 From Tarskian to Possible World Semantics. The truth  conditional clauses for the three logical operators directly reflect  the meanings of the natural language expressions they symbolize:  ‘¬’ means not; ‘→’ means  if...then; ‘∀’ means all. It is  easy to see, however, that we cannot expect to add an equally simple  clause for sentences containing an operator that symbolizes  necessity. For a Tarskian interpretation fixes the domain of quantification and the extensions of all the predicates. Pretty  clearly, however, to capture necessity and possibility, one must be  able to consider alternative “possible” domains of  quantification and alternative “possible” extensions for  predicates as well. For, intuitively, under different circumstances,  fewer, more, or other things might have existed and things that  actually exist might, in those circumstances, have had very different properties. (6), for example, is false because John could have had  non-mammalian pets: a canary, say, or a turtle, or, under  very different circumstances, a dragon. A bit more formally  put: Both the domain of quantification and the extension of the  predicate ‘P’ could, in some sense or other, have  been different. 