First is the implication relation symbolised by the hook operator, "⊃" or perhaps some relevant implication operator.
The conclusion of a rule appears to the left of the implication operator ":-".
The implication operator is represented using the combination of a colon and a dash (i.e., ":-").
Note that, a special class of fuzzy implication can be naturally obtained by generalizing the implication operator from the quantum logic, namely p→q⇔¬p∨(p∧q).
Also notable are the discussions of defeasibility and conditionality in Alchourron 1993, 1996, where a revision operator (operating on antecedents of conditionals) is relied on in conjunction with a strict implication operator and a strictly monadic deontic operator.
The implication operator is used to connect the conditions and conclusion of a rule, where the conditions appear to the right of the operator and the conclusion appear to the left of the operator.
Hook, or a relevant implication operator, seems to capture one of the relations encountered in the sea battle case, which can be thought of as holding paradigmatically between bearers of truth values, but can be loosely thought of in terms of states of affairs.
Two expressions are joined with an implication operator (⇒ or ⇔ corresponding to "if" and "if and only if"), to form a rule.
Then, the min operator is used for the conjunction and for the implication operations.
The support relation is represented as curved filled-in arrow, and the implication logical operators and, or are represented as black triangles, white triangles, filled-in arrows, respectively.
