Definition generalized logics and stationary logics

Question

Can someone explain or give a definition of what is "generalized logics" and what is "stationary logics"? From the text where these terms appeared I know, that generalized logics is not first order logic.

Answer 1

There is no formal definition of a generalized logic (as far as I'm aware) but the informal description is to start with first order logic and add to it by allowing

• infinite sequences of quantifiers,
• infinite disjunctions and conjunction or
• special predicates - typically interpreted in $V$ - our true background universe.

A particularly nice example of a generalized logic is $\mathcal L(aa)$ - the stationary logic. $\mathcal L(aa)$ is just first order logic with an additional second order quantifier $\operatorname{aa}$. We have, for any given model $\mathcal L(aa)$-model $\mathcal M$ $$ \mathcal M \models \operatorname{aa}s \phi(s) : \iff \{ A \in [M]^{\le \omega} \mid \mathcal M \models \phi(A) \} \text{ contains a club (in } V \text{).} $$

The meaning of this additional quantifier becomes more apparent in specific examples. Say that $\phi$ is a $2$-ary formula in the language of set theory and $\mathcal M$ is an $\mathcal{L}(\operatorname{aa})$-model such that $$ \phi(\mathcal{M}) := \{ (x,y) \mid x,y \in \mathcal{M} \wedge \mathcal{M} \models \phi(x,y) \} $$ is (in $V$) a linear order. Then we can express the statement "$\phi(\mathcal M)$ has confinality $\le \omega$" in $\mathcal{L}(\operatorname{aa})$ as follows: $$ \mathcal M \models \operatorname{aa} s \forall x \exists y (\phi(x,y) \wedge s(y) ). $$ We can even express the fact that $\mathcal M$ itself is at most countable as $$ \mathcal{M} \models \operatorname{aa}s \forall x (s(x)). $$ Both examples demonstrate that $\mathcal{L}(\operatorname{aa})$ has strictly stronger expressible power than first order logic. And still $\mathcal{L}(\operatorname{aa})$ satisfies many desirable model theoretic properties. It satisfies a version of the Löwenheim-Skolem theorem, countable compactness, has a nice proof theory, ...

You can find all of this and much more in

• Barwise, Kaufmann. Stationary Logic and
• Kennedy, Magidor, Väänänen. Inner Models from Extended Logics.