logical consistency

A set of formulas is logically consistent if there is no proof of contradiction from these formulas.

In other word, a set of formulas is inconsistent iff we can derive a contradiction from it in the proof system: , where represents a contradiction.

Relation with Satisfiability

Note that consistency is defined syntactically, which is different from satisfiability, which is a semantic property. In propositional logic and first-order logic, the two notions are equivalent because the logic is sound and complete. But for other logics, this may not be the case. ¹

Footnotes

1. logic - Difference between consistency and satisfiability - Mathematics Stack Exchange ↩