Is consistency the same as sound or complete, in mathematical logic?

Question

In a propositional logic system for example, it is sound iff $\Gamma \vdash \varphi \implies \Gamma \vDash \varphi$ and complete iff $\Gamma \vDash \varphi \implies \Gamma \vdash \varphi$ (I don't know if this definition also happens to be the same in higher order logics).

Anyhow, what then does it mean for a system to be "consistent"? Is this the same as a logic system being complete or sound, or does it mean something else? Is it a syntactic claim? A semantic one?

Answer 1

A set of sentences or a theory is consistent if it does not contain a contradiction.

Consistency can be defined in either semantic or syntactic terms. The semantic definition states that the set is consistent if and only if it has a model, i.e., there exists an interpretation under which all formulas in the set are true.

In this case, we say also that the set is satisfiable

In the context of mathematical logic, where we consider a proof system and the corresponding relation of derivability ($\vdash$), we say that

a set $\Gamma$ of sentences is consistent if and only if there is no formula $\varphi$ such that both $\Gamma \vdash \varphi$ and $\Gamma \vdash \lnot \varphi$.

If $\bot$ is part of the language, the above amounts to saying that $\Gamma \nvdash \bot$.

To say that a proof system is sound means that only true formulas can be derived with it.

Thus, soundness implies consistency, because $\bot$ is not true.

Regarding completeness, we have that, in classical logic, where Ex falso holds, an inconsistent proof system is trivially complete : being inconsistent, it proves every formula, and thus also the valid ones.