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 from the conclusion.

The idea of division enables us to define a relation of conditional 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 Σ as Γ ⊨ Σ. Entailment is the special case of this where Σ has only one member. When Σ has more than one member, its members will be referred to as alternatives because a relation of conditional 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, conditional 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). The way a property or relation is expressed using conditional exhaustiveness is tied directly to the negative form of the definition of the property or relation. When no relation of conditional 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 conditional exhaustiveness can be converted into definitions in terms of entailment by replacing empty sets of alternatives with ⊥ and reducing the size of multiple sets of alternatives by replacing members by adding assumptions that are contradictory to them (using the basic law for conditional exhaustiveness).

Entailment obeys analogues to the principles of reflexivity and transitivity for implication. In the case of reflexivity, the analogy is with the law for premises; and, in the case of transitivity, it is with the chain law. Taken together, these principles yield all laws of entailment. Two principles for entailment that follow from them—monotonicity and the law for lemmas—state conditions under which we may add and drop assumptions. The second principle licenses the use of lemmas, valid conclusions that are of interest only as premises in further arguments. The first tells us that entailment is monotonic in the sense that it will never stop holding because of additions to the set of assumptions. This principle is significant in distinguishing entailment from other forms of good inference, whose riskiness means that they are non-monotonic (because adding information telling us that the risk does not pay off will undermine their quality).

The laws describing the behavior of ⊤ and ⊥ in the context of conditional 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 turnstile) in the sense that replacing each such term in a law by the one dual to it will produce another law.