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.

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.

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.

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 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 ⊤.

Sentences can also be compared by describing 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 (a relation for which we use the sign ⋈). Contradictory sentences always have opposite truth values and thus make claims that are opposite in a way that permits no third alternative.

The relations of entailment, mutual exclusiveness, and joint exhaustiveness along with the properties of tautologousness and absurdity enable us to describe any deductive property or relation of two sentences. There are connections among entailment, mutual exclusiveness, and joint exhaustiveness that can be displayed by a square of opposition. Sentences that are neither mutually exclusive nor jointly exhaustive and neither or which implies the other are logically independent.