What is the intuitive meaning of the Routley-Meyer 3-place predicate used to give a semantics of relevance logic? What's a good way to visualize what it's doing?
I have been trying off and on for a few weeks to understand the three-place accessibility relation for relevance logics. The corresponding two-place accessibility relation for modal logics is by contrast much easier to understand; it shows which worlds are visible from the current world for the purposes of the notions of possibility (holding in a visible world) and necessity (holding in all the visible worlds).
What follows is an explanation of what a Routley-Meyer model is and some comments describing my attempts to understand what $R$ does.
I'm trying to understand relevance logics by rewriting the various connectives using different syntax to make the role of the first argument to $R$ as the vantage point of sorts more obvious, hopefully. I'm specifically looking at the general semantics of relevant logic given here. The SEP article on relevant logic talks a little about possible interpretations of the worlds as channels or as bundles of information or as situations, but presenting multiple semantics for relevant implication back to back is difficult to understand if you don't have a solid first interpretation to latch onto.
I'll present the content from the Wikipedia article rewritten into a slightly different notation, with some commentary about my attempts to understand it.
A Routley-Meyer model $M$ consists of a frame $F$ and $\Vdash$, a three-place predicate taking a model, a designated world, and a well-formed formula. A Routley-Meyer frame $F$ is a quadruple $(W, R, *, 0)$. $W$ is the set of worlds and the domain of discourse. $R$ is a 3-place predicate where $Rabc$ appears to mean something like: to an ideal observer at point $a$, $b$ implies $c$. $*$ is a 1-place function written in a postfix way called Routley negation; $*$ is an involution. $0$ is a constant symbol, representing a designated world. $\Vdash$ takes a world and assigns truth values to propositions.
I define the following abbreviations
$$ a \le b \;\;\text{iff}\;\; R(0,a,b) $$
$$ a\, (\le_w)\,b \;\;\text{iff}\;\; R(w,a,b) $$
$R$ and $*$ are subject to the following constraints. These constraints are table stakes to be a Routley-Meyer model.
$$ a \le a $$
$$ a \le b \;\text{and}\; b \le c \;\;\text{implies}\;\; a \le c $$
Routley negation is an involution. I do not know whether the absence of fixed points of Routley negation is implied by the other axioms or not.
$$ a^{**} = a $$
$\le$ satisfies a contrapositive-like property
$$ a \le b \;\text{implies}\; b^* \le a^* $$
Additionally, the valuation predicate $\Vdash$ satisfies the following constraints. I will write $M\,(\Vdash_a)\,A$ where Wikipedia uses $M,a\Vdash A$.
We have the rule.
For all well-formed formulas $A$, if $M \, (\Vdash_a)\, A$ and $a \le b$ then $M \,(\Vdash_b)\, A$.
$\Vdash$ is defined inductively on well-formed formulas.
$$ M \,(\Vdash_a)\, A \land B \;\;\text{iff}\;\; M\,(\Vdash_a)\,A \;\text{and}\; M\,(\Vdash_a)\,A $$ $$ M \,(\Vdash_a)\, A \lor B \;\;\text{iff}\;\; M\,(\Vdash_a)\,A \;\text{or}\; M\,(\Vdash_a)B $$ $$ M \,(\Vdash_a)\, A \to B \;\;\; \text{iff}\;\;\; \text{for all worlds $b$ and $c$ where $b\,(\le_a)\,c$, $M\,(\Vdash_b)\,A$ implies $M\,(\Vdash_c)\,B$} $$ $$ M \,(\Vdash_a)\, \lnot A \;\;\text{iff}\;\; M\,(\nVdash_{a^*})\,A $$
The semantics for $\land$, $\lor$, and $\lnot$ are all straightforward and easy to follow. It's a little weird that determine the truth of a negated proposition in a world $a$ in another world that isn't $a$, but that's fine.
The truth conditions for $A \to B$ is really the only one to use $R$ in a significant way, but I can't figure out what it means. Taking the negation of both sides yields the following.
$$ M \,(\nVdash_a)\,A \to B \;\;\text{iff}\;\; \text{there exist worlds $u$ and $v$ such that $u\,(\le_a)\,v$ and $M\,(\Vdash_u)\,A$ and $M\,(\nVdash_v)\,B$} $$
Negating both sides almost makes sense. Given a model $M$ with a distingushed world $a$ in the background, $A \to B$ fails if there exists an accessible ordered pair of worlds $(u, v)$ such that $A$ is true at $u$ and $B$ is false at $v$. So all the accessible pairs of worlds are our hunting grounds for counterexamples. Or, equivalently, we have a set of all counterexample-bearing world pairs, $\{ (w_1, w_2) \mathop| M\,(\Vdash_{w_1})\,A \land M\,(\nVdash_{w_2})\,B \}$ and $A \to B$ fails at our distinguished world $a$ if and only if we can see one of them. I can also restrict my set of counterexample world-pairs to just world-pairs that are known to be related to some observer somewhere $\{ (w_1, w_2) \mathop| M\,(\Vdash_{w_1})\,A \;\text{and}\; M\,(\nVdash_{w_2})\,B \;\text{and}\; (\exists w_3 \mathop. w_1 \,(\le_{w_3})\, w_2) \}$. So, now, we win and successfully refute $A \to B$ if we're in one of the lucky worlds that can see a counterexample. But constructing counterexamples with propositions in different worlds doesn't really seem to make sense. I'm stuck on how to interpret $R$.