Journal of Applied Logic 13 (3):188-196 (2015)

Gillman Payette
University of British Columbia
The purpose of this brief note is to prove a limitative theorem for a generalization of the deduction theorem. I discuss the relationship between the deduction theorem and rules of inference. Often when the deduction theorem is claimed to fail, particularly in the case of normal modal logics, it is the result of a confusion over what the deduction theorem is trying to show. The classic deduction theorem is trying to show that all so-called ‘derivable rules’ can be encoded into the object language using the material conditional. The deduction theorem can be generalized in the sense that one can attempt to encode all types of rules into the object language. When a rule is encoded in this way I say that it is reflected in the object language. What I show, however, is that certain logics which reflect a certain kind of rule must be trivial. Therefore, my generalization of the deduction theorem does fail where the classic deduction theorem didn’t.
Keywords Logical Consequence  substructural Logic  admissible rules
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DOI 10.1016/j.jal.2015.03.001
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References found in this work BETA

Deflating Logical Consequence.Lionel Shapiro - 2011 - Philosophical Quarterly 61 (243):320-342.
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