Bricken's boundary logic

William Bricken's boundary logic (BL) (see also) is interesting dual to Peirce's existential graphs. In his system blank sheet represents FALSE (Peirce: TRUE) and composition of structures represents disjunction (Peirce: conjunction). Accordingly basic logical operations are represented as follows:

‹void› -- FALSE
() -- TRUE
(p) -- NOT p
p q -- p OR q
((p) (q)) -- p AND q
(p) q -- IF p THEN q

Basic transformation rules:

()() = () -- calling [T v T = T]
(()) = ‹void› -- crossing [~T = F]
(() p) = ‹void› -- occlusion [~(T v p) = F & ~p = F]
() p = ()  -- dominion [T v p = T]
((p)) = p -- involution (~(~p) = p)
p {q p} = p {q} -- pervasion (from any depth) [p v ~(... q v p...) = p v ~(...q...)]

As parentheses operate mostly as negation, they may be replaced by ‹›:
‹p› -- NOT p
‹p q› -- NOT (p OR q)
‹‹p›‹q›› -- NOT (NOT p OR NOT q) = p AND q
‹p› q -- NOT p OR q = IF p THEN q

Hence, it is similar to operation in my notation inside brackets, where negation applies to disjunction:
‹p q› -- [‹p q›] = [-[p+q]] = [(-p-q)]
‹‹p›‹q›› -- [‹‹p›‹q››] = [-[-p-q]] = [(+p+q)]
‹p› q -- [‹p› q] = [-p+q]