Definition:Hilbert Proof System/Predicate Logic/Instance 1

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



Let $\LL$ be the language of predicate logic.

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

Axioms
Let $x, y, z$ be variables of $\LL$.

Let $\tau$ be a term of $\LL$.

Let $f_n$ be an $n$-ary function symbol of $\LL$ with $n > 0$.

Let $p_n$ be an $n$-ary relation symbol of $\LL$ with $n > 0$.

Let $\mathbf A, \mathbf B$ be WFFs.

Let $\map{ \mathbf A } {x \gets \tau}$ be the substitution instance of $\mathbf A$ substituting $\tau$ for $x$.

The universal closures of the following WFFs are axioms of $\mathscr H$:

Rules of Inference
The sole rule of inference is Modus Ponendo Ponens:


 * From $\mathbf A$ and $\mathbf A \implies \mathbf B$, one may infer $\mathbf B$.

Derived Rules

 * Proof by Contradiction for Hilbert Proof System Instance 1 for Predicate Logic
 * Reductio ad Absurdum for Hilbert Proof System Instance 1 for Predicate Logic