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.

Propositional modal logic over relational frames is naturally extended with propositional quantifiers by letting them range over arbitrary sets of worlds of the relevant frame. This is also known as second-order propositional modal logic. The propositionally quantified modal logic of a class of relational frames is often not axiomatizable, although there are known exceptions, most notably the case of frames validating the strong modal logic $\mathrm {S5}$. Here, we develop new general methods with which many of the open questions in this area can be answered. We illustrate the usefulness of these methods by applying them to a range of examples, which provide a detailed picture of which normal modal logics define classes of relational frames whose propositionally quantified modal logic is axiomatizable. We also apply these methods to establish new results in the multimodal case.

Fix a language extending the language of ordered fields by at least one newpredicate or function symbol. Call an L-structure Rpseudo-o-minimal if it is (elementarily equivalent to) anultraproduct of o-minimal structures. We show that for any recursive list ofL-sentences , there is a real closed field satisfying whichis not pseudo-o-minimal. This shows that the theory To−minconsisting of those -sentences true in all o-minimal-structures, also called the theory of o-minimality (for L), isnot recursively axiomatizable. And, in particular, there are locally o-minimal,definably complete expansions of real closed fields which are notpseudo-o-minimal.