Concept

Kripke model

A set of worlds with arrows between them, together with an assignment saying which statements hold at each world. It gives the modal operators their meaning: necessary at a world means true at every world that world points to.

Named by 2 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

Which axioms hold on which frames. A table of frames against modal axioms, each cell decided by checking the axiom under every valuation.

The axiom is the shape of the graph

Add one operator meaning necessarily and the choice of which axioms to accept stops being a matter of taste. Each candidate axiom is true of exactly those worlds-and-arrows diagrams whose arrows have a stated property, and a logic is a class of graphs.

logic · modal logic

Named alongside it

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

Accessibility Axiom Constructive proof Double negation Excluded middle Exhaustive search Frame Heyting algebra Intuitionistic logic Modal logic Necessity Open set

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