Hope you're all doing well. My question is about non-axiomatizable logics. My understanding is that a "logic" (the mathematical structure) is just another word for a "propositional calculus" as in http://en.wikipedia.org/wiki/Propositional_calculus. (see the formal defn. under "general definition of a propositional calculus.") It seems like this definition inherently depends on axioms, and thus is intuitively "axiomatized."
Background:
Say we look at the logic S4, which we can think of(alternatively to the rigorous definition in the above wiki article) as a set of wffs in a specific language $L$ ($L$ is basically the propositional language augmented with some modal operators $\Box$ and $\Diamond$ and with a suitable definition of grammatically correct expressions, i.e. wffs in the language). This set of wffs (S4) contains the tautologies of propositional logic, but also some additional axioms and any wffs in $L$ that can be obtained by using the inference rule $\textit{modus ponens}$. (See http://plato.stanford.edu/entries/logic-modal/ for these additional axioms).
Please correct me if i'm wrong about any of this; I'd appreciate it. In my reading I've come across the idea of a non-axiomatizable logic. (under definition 17 page 139 of http://individual.utoronto.ca/philipkremer/onlinepapers/DTL.pdf) It looks like the concept is quite common, i.e. http://onlinelibrary.wiley.com/doi/10.1002/malq.19610070113/abstract. As I said, the wiki definition of a logic seems to be inherently axiomatizable.
My question:
Is there a (relatively simple) example of a non-axiomatizable logic? I've tried googling it, but nothing too useful has come up yet. Is there somewhere I can learn more about it? Thank you for any help/clarification!
Sincerely,
Vien
In fact, there is. One such classic example is second-order logic, whose semantics can be given, but for which you can prove there is no axiomatization. I posted some resources here in case you're interested (but just FYI, you need to already be familiar with first-order semantics to understand the significance of second-order logic).