 As we shall see below, \(K_{a}\varphi\) is read as stating that \(\varphi\) holds in all of the worlds accessible to a. In this sense, K can be regarded as behaving similarly to the ‘box’ operator, \(\square\), often used to denote necessity. In evaluating \(K_{a}\varphi\) at a possible world w, one is in effect evaluating a universal quantification over all the worlds accessible from w. The universal quantifier \(\forall\) in first-order logic has the existential quantifier \(\exists\) as its dual: This means that the quantifiers are mutually definable by taking either \(\forall\) as primitive and defining \(\exists x\varphi\) as short for \(\neg\forall x\neg\varphi\) or by taking \(\exists\) as primitive and defining \(\forall x\varphi\) as \(\neg\exists x\neg\varphi\). In the case of \(K_{a}\), it may be seen that the formula \(\neg K_{a}\neg\varphi\) makes an existential quantification: It says that there exists an accessible world that satisfies \(\varphi\). In the literature, a dual operator for \(K_{a}\) is often introduced. The typical notation for \(\neg K_{a}\neg\) includes \(\langle K_{a}\rangle\) and \(\widehat{K}_{a}\). This notation mimics the diamond-shape \(\lozenge\), which is the standard dual operator to the box \(\square\), which in turn is standard notation for the universally quantifying modal operator (see the entry on  modal logic).