Hilbert’s Axioms
BACKGROUND
APPROACH
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)
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.
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.
The sixteen possible combinations start at the top of this ‘tree’. Each layer down simplifies each combination. Ultimately, they all boil down to truth, represented here by a single blue dot.
Breakdown of the visual map.
A visual representation of Hilbert’s Second Axiom System.