Residuals and conjugates in positive substructural logic

Abstract

In substructural, fuzzy, and many other logics, relations of residuation between conjunction-like and conditional-like connectives are central (Ono et. al. 2007). For instance, in the context of frame semantics, relations of residuation between connectives allows for those connectives to be interpreted by a single relation, as has been studied in the context of Gaggle Theory (Bimbó & Dunn 2008). In Boolean algebras with operators (BAOs), residuation has a second face in the form of relations of conjugation (Jónsson & Tsinakis 1993; Mikulás 1996) – the residuals of an operator are definable in terms of its conjugates, and vice versa, by means of Boolean negation. An immediate result of this is that in BAOs, a collection of operations all conjugated with respect to each other may be interpreted by a single relation in the frame semantics. This talk concerns relations of residuation and conjugation in a positive context – in particular, in the context of logics extending the positive non-associative Lambek calculus with distributive lattice operations. This logic is the extension of distributive lattice logic by means of a binary operator – sometimes called fusion – with left and right residuals, where fusion is assumed only to be monotone (or, equivalently when the residuals are present, to distribute over (and into) the lattice join). The extension of this logic with which we’re concerned is that resulting from the addition of two additional fusionlike connectives, where each fusion is conjugated with respect to the others, and left and right residuals for each additional fusion.