Linear Logic Displayed

Linear logic" (LL; see Girard [6]) was proposed to be of use in computer science, but it can be formulated as a "display logic" (DL; see Belnap [2]), which is a kind of Gentzen calculus admitting easy proof of an Elimination Theorem. Thus LL is naturally placed within a wider prooftheoretical framework that is known to include relevance, intuitionist, and modal logics, etc., and that permits coherent variations on LL itself— including the definition of "punctual logic". In order to accommodate LL, two independently useful modifications of DL are made. First, DL possessed an unmotivated identification of two of its structural constants. This identification is dropped in order to make room in DL for the several propositional constants in LL. Second, DL possessed an unmotivated bias towards connectives that, when they are introduced as consequents, have restrictions put on their antecedents. This bias is abandoned in order to make room in DL for a dual pair of modal-like "exponential" connectives of LL. The latter modification requires restructuring the proof of the Elimination Theorem for DL, rendering it perfectly symmetrical in antecedent and consequent. / Introduction To "display" any logic is to exhibit it as a display logic, that is, as the special sort of Gentzen consecution calculus defined in Belnap [2] (DL), and thus to place that logic within a certain significant proof-theoretical framework. The aim of this paper is to display "linear logic", which is a logic proposed by Girard [6] in connection with some important computer science considerations. It turns out that the display of linear logic requires some healthy adjustments in the universal features of display logic itself. One set of adjustments is required in order to treat the "exponentiation" connectives of linear logic, and another to treat its four (instead of two) propositional constants while keeping to a single "family" of display logic. After we are done, we will be able to see how displaying linear logic permits a well-organized consideration of some of its essential features and of some of its variants. Received February 15, 1989