User:Lord Farin/Sandbox/Equivalence of Definitions of Consistent

Theorem
Let $\LL$ be a logical language.

Let $\mathscr P$ be a proof system for $\LL$.

The following definitions of consistent are equivalent:

Proof
Let $\FF$ be a set of logical formulas in $\LL$.

Definition 1 implies Definition 2
Let $\FF$ be consistent by definition 1.

Suppose $P$ derivable from $\FF$.

Hence, by definition, $P$ is a tautology.

Then by Tautology is Negation of Contradiction, $\neg P$ is a contradiction.

So, by definition, $\neg P$ is not derivable from $\FF$.

Similarly, suppose $\neg P$ is derivable from $\FF$.

Hence, by definition, $\neg P$ is a tautology.

Then by Contradiction is Negation of Tautology, $P$ is a contradiction.

So, by definition, $P$ is not derivable from $\FF$.

Thus either $P$ or $\neg P$ is not derivable from $\FF$.

Hence $\FF$ is consistent by definition 2.

Definition 2 implies Definition 3
Let $\FF$ be consistent by definition 2.

Let $P$ be derivable from $\FF$.

Then, by definition, $\neg P$ is not a derivable formula.

That is, there exists a formula that is not derivable from $\FF$.

Hence $\FF$ is consistent by definition 3.

Definition 3 implies Definition 2
Let $\FF$ be consistent by definition 3.

that there exists $P$ such that both $P$ and $\neg P$ are derivable from $\FF$.

Let $Q$ be a logical formula.

From the Rule of Explosion:
 * $\apren {P \land \neg P} \implies Q$

and thus it follows that $Q$ is a derivable formula.

Thus, whatever $Q$ is, it is derivable from $\FF$.

That is, $\FF$ is not consistent by definition 3.

This contradicts the hypothesis.

Hence $\FF$ must be consistent by definition 2.