There are reasonably nice logics

Abstract

A well-known question of Feferman asks whether there is a logic which extends the logic , is ℵ0-compact and satisfies the interpolation theorem. (Cf. Makowsky [M] for background and terminology.)The same question was open when ℵ1 in is replaced by any other uncountable cardinal κ. We shall show that when κ is an uncountable strongly compact cardinal and there is a strongly compact cardinal > κ, then there is such a logic. It is impossible to prove the existence of uncountable strongly compact cardinals in ZFC. However, the logic that we describe has a simple and natural definition, together with several other pleasant properties. For example it satisfies Robinson's lemma, PPP (pair preservation property, viz. the theory of the sum of two models is the sum of their theories), versions of the elementary chain lemma for chains of length < λ, and isomorphism of (suitable) ultralimits.This logic is described in §2 below; we call it 1. It is not a new logic—it was introduced in [Sh, Part II, §3] as an example of a logic which has the amalgamation and joint embedding properties. See the transparent presentation in [M]. But we shall repeat all the definitions. In [HS] we presented a logic with some of the same properties as 1, also based on a strongly compact cardinal λ; but unlike 1, it was not a sublogic of λ,λ.