Completeness

Completeness is a property that theories can have when treated using the tools of modern mathematical logic and its mathematically precise treatment of logical consequence. This entry discusses two senses in which theories may be complete. Given the idealized view of theories logic employs, completeness in one sense has only been studied in relation to mathematical theories. But there is also a sense of completeness that applies to a theory of logical consequence, it is the completeness of a deductive system of logic.

Theories are a form of knowledge in which core concepts and assumptions are systematized so as to organize the understanding of the theory’s subject matter. The composition of theories in science can be very complicated. They may consist of data, experiments, models, and even research programs; they are not expressed in a uniform way. However, in logic theories are treated in an idealized and uniform way: All theories consist only of statements. Insofar as we can paraphrase the core elements of a theory into statements, this idealization is not misguided. Thinking of theories in these terms can help organize and reduce the number of core elements of a subject, since many other facts about the subject matter will follow directly from those core concepts. Logic studies a particular way that consequences can be drawn from statements, namely logical consequence. Statements are related by logical consequence when one statement follows from a set of statements with certainty. For example, the Law of Gravitation predicts the velocity at which a stone hits the ground when it falls from a cliff. This is an elementary instance of one statement about a particular stone’s velocity following, with certainty, from a general statement together with other statements about the particular height of the cliff.