## Symbolic Logic

Mapping, Data Visualization

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.

Hilbert’s second axiom system.