 Beth’s semantic tableaux, inspired by Brouwer’s notion of spread, predated Kripke’s semantics;  Troelstra and van Ulsen  give an authoritative account of the history. For a modern version of Beth semantics which facilitates comparison with Kripke semantics, a Beth structure is a Kripke structure in which the partially ordered set \(K\) is a rooted tree with \(k_0\) as the root, and the forcing conditions in a Beth model are the same as those in a Kripke model with two exceptions. The forcing conditions for \((A \vee B)\) and \(\exists x A(x)\) in a Beth model are as follows, where a branch of \(K\) is a maximal linearly ordered subset \(k_0 \le k_1 \le k_2 \le ...\) of \(K.\)