The idea that the norms of deductive reasoning reflect a system of relations among propositions fits into a simplified picture of the function of language. According to this picture, a person’s beliefs amount to a proposition that rules out a certain range of possibilities for the actual history of the universe. The desire to know more is in part the desire to narrow the range of possibilities that are left open. When language is used cooperatively, we share our abilities to rule out possibilities by using assertions to convey propositions. The sentences we can sincerely assert are the ones that are entailed by the sum total of our beliefs, and we accommodate someone else’s assertion by adjusting our beliefs so that what they asserted is now entailed by our beliefs.

This picture is oversimplified and something must be said about several respects in which the actual operation of language is more complex. Each is associated with an aspect of meaning:

While an account of how sentences express propositions is the province of semantics, these complicating phenomenon are usually said to be the subject matter of pragmatics.

Although assertion is the only speech act we will study, not even all declarative sentences have this force. J. L. Ausin estimated that assert was only one of thousands of performative verbs that can be used to both perform and describe speech acts. Although many of these speech acts do not serve to convey propositions, their force can often be described with reference to propositions.

We will consider only what is implied by a sentence as part of its truth conditions and not further information that may be implicated as conditions for appropriate assertion beyond the requirements for truth. A false implicature will make a sentence misleading but may leave it true. One indication of this sort of case is a yes-but answer to the yes-no question corresponding to the sentence.

Indexicality means that the propositions expressed by sentences—and thus their deductive properties and relations—can depend on the contexts in which they are used. It would be possible to compare sentences only when each was associated with a specified (but perhaps different) context—such sentences-in-context are sometimes called statements. However, we will compare sentences only within a single context of use and consider only properties and relations of sentences that hold no matter what that context is. As with implicature and presupposition, accommodating sentences to the rules governing indexical phenomena provides a way of extracting information that goes beyond entailment.

Vagueness poses problems analogous to those posed by indexicality and presupposition. As with indexicality, we will assume a context of use; and, as with presupposition, we will assume supplementary specifications of truth value (in this case precise delineations of the boundaries of vague terms). Deductively valid conclusions will not rely on information about these factors, but accommodation to vague assertions can support non-deductive inference to extract further information. One way of explaining the sorites paradox is to suppose that it rests on a failure to distinguish this sort of inference from deductive inference.

Since a semantic presupposition is something that must hold in order for a sentence to have a truth value at all, sentences with non-tautologous presuppositions can fail to have truth values. The pervasiveness of definite descriptions—which can fail to refer to anything if the facts are not right—makes it hard to simply ignore sentences with non-trivial presuppositions. Instead, we will treat all terms as if they refer, simply stipulating reference values and truth values in other cases (eventually distinguishing an empty reference value) but considering only relations between sentences that hold for all such stipulations (the method of super-valuations).