Für eine Aussagenvariablenmenge ${\displaystyle {}V}$ und beliebige Ausdrücke ${\displaystyle {}\alpha ,\beta ,\gamma }$ legt man folgende (syntaktische) Tautologien axiomatisch fest.

1. ${\displaystyle \vdash \alpha \rightarrow (\beta \rightarrow \alpha ).}$

2. ${\displaystyle \vdash (\alpha \rightarrow \beta )\wedge (\beta \rightarrow \gamma )\rightarrow (\alpha \rightarrow \gamma ).}$

3. ${\displaystyle \vdash (\alpha \rightarrow \beta )\wedge (\alpha \rightarrow \gamma )\rightarrow (\alpha \rightarrow \beta \wedge \gamma ).}$

4. ${\displaystyle \vdash (\alpha \wedge \beta \rightarrow \gamma )\rightarrow (\alpha \rightarrow (\beta \rightarrow \gamma ))}$

   und

   ${\displaystyle \vdash (\alpha \rightarrow (\beta \rightarrow \gamma ))\rightarrow (\alpha \wedge \beta \rightarrow \gamma ).}$

5. ${\displaystyle \vdash \neg \alpha \wedge \alpha \rightarrow \beta .}$

6. ${\displaystyle \vdash (\alpha \rightarrow \beta )\wedge (\neg \alpha \rightarrow \beta )\rightarrow \beta .}$