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Concept

Excluded middle

The axiom that every statement holds or fails, which the constructive reading refuses because a proof of a disjunction has to say which side. Dropping it leaves a coherent logic whose models are the open sets of a space, where a set and its negation miss the boundary between them.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

A set, its negation, and its double negation. Four bars on one number line showing an open set, its negation, their union, and the double negation.

The middle that is not excluded

Either it is raining or it is not. Drop that as an axiom and what is left is still a logic — one with models made of open sets and of stages of knowledge, in which a set and its negation between them miss the boundary.

logic · Non classical logic 4 axioms against 4 finite algebras. A table of candidate axioms against finite Heyting algebras built from small orders, marking which algebras validate which axiom at every valuation.

Not one step but a continuum

Classical logic is the constructive system plus one axiom, which makes it sound as though there are two logics and one gap. There are uncountably many logics in that gap, each one a class of algebras, and the smallest separations between them fit in five elements.

logic · Non classical logic The smallest algebra that refutes each formula. A table of formulas against the smallest finite Heyting algebra refuting each, found by searching every order on a few points, with the formulas no such algebra refutes marked.

Refutable in something small

A formula that is not a theorem of the constructive system fails in some finite algebra, and the algebra can be found by search. That single property is what makes the propositional logic decidable — and the predicate version loses the property and the decidability with it.

logic · Non classical logic Gluing two countermodels: p ∨ ¬p. Two small Kripke models, each refuting one half of a disjunction, and the model made by placing both above a new first stage, at which neither half is forced.

A proof that says which half

Classical logic proves p ∨ ¬p without any idea which half is true. The constructive system never does that: whenever it proves a disjunction, it proves one of the two halves. The reason is a picture — two countermodels placed side by side above a new first stage that forces neither — and the logics between the two lose the property exactly when their pictures are not allowed to be glued.

logic · Non classical logic Three truth values, two of which count as holding. Truth tables of negation, conjunction and disjunction in the logic of paradox, with the values that count as holding — both and true — shaded.

A contradiction that stays where it is

In classical logic one contradiction proves everything: from p and not-p, any q at all. Add a third truth value — both true and false — and count it as holding, and the explosion stops. The classical laws all survive as laws; what goes is exactly the reasoning that carries a contradiction somewhere it was not. Run the other way, the same three tables make a logic of gaps instead of gluts.

logic · Non classical logic

Named alongside it

The objects these essays reach for when they reach for this one.

Heyting algebraIntuitionistic logicDouble negationExhaustive searchKripke modelOpen setTopology of logicConstructive proofConsistencyCounterexampleDecidabilityDisjunction property

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