Axiom
P → P
The law of identity: a proposition implies itself. The simplest truth, the foundation upon which all reasoning rests.
Where thought finds its architecture
Before structure, there is only the raw material of thought — scattered, unexamined, waiting.
Ideas find their place — stacked, interlocking, each supporting the next like basalt columns.
The law of identity: a proposition implies itself. The simplest truth, the foundation upon which all reasoning rests.
Non-contradiction — no statement can be simultaneously true and false. The bedrock principle that makes discourse possible.
Excluded middle. Every proposition is either true or its negation is true. There is no third possibility in classical logic.
Contraposition. If the consequent falls, so falls the antecedent. Every implication carries its shadow.
Universal generalization — what holds for all, holds for each. The bridge from the specific to the general.
Existential equivalence. To say something exists is to deny universal absence.
Modus ponens — the engine of deduction. Given an implication and its premise, the conclusion follows inexorably.
The principle of simplification. A true statement remains true regardless of what precedes it in implication.
Distribution of implication. The chain of reasoning can be decomposed and recomposed without loss of truth.
Biconditional — the symmetry of equivalence. Two truths that entail each other become indistinguishable in logical space.
If not everything satisfies a predicate, then there exists a counterexample. The power of negating universals.
Hypothetical syllogism — the transitivity of implication. Chains of reasoning link into unbroken sequences of truth.
From axioms to theorems, thought flows like water — each step following from the last with the certainty of gravity.
The logical day begins not with sunrise, but with the moment a mind recognizes its own capacity for structured thought. Every proposition examined, every inference validated, every contradiction resolved — these are the hours of a logical day.
Therefore, we reason.