   A very important construction, which carries over to many logics and many algebras other than Boolean algebras, is the construction of a Boolean algebra associated with the sentences in some logic. The simplest case is sentential logic. Here there are sentence symbols, and common connectives building up longer sentences from them: disjunction, conjunction, and negation. Given a set \(A\) of sentences in this language, two sentences \(s\) and \(t\) are equivalent modulo \(A\) if and only if the biconditional between them is a logical consequence of \(A\). The equivalence classes can be made into a BA such that + corresponds to disjunction, \(\cdot\) to conjunction, and \(-\) to negation. Any BA is isomorphic to one of this form. One can do something similar for a first-order theory. Let \(T\) be a first-order theory in a first-order language \(L\). We call formulas \(\phi\) and \(\psi\) equivalent provided that \(T \vdash \phi \leftrightarrow \psi\).  The equivalence class of a sentence \(\phi\) is denoted by [\(\phi\)]. Let \(A\) be the collection of all equivalence classes under this equivalence relation. We can make \(A\) into a BA by the following definitions, which are easily justified: