In English we can use the word "or" to allow the possibilty of two choices being true (as in saying someone will take cream or sugar with her coffee) or to suggest that one choice in fact rules out the other (as in saying someone will have either coffee or tea). The signal O indicates the first, which we call the inclusive use.

The truth-table pattern is
P Q || PQO
T T........T
T F........T
F T........T
F F........F

The word "unless" can also be used to indicate disjunction, and in most cases we assume that it is in the inclusive sense. All of the following would then be symbolized with PQO, given the code that P stands for the idea that Jane studies and Q that she is lazy:
--Either Jane studies or she is lazy.
--Unless Jane studies we'll know she is lazy.
--Jane studies unless she is lazy.

The key rules for working with O are (1) that ruling out one choice when only two are provided proves the truth of the other, and (2) that given one thing as true we can always add another. We refer to these patterns as O elimination (Either P or Q, Not-P, therefore Q) and O introduction (P, therefore P or Q).

O, like A, is both commutative and associative: neither the order nor the grouping of the variables matters.

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