I'm reading up on nonclassical-logic.
In Priest's nonstandard $N$ of his "Introduction to Nonclassical Logic [. . .], Second Edition", it is an exercise to
show
$$\not\vdash_N \square p\supset p$$
and exhibit a counterexample in the style of his examples of the logic $N$ ibid.
My Attempt:
Assume $N$. Then the tableau might be
$$\begin{align} \lnot(\square p &\supset p), 0\\ \square p, & 0\\ \lnot p, & 0\\ p, & 0, \end{align}$$
with the diagram for the counterexample being
$$\stackrel{p, \lnot p}{\stackrel{\curvearrowright}{\boxed{w_0}}}.$$
This ought to be a simple exercise for me but, alas, I'm stuck; I think I did it wrong.
Why?
Well, I'm on page 97 ibid and I hadn't done the exercises necessary from the previous chapter, $\S 4$.
I don't have the time to do every exercise of the book. I picked this exercise because it seemed easy.
Please help :)
Edit: It appears that I have shown the negation of the statement in question by mistake. Exactly where did I mess up? Or does the principle of explosion not hold in $N$?
$\square p\supset p$ is not provable in K: it is an instance of the additional axiom we add to get the stronger system T. Since any normal model is also a non-normal model (i.e. K is stronger than N), the usual countermodel for showing $\square p\supset p$ is not a theorem of K shows that it is not a theorem of N either.
The usual countermodel has a single successor-less (normal) node at which $p$ is false. Since the node is normal and has no successors, $\square p$ is true here, so since $p$ is false, $\square p \to p$ is false.
The first few of these statements in this problem are just statements that are well-known to not be provable in K so really belong in the previous chapter: I think the real purpose of this exercise is as a warm-up the next part where it asks which ones hold if you add reflexivity / transitivity. For instance, we know that in normal modal logic $\square p \supset p$ becomes valid if we add the reflexivity condition... is that still the case if we drop the normality? It is, but we have to be careful to make sure the argument from last chapter generalizes.
(And by contrast, I don't believe the transitivity axiom $\square p\supset \square\square p$ for normal modal logic is valid for transitive non-normal models... don't take my word for that though, I've never seen a presentation of semantics of non-normal modal logic until right now.)