Equivalence of Definitions of Consistent Proof System

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Theorem

The following definitions of the concept of Consistent Proof System for Propositional Logic are equivalent:


Let $\mathcal L_0$ be the language of propositional logic.

Let $\mathscr P$ be a proof system for $\mathcal L_0$.

Definition 1

Then $\mathscr P$ is consistent if and only if:

There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$

That is, some logical formula $\phi$ is not a theorem of $\mathscr P$.

Definition 2

Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds.


Then $\mathscr P$ is consistent if and only if:

For every logical formula $\phi$, not both of $\phi$ and $\neg \phi$ are theorems of $\mathscr P$


Proof

Definition 1 implies Definition 2

Suppose that $\neg\vdash_{\mathscr P} \phi$.

Suppose additionally that there is some logical formula $\psi$ such that:

$\vdash_{\mathscr P} \psi, \neg \psi$

By the Rule of Explosion:

$\psi, \neg \psi \vdash_{\mathscr P} \phi$

By Provable Consequence of Theorems is Theorem, we conclude:

$\vdash_{\mathscr P} \phi$

in contradiction to our assumption.

$\Box$


Definition 2 implies Definition 1

Suppose either $\phi$ or $\neg\phi$ is not a theorem of $\mathscr P$.

The implication follows trivially.

$\blacksquare$


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