# Multiple Conclusion Linear Logic: Cut Elimination and More

• Valeria de Paiva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9537)

## Abstract

Full Intuitionistic Linear Logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par, and implication, into an intuitionistic linear logic. It was shown that their formalization of FILL did not enjoy cut-elimination by Bierman, but Bellin proposed a change to the definition of FILL in the hope to regain cut-elimination. In this note we adopt Bellin’s proposed change and give a direct proof of cut-elimination. Then we show that a categorical model of FILL in the basic dialectica category is also a LNL model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. Lastly, we give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor.

## Keywords

Full intuitionistic linear logic Classical linear logic Dialectica category Cut-elimination Tensorial logic Linear/non-linear models Categorical model Proof theory Par

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