Hilbert’s Axioms
![](https://freight.cargo.site/t/original/i/61cef79b1ee970512b31fe063be43f15e26060cfb5f69d0b27b35de2a91dfc68/hilbert-s-axioms_13x.png)
Hilbert’s axioms consist of three tautologies — logical sentences that always return as true, no matter what the truth values of the letters are.
Hilbert’s axioms are:
A → (B → A)
(A → (B → C)) → ((A → B) → (A → C))
(¬A → ¬B) → (B → A)
Hilbert’s axioms are:
A → (B → A)
(A → (B → C)) → ((A → B) → (A → C))
(¬A → ¬B) → (B → A)
These axioms, along with the rule of Modus Ponens, construct Hilbert’s proof system for classical logic.
Letters can either be true or false — here, blue dots represent “true” letters and orange diamonds represent “false” ones. This visualization takes every possible combination for each axiom — a total of sixteen different possibilities — and traces their paths to truth.
Letters can either be true or false — here, blue dots represent “true” letters and orange diamonds represent “false” ones. This visualization takes every possible combination for each axiom — a total of sixteen different possibilities — and traces their paths to truth.
![](https://freight.cargo.site/t/original/i/3dcad08d875f96aeb205076b6926494ef6088455f2dd10e16a0b7364cc3c31eb/hilbertsportfoliozoom.png)
![](https://freight.cargo.site/t/original/i/b842410317cc36144f27985aff804e780b0133c3cb61ff362c26a1d6c9add0a6/hilbert-s-axioms_13x-copy.png)
![](https://freight.cargo.site/t/original/i/1f942261cb9236ffb7a51ccaea4d98c962524383ab718630ad1a956ae4ab8c3b/hilberts2ndaxiom.jpg)
Hilbert's Second Axiom System
![](https://files.cargocollective.com/c1748463/hilbert-s-axiom-for-animation-2.gif)