A type of construction frequently employed when proving completeness of a modal logic in which the syntax of a language—its theories, formulae, etc.—are interpreted as the very semantic objects to which they are intended to refer. For example, in normal modal logics, canonical Kripke models equate maximal, deductively closed sets of sentences (i.e., maximal theories) with possible worlds. Truth of an atom $p$ at a world $T$ then becomes equivalent to $p$’s appearing in the theory $T$ and the accessibility relation between worlds is defined by an appropriate syntactic relation between these sets of sentences, i.e.:

• • $TR{T}^{\prime }\text{iff}\text{{}\phi |\square \phi \in T\text{}}\subseteq T^{\prime }$

This guarantees that for each maximal theory $T$, $T$ acts as a possible world in the canonical model such that $\phi$ is true at $T$ if and only if $\phi \in T$, entailing that whenever $\Gamma ⊬\phi$ there exists a possible world in the canonical model at which all formulae $\psi \in \Gamma$ are true but $\phi$ fails. A related notion in first-order logic is that of a term model, in which a model is constructed with the syntactic constants of the language serving as elements of the domain.