The Landscape of Deduction

Abstract

Abstract Recent logical research is showing an increasing sensitivity to the existence of what may be called different ‘styles of deduction’, organized in logical proof calculi that may vary even in the basic structural rules governing their general handling of premises and conclusions. Standard structural rules, such as Monotonicity, Contraction, Cut or Permutation, may be either lost completely in the process, or be modified to subtler variants (such as ‘leftward’ or ‘cautious’ forms of Monotonicity). Any system of structural rules may be seen as characteristic of a certain deductive style: ‘classical’, ‘relevant’, ‘dynamic’, or ‘categorial’. In general, these styles will be more sensitive to the syntactic presentation of premises, whose ordering and multiplicity become crucial: most inference is ‘occurrence-oriented’. Starting from different motivations, students of categorial logic, relevance logic and linear logic have been exploring this space of possibilities, using various formats, including Hilbert-style axiomatic systems and Gentzen-style calculi of sequents (cf. [Lambek 1958], [Dunn 1985], [Girard 1987], [van Benthem 1991]). In particular, the latter format seems to be gaining the upper hand these days as a natural form of organization, bringing out what are perceived to be the major options in choosing one inferential style or another. There are also other research lines, however, which tend in the same direction, such as the current work on varieties of inference in artificial intelligence and their logical properties (cf. [Makinson 1991]), which has brought in ‘preferential’ or ‘default’ styles of heuristic reasoning, again with a spectrum of different structural rules.