Entailment may be defined in two equivalent ways, negatively as the relation that holds when the conclusion is false in no possible world in which all the premises are true or positively as the relation which holds when the conclusion is true in all such worlds. The negative form has the advantage of focusing attention on the sort of possible world that serves as a counterexample to a claim of entailment. The positive form characterizes a relation of entailment as a conditional guarantee of the truth of the conclusion, a guarantee conditional on the truth of the premises.

The requirements for a world to serve as a counterexample to entailment suggest the general idea of dividing a pair of sets by making all members of the first true and all members of the second false. A world will be said to divide an argument when it divides the premises and conclusion.

The idea of division enables us to define a relation of relative exhaustiveness between sets: one set renders another exhaustive when there is no possible world that divides the two sets. We will extend the notation for entailment to express this relation between sets Γ and Δ as Γ ⇒ Δ. Entailment is the special case of this where Δ has only one member. When Δ has more than one member, its members will be referred to as alternatives because a relation of relative exhaustive provides a conditional guarantee only that at least one member of the second set it true.

Since a set of alternatives can have more than one member or be empty, relative exhaustiveness encompasses all the deductive properties and relations we have considered (as well as an extension of the idea of joint exhaustiveness to any set of sentences). We way a property or relation is expressed using relative exhaustiveness is tied directly to the negative form of its definition. When no relation of relative exhaustiveness holds no matter how a set is divided into two parts, all patterns of truth values for its members are possible and the set is logically independent. A single sentence that forms a logically independent set is logically contingent.

Definitions in terms of relative exhaustiveness can be converted into definitions in terms of entailment by replacing empty sets of alternative with ⊥ and reducing the size of multiple sets by replacing members with contradictory sentences among the assumptions (using the basic law for relative exhaustiveness).

Relative exhaustiveness satisfies three basic principles: repetition (the relation holds whenever an assumption is repeated as an alternative), cut (if the relation holds whether a sentence appears as an assumption or as an alternative, the sentence need not appear as either), and monotonicity (when the relation holds, it will continue to hold with added assumptions or alternatives). The term monotonic reflects the fact that the number of cases of relative exhaustiveness never decreases when the set of assumptions or set of alternatives increases. The cut law may be generalized to say that, if the relation holds no matter how sentences from some set are distributed among assumptions and alternatives, it holds when these sentences do not appear as either. Generalized cut, repetition, and monotonicity together suffice to imply all the principles governing relative exhaustiveness.

The basic principles governing entailment are closely related to those governing relative exhaustiveness. Two of these—the law for premises (any premise is a valid conclusion), monotonicity (adding premises never damages validity)—are special cases of laws for relative exhaustiveness. A third is a slight variation on an instance of the cut law: the law for lemmas says that a premise may be dropped if it is entailed by the other premises. The latter licenses the use of lemmas, valid conclusions that are of interest only as premises in further arguments. A more general law, called the chain law, says that anything entailed by a set of valid conclusions from given premises is itself a valid conclusion. This, together with a law for premises, yields all laws of entailment, and these two principles amount to principles of reflexivity and transitivity for the relation between sets that holds when one set entails each member of the other. Although this places monotonicity in the background, it is significant in distinguishing entailment from other forms of good inference, whose riskiness means that they are non-monotonic because adding information that the risk has not paid off will undermine their quality.

The laws describing the behavior of ⊤ and ⊥ in the context of relative exhaustiveness exhibit a kind of symmetry that we will see in other laws later. The sentences ⊤ and ⊥ are dual as are the terms premise and alternative (or the left and right of an arrow) in the sense that replacing each such term in a law by the one dual to it will produce another law.