 The formal study of natural language semantics started with Montague and gave rise to a flourishing field of inquiry (see entry on  Montague semantics).  The basic idea in this field is to associate to natural language sentences wffs of a formal language, in order to represent sentence meanings in a logically perspicuous manner. The association reflects the compositionality of meanings: different syntactic subcomponents of sentences correspond systematically to syntactic subcomponents of the wffs; subcomponents of wffs thus represent the meanings of the subcomponents of the sentences. The formal language eschews ambiguities and has its own formal semantics, which grants that formulas have logical properties and relations, such as logical truth and entailments, so that in particular certain sequences of formulas count as logically valid arguments. The ambiguities we normally find in natural language sentences and the entailment relations that link them are captured by associating ambiguous sentences to different unambiguous wffs, in such a way that when a natural language argument is felt to be valid there is a corresponding sequence of wffs that count as a logically valid argument. In order to achieve all this, Montague appealed to a higher-order logic. To see why this was necessary, one may focus on this valid argument: