A multimodal logic for reasoning about complementarity

Two objects o1, o2 of an information system are said to be complementary with respect to attribute a if a(o1) = −a(o2), where a(o) is the set of values of attribute a assigned to o. They are said to be complementary with respect to a set of attributes A if they are complementary with respect to each attribute a ∈ A. A multi-modal logical language for reasoning about complementarity relations is presented, with modalities [A] and 〈A〉 parameterised by subsets of a given set Attr of attributes. Two complete deduction system for the language are presented: a Rasiowa Sikorski style system using signed formulae, and an equivalent sequent calculus system in Gentzen style complete for theories.