The relation of entailment concerns the possibilities of truth and falsity for premises and conclusions; that is, it concerns the truth values of these sentences in various possible worlds. The possibilities in question are logical possibilities, which may be understood as the situations whose description is permitted by the semantic rules of the language.

Entailment by a single premise, or implication, is a relation between sentences that orders them by their content. More precisely, φ ⇒ ψ when φ says everything that is said by ψ. When sentences imply each other, they say the some thing, and we say they are equivalent, a relation for which we use the sign ⇔. When φ ⇒ ψ but these sentences are not equivalent, φ says more than ψ and we will often say that φ makes a stronger claim and ψ a weaker one.

Implication is a reflexive and transitive relation. The latter property tells us that implication can be transferred by implication: implication is preserved if we weaken the conclusion to something it implies or strengthen the premise to something that it is implied by. Equivalence is reflexive and transitive and also symmetric, but it is not preserved when sentences are strengthened or weakened.

Sentences can also be compared to describe differences in what they say. Sentences that cannot both be true are mutually exclusive (a relation for which we use the sign ×). The claims made by such sentences are opposite but opposite in a way that permits a third alternative. Sentences which are complementary in the sense that each must be true if the other is false are jointly exhaustive (for which our notation is ◯). When these two relations both hold, sentences are contradictory (and we use the combined sign ⊗). Contradictory sentences always have opposite truth values make claims that are opposite in a way that permits no third alternative. Sentences that are neither mutually exclusive nor jointly exhaustive and neither or which implies the other are logically independent.

The relations ×, ◯, and ⊗ are all symmetric. Mutual exclusion is preserved when sentences are strengthened; since this change from conclusion to premise is in a direction opposite to the entailment arrow, we will say that mutual exclusion is contravariant. On the other hand, joint exhaustiveness is preserved when sentences are weakened, and we will say it is covariant. Contradictoriness is not preserved when sentences are either strengthened or weakened, nor is it transitive; but it satisfies a doubling law to the effect that anything contradictory to something contradictory to φ is equivalent to φ.

The deductive relations a sentence stands in depend on its truth values in various possible worlds. That is, they depend on its truth conditions. These truth conditions are encapsulated in the proposition it expresses, which can be thought of as a way of dividing all possibilities into those it rules out and those it leaves open. This means that a proposition can be depicted as a division of space into two regions.

At one extreme are tautologies, which rule out no possibilities and thus have no content. All tautologies are equivalent and we will distinguish one, Tautology, for which we use the notation ⊤. At the other extreme are sentences that rule out all possibilities. Such sentences are absurd and all are equivalent to the single representative Absurdity, for which we use the notation ⊥.

An argument with a tautology as a conclusion, is always valid; but a tautologous premise contributes nothing to the validity of an argument. An argument with an absurd premise is always valid but by default since its premises cannot all be true. An argument with an absurd conclusion is valid when and only when its premises form an inconsistent set, and this will enable us to study inconsistency by way of entailment.

Although certain groups of sentences can be ordered linearly between ⊥ and ⊤ as a series of claims with steadily increasing content, the full range of propositions expressed by sentences are better thought of as inhabiting a much more complex logical space. This space might be a space of possibilities with propositions appearing as ways of dividing the space into regions, or it might be a space that has as its points propositions themselves. Logical space in this second sense has a bottom in the proposition expressed by ⊥ and a top provided by ⊤. When there are a significant number of possible worlds, there will be many more propositions with intermediate content than there are strong propositions near ⊥ or weak ones near ⊤.