Deductive closure

In mathematical logic, 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 that is deductively closed. The deductive closure of a theory 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 .

Deductive closure

In mathematical logic, 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 that is deductively closed. The deductive closure of a theory 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 .