reasoner.dev
the architecture of inference, made navigable
forall p in P : (axiom(p) -> derivable(p))
AXIOM
A starting point assumed without proof. The ground beneath the argument; the bedrock from which every consequence will be drawn.
An axiom is not a claim about the world but a stipulation about the rules of a game. Choose your axioms and you have chosen which truths are reachable. Reasoner begins where Euclid began — with a small, severe set of statements held to be self-evidently admissible, and a willingness to accept whatever cathedral they produce.
We adopt three: that identity is preserved under inference, that contradictions terminate a branch, and that every proposition has a negation.
A1. forall x : x = x
A2. not (P and not P)
A3. P or not P
HYPOTHESIS
A provisional commitment. The argument's wager: assume this, and let us see what follows.
A hypothesis is the engine of conditional reasoning. Unlike an axiom, it is not held to be true — only entertained. Beneath every if … then … is the silent agreement that we will proceed as if the antecedent held, and that whatever conclusion we reach will inherit that same conditional weight.
Hypotheses fork the proof. Each branch is a possible world, kept alive only as long as it remains contradiction-free.
assume: P(x) -> Q(x)
witness: exists x : P(x)
DERIVATION
The mechanical phase. Rules of inference applied to the premises until something new appears on the page.
Derivation is the part of a proof that a machine can do. Modus ponens, universal instantiation, conditional proof — each is a small, dependable transformation that takes one or more lines and yields a new line, justified by the rule above.
Watched in slow motion, derivation has the texture of weaving: the threads are propositions, the loom is the rulebook, and the cloth that emerges is a proof in canonical form.
1. P(a) -> Q(a) [from H]
2. P(a) [witness]
3. Q(a) [MP, 1, 2]
LEMMA
An intermediate result. Not the destination, but a load-bearing wall. Without it, the theorem cannot stand.
A lemma is the proof's modesty: an admission that the argument is too long to hold in a single breath, that some sub-claim must be settled and stored before the main line can proceed. Mathematicians often remember the lemmas long after the theorems they were summoned to support — Zorn's, Yoneda's, König's — because the lemma is where the work happens.
To name something a lemma is to declare it useful elsewhere.
Lemma L_1: forall x : (P(x) -> Q(x)) -> (exists y : Q(y))
THEOREM
The conclusion the entire descent has been preparing. Discharged of its hypotheses, returned to the world as a free-standing truth.
A theorem is a hypothesis that has paid its debts. The conditional weight it once carried has been absorbed into its statement, and what remains can be lifted, transported, and reused without carrying the proof along. This is the strange economy of mathematics: a fifty-page argument compresses into a single line, and the line moves freely between minds.
The proof closes. The branch that survived now bears its name.
Theorem (Reasoner):
forall P, Q : (A1 and A2 and A3) and H |- P(a) -> Q(a)
Q.E.D.
end of proof.
the argument is a path; the path is the argument.