mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, a set of logical formulae is deductively closed if it contains every formula that can be logically deduced from , formally: if always implies . If is a set of formulae, the deductive closure of is its smallest

superset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

that is deductively closed. The deductive closure of a

theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...

is often denoted or . This is a special case of the more general mathematical concept of closure — in particular, the deductive closure of is exactly the closure of with respect to the operation of logical consequence ().

Examples

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

, the set of all true propositions is deductively closed. This is to say that only true statements are derivable from other true statements.

Epistemic closure

epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Epis ...

, many philosophers have and continue to debate whether particular subsets of

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

s—especially ones ascribing

knowledge Knowledge can be defined as Descriptive knowledge, awareness of facts or as Procedural knowledge, practical skills, and may also refer to Knowledge by acquaintance, familiarity with objects or situations. Knowledge of facts, also called pro ...

or justification of a

belief A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take ...

to a subject—are closed under deduction.

References

{{Mathlogic-stub Concepts in logic Closure Logical consequence Propositional calculus Set theory