Doty, [in the paper I link above](https://arxiv.org/pdf/0902.1950.pdf), Ellerman gives partition logic as a Boolean logic. A Boolean logic is one where a left and right adjoint laws hold,

\$x \setminus y \leq x \land \neg y, \\\\ \neg x \lor y \leq x \Rightarrow y. \$

I think what confuses most people is that Ellerman uses the dual poset.

But if you look at page 19, we have the following:

\$[x] \cap [y] \subseteq [z] \Leftrightarrow x \leq y \multimap z, \$
and,
\$y \multimap z = \bigvee \lbrace x \mid [x] \cap [y] \subseteq [z]\rbrace. \$