Equivalence of Definitions of Inconsistent (Logic)

Theorem

The following definitions of the concept of Inconsistent in the context of Logic are equivalent:

Let $\LL$ be a logical language.

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

Definition $1$

A set $\FF$ of logical formulas is inconsistent for $\mathscr P$ if and only if:

For every logical formula $\phi$, $\FF \vdash_{\mathscr P} \phi$.

That is, every logical formula $\phi$ is a provable consequence of $\FF$.

Definition $2$

A set $\FF$ of logical formulas is inconsistent for $\mathscr P$ if and only if:

There exists a logical formula $\phi$ such that both

$\FF \vdash_{\mathscr P} \paren {\phi \land \neg \phi}$

Proof

Definition $(1)$ implies Definition $(2)$

Let $\FF$ be an inconsistent set of logical formulas by definition $1$.

Let $\phi$ be an arbitrary logical formula in $\FF$.

Then by hypothesis:

$\phi \land \lnot \phi$

is a logical formula in $\FF$.

Thus $\FF$ is an inconsistent set of logical formulas by definition $2$.

$\Box$

Definition $(2)$ implies Definition $(1)$

Let $\FF$ be an inconsistent set of logical formulas by definition $2$.

Then by definition:

$\exists \phi \in \F: \FF \vdash_{\mathscr P} \paren {\phi \land \neg \phi}$

That is, $\phi \land \neg \phi$ is a logical formula in $\FF$.

Let $\psi$ be an arbitrary logical formula.

By the Rule of Explosion:

$\forall \psi: \paren {\phi \land \neg \phi} \implies \psi$

That is:

$\forall \psi: \FF \vdash_{\mathscr P} \psi$

That is:

$\psi \in \FF$

As $\psi$ is arbitrary, it follows that every logical formula is a provable consequence of $\FF$.

Thus $\FF$ is an inconsistent set of logical formulas by definition $1$.

$\blacksquare$