Underlying functions

We draw upon some of the methods of symbolic logic for analyzing sentences and reverse them as well to synthesize or generate sentences. Thus John has a dog may be analyzed into the existential quantifier and functions as ‘ there is a B ’ and ‘A has B’ and ‘A = John’ and ‘B is a dog’. Dog is in fact a one-place function, in contrast with functions of two or more places. To show this its representation as a lexical item in a theory (description) of English would not be as one word, but as A is a dog. A relational noun such as son is a two-place function A is B’s son. A gives B to C has three places. Apparently homonymous items that actually differ in the number of places would show this fact explicitly, such as A is a child and A is B’s child. (Note that ‘A is B’s child’ does not necessarily imply ‘A is a child’.) Since difference in number of places correlates with differences in syntactic behavior, such a representation of lexical items as schematic sentences shows the syntactic differences and facilitates the application of appropriate rules to generate utterances. It also does so, for example, for mass nouns vs. count nouns vs. adjectives, e.g. A is sugar, A is a substance, A is sweet. Thus, to amend what was said above, the unit to be defined is a lexeme as a function.

In the definitions of items, their semantic components are also in the form of schematic sentences or functions. In a theory of a language, then, the definition or meaning of an item is seen as a set of sentences which together translate, or paraphrase, the sentence to be defined (Peirce 1933: pars. 427, 569). A definition is thus a statement of equivalence between the defined sentence and the defining sentences. It corresponds roughly to a similar statement, or schema of statements, in the object language of whose truth native speakers are competent to judge (Weinreich 1962: 42 ff.). It is also the covert major premise in various logical arguments phrased in the object language (Peirce 1933: pars. 176, 179). We can therefore involve informants in testing putative definitions as shown in the discussion of tests.