Bridging the gap between natural logics and description logics

Description logics and logics in the newer area of ‘natural logics’ are typically weak but interesting and tractable fragments of first-order logic. The logics involved are sometimes called ‘term logics’, where a term is something interpreted as a subset of a given model, and they usually lack individual variables. They are extensions of syllogistic logics and weakenings of first-order logic. This paper discusses description logics and natural logics and then presents logics which connect the two areas. The emphasis is on completeness results for the logics at hand. So we bring to the description logic area logics that are loosely based on natural language constructions (as is typical of natural logics), and completeness results for them. In the other direction, we bring to natural logics the distributive lattice operations on concepts (nouns) that have so far been missing and that are typical of description logics.