Constructing Natural Extensions of Propositional Logics

Abstract

The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to Łoś and Suszko. However, it was recently observed by Cintula and Noguera that both of these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the Łoś–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality ${\kappa}$ may have more than one conservative extension of cardinality ${\kappa}$ by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana.