Common
Consistency
Non-contradiction of a theory
Consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula φ such that both φ and its negation ¬φ are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and ⟨A⟩ the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when there is no formula φ such that φ∈⟨A⟩ and ¬φ∈⟨A⟩. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent...
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