Which logic do you get by combining the topological semantics for IPC with a binary accessibility relation?
I'm trying to come up with a semantics for a relevance logic that's simpler than a Routley-Meyer model. The main thing I'm trying to remove is the ternary accessibility relation because I find it tricky to think about.
I tried taking a model with a one-place accessibility relation (like a modal logic frame), but replacing the classical truth values with the topological semantics for IPC. I take $\tau^R$, the standard topology on $\mathbb{R}$ as my truth values and make $\mathbb{R}$ the sole designated truth value.
If $w$ is a world, let $[\varphi]_w \in \tau^R$ denote the truth value of $\varphi$ at world $w$. In addition, let $0$ be a designated world.
Let $[a]_w$ be the value of $a$ at the world $w$ when $a$ is primitive.
Let $[\lnot a]_w$ be equal to $(\mathbb{R} \setminus [a]_w)^o$.
Let $[a \lor b]_w$ be equal to $[a]_w \cup [b]_w$.
Let $[a \land b]_w$ be equal to $[a]_w \cap [b]_w$.
Let $[a \to b]_w$ be equal to the following. It is the interior of the intersection of normal intuitionistic $\to$ evaluated at each accessible world.
$$ \left(\bigcap \left\{ \,([a]^c_u \cup [b]_u)^o \mathop{:} Rwu \, \right\} \right)^o $$
Additionally, $[a \to b]_w$ is $\mathbb{R}$ if there are no worlds accessible from $w$.
I impose the restriction that $(0, 0) \in R$. This gives us modus ponens $\frac{a \to b \;\;\text{and}\;\; a}{b}$.
Additionally, a given wff $\varphi$ is true if and only if it holds at the designated world $0$. This is like a Routley-Meyer model, which also has a notion of a "starting point".
$$ W, R, 0, [\cdot] \models \varphi \;\;\text{if and only if}\;\; [\varphi]_0 = \mathbb{R} $$
This logic is nontrivial.
$a \to a$ is a tautology, since $(a \to a)$ is intuitionistically true at every world accessible from $0$.
$a \lor \lnot a$ is not a tautology. It fails for $a = (0, 1)$ in world 0.
This logic seems to have a relevance-like flavor to it.
For example, $a \to (b \to a)$ is falsified by the following model.
Suppose I have three worlds, $0 \to w$. $R$ is exactly $\{(0, 0), (0, w)\}$.
Let $b$ have the value $(0, 1)$ in world $w$.
Let $a$ have the value $[0, 1]^c$ in world $w$.
So $b \to a$ is $\mathbb{R}$ at world $w$.
Let $b$ have the value $(1, 2)$ in world $0$.
Let $a$ have the value $[1, 2]^c$ in world $0$.
So, $b \to a$ has the value $[0, 2]^c$ in world $0$.
This means that $a \to (b \to a)$ has value $((1, 2) \cup [0, 2]^c)^o$ at world $0$, which is not $\mathbb{R}$.
