Display logic

I formulate a Gentzen consecution calculus for an i n d e f ~ t e number of logics all mixed together, including boolean (two valued), intuitionistic, relevance, and (various) modal logics. This is accomplished by augmenting and refining the structural ideas of Gentzen (1934). The key feature of the calculus permitting control in the presence of multiple logics is this: every "positive part" of a consecution can he displayed as the consequent, standing alone, of an equivalent consecution; and every "negative part" can be displayed as the antecedent, standing alone, of an equivalent consecution; such a calculus I call a "Display logic". The first spin-off is a generalized Elimination theorem. (Some terminology, such as "consecution calculus", which translates "Sequenzen Kalkiil", comes from Anderson and Belnap (1975); other terminology comes from Curry (1963). Section 2 outlines the grammatical structures needed for Display logic, emphasizing the ways in which it develops themes in Gentzen. Section 3 explains the system itself in a wholesale way, but without attention to the special features of the various calculuses that Display logic can treat. Then the rules are divided into (1) display equivalences, (2) structural rules, and (3) connective postulates. Section 4 states and proves a wholesale (cut) Elimination theorem. Section 5 discusses in more detail how various logics appear in the context of Display logic. Section 6 elaborates a series of possibilities. The grammar of any consecution calculus is complex (though the complexity is frequently played down by means of reliance on geometrical intuitions; see Section 6.8 for a related point), and because we are treating so many logics simultaneously, the grammar of Display logic is even more so.