In this note we study a counterpart in predicate logic of the notion of logical friendliness, introduced into propositional logic in [15]. The result is a new consequence relation for predicate languages with equality using first-order models. While compactness, interpolation and axiomatizability fail dramatically, several other properties are preserved from the propositional case. Divergence is diminished when the language does not contain equality with its standard interpretation.

In the substitutional framework, validity is truth under all substitutions of the nonlogical vocabulary. I develop a theory where □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition.

When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and ${\mathcal {L}}_{\omega _1, \omega }$ -elementary? We find that these are exactly the classes (...) that can be defined by an infinitary formula that has no infinitary disjunctions. (shrink)