Definition:Hilbert Proof System/Instance 2

Definition
 This instance of a Hilbert proof system is used in:



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

$\mathscr H$ has the following axioms and rules of inference:

Axioms
Let $p, q, r$ be propositional variables.

Then the following WFFs are axioms of $\mathscr H$:

Rule of Uniform Substitution
Any WFF $\mathbf A$ may be substituted for any propositional variable $p$ in another WFF $\mathbf B$.

Rule of Substitution by Definition
The following expressions are regarded definitional abbreviations:

Rule of Detachment
If $\mathbf A \implies \mathbf B$ and $\mathbf A$ are theorems of $\mathscr H$, then so is $\mathbf B$.

That is, Modus Ponendo Ponens.

Rule of Adjunction
If $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H$, then so is $\mathbf A \land \mathbf B$.

(This rule can be proved from the other three and so is only a convenience.)