 Summary: Intensionality and Possible Worlds. Analyzed in terms of possible world semantics, then, the general failure of classical  substitutivity principles in modal logic is due, not to an  irreducibly intensional element in the meanings of the modal  operators, but rather to a sort of mismatch between the surface  syntax of those operators and their semantics: syntactically, they  are unary sentence operators like negation; but semantically, they  are, quite literally, quantifiers. Their syntactic similarity to  negation suggests that, like negation, the truth values of  ⌈□φ⌉ and  ⌈◇φ⌉, insofar as they are determinable at all, must be determined by the truth value of φ.  That they are not (in general) so determined leads to the distinctive substitutivity failures noted above. The possible worlds analysis of  the modal operators as quantifiers over worlds reveals that the unary syntactic form of the modal operators obscures a semantically  relevant parameter. When the modal operators are interpreted as  quantifiers, this parameter becomes explicit and the reason  underlying the failure of extensionality in modal logic becomes  clear: That the truth values of  ⌈□φ⌉ and  ⌈◇φ⌉ are not in general  determined by the truth value of φ at the world of evaluation is, semantically speaking, nothing more than the fact that the truth  values of ‘∀xFx’ and  ‘∃xFx’ are not in general determined by the  truth value of ‘Fx’, for any particular value of  ‘x’. Possible world semantics, therefore,  explains the intensionality of modal logic by revealing that the syntax of the modal operators prevents an adequate expression of  the meanings of the sentences in which they occur. Spelled out as  possible world truth conditions, those meanings can be expressed in a wholly extensional fashion.   (For  a more formal exposition of this point, see the supplemental article   The Extensionality of Possible World Semantics.)  