In
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, 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, a 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 ...
that is deductively closed.
The deductive closure of a
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
is often denoted or . Some authors do not define a theory as deductively closed (thus, a theory is defined as any set of
sentences), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as a ''deductively closed theory'' to emphasize it is defined as a deductively closed set.
Deductive closure 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
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...
().
Examples
In
propositional logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
, 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
In
epistemology
Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowle ...
, many philosophers have and continue to debate whether particular subsets of
proposition
A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
s—especially ones ascribing
knowledge
Knowledge is an Declarative knowledge, awareness of facts, a Knowledge by acquaintance, familiarity with individuals and situations, or a Procedural knowledge, practical skill. Knowledge of facts, also called propositional knowledge, is oft ...
or
justification of a
belief
A belief is a subjective Attitude (psychology), attitude that something is truth, true or a State of affairs (philosophy), state of affairs is the case. A subjective attitude is a mental state of having some Life stance, stance, take, or opinion ...
to a subject—are closed under deduction.
References
{{Mathlogic-stub
Concepts in logic
Closure
Logical consequence
Propositional calculus
Set theory