I'm reading an interesting on modal logic by Timothy Surendonk [1], where he defines the concept of canonical logic as the following.
A logic $L$ is defined to be canonical if there's a frame $\left< X, N \right>$ satisfying the following [1]
- $X$ is the set of all maximal $L$-consistent sets of formulae.
- $N(x) \supseteq \left\{ \left| A \right| \mid \square A \in c \right\}$ where $\left|A\right| = \left\{ x \in X \mid A \in x \right\}$
- $\left< X, N \right> \models L$
The author then proves that all simple models with axioms of a certain type are canonical.
For $E$, $E4$, $E5$, and logics formed from axioms of the form $M_1 p \to M_2 p$ (where $M_1$ and $M_2$ are positive modalities) are all canonical.
These results are easy since we can simply deal with the non-effable sets $Y$ (not of the form $\left| A \right|$) by placing $Y \in N(x) \iff x \in Y$.
I cannot understand the last line of this proof. In particular, I cannot get the intuition of how to build a set $Y$ that's consistent with any set of axioms of that form while still following point 2. in the definition of canonical sets.
Can anyone with more experience in this area give me a hand here?
[1]: Surendonk, Timothy J.. ““ Does EK 4 have the Finite Model Property ? ” and related open questions.” (2007). Legal PDF link.