross T. Brady
Studia Logica: An International Journal for Symbolic Logic
Vol. 52, No. 4 (Nov., 1993), pp. 565-585
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015698
Page Count: 21
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Rules in Relevant Logic. II: Formula Representation
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Abstract

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form, $A_{1}\rightarrow .A_{2}\rightarrow.....A_{n}\rightarrow B$, the conjunctive form, $A_{1}\&A_{2}\&...A_{n}\rightarrow B$, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form, $A_{1}\&A_{2}\&...A_{n}\supset B$ The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either $(A\rightarrow B)\&(B\rightarrow C)\rightarrow .A\rightarrow C$ or $A\rightarrow B\rightarrow .B\rightarrow C\rightarrow .A\rightarrow C$.

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