Abstract

The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential !) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL + , is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL + , instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different introduction rule for the exponential, !I + , which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph "Natural Deduction". It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL + ; in particular one can formulate a notion of strong validity (as in Prawitz's "Ideas and Results in Proof Theory") permitting a proof of strong normalization. T...

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