A disjunction φ ∨ ψ is false only when its disjuncts are both false, and it thus says only what both of them say. The law for disjunction as a premise tell us that we can establish a conclusion using such a premise by showing that it is entailed by each of the disjuncts (given our other premises). This way of exploiting a disjunction is known as an proof by cases and it appears in our system of derivations as the rule Proof by Cases (PC) that leads us to divide a gap into two case arguments, each of which takes over the original goal and adds one of the two disjuncts as a supposition.

To show that a disjunction is a valid conclusion, we must show that its disjuncts are rendered jointly exhaustive by the premises. We can do this by showing that one of the disjuncts will follow if we add the contradictory of the other to our premises. We use the notation ¬^± φ to indicate the result of either negating or de-negating φ. The law for disjunction as a conclusion then tells us that we can conclude a disjunction if we can conclude one disjunct provided we take the negation or de-negation of the other disjunct as a premise. The rule implementing this idea is Proof of Exhaustion; it enables us to conclude a disjunction from an argument that may be called hypothetical since it bases a disjunct on an assumption (of the negation or de-negation of the other disjunct) that we may not be prepared to assert categorically. It does not matter for the soundness or safety of PE which disjunct figures as the goal of this hypothetical argument and which is negated or de-negated in its supposition.

Derivations, especially those that have a disjuction as a goal as well as a premise can often be developed in different ways. Some of these can be significantly longer than others but the choice between forms of PE will usually have only a limited impact on the length.

Conjunction and disjunction are opposite in the sense of being dual. One manifestation of this relation is in De Morgan’s laws, which tell how to restate the denial of a conjunction or disjunction as an assertion of the other form of compound. Another manifestation is a pattern in laws of relative exhaustiveness which allows us to interchange conjunctions and disjunctions if at the same time we interchange ⊥ and ⊤ and also premises and alternatives.