Translating natural language into a metalanguage
Predicate calculus, the logical metalanguage into which formal semanticists translate natural languages like English, contains a range of expressions. These expressions represent the meaning expressed by units of language like nouns, verbs and adjectives by means of terms. There are two kinds of terms: individual constants and predicates. Constants are expressions that relate to specific entities (like James Bond or the spy) and are represented by lower-case letters of the alphabet like a, b, c and so on. Predicates are expressions that rep resent processes (expressed by verbs like eat), properties (expressed by adjectives like funny), roles (expressed by nouns like a top British spy) and relations (expressed by prepositions like under). One-place predicates like funny, die or a top British spy only require a single participant to complete their meaning (e.g. James Bond is funny; James Bond died; James Bond is a top British spy), while two-place predicates like appreciate or under require two participants (e.g. James Bond appreciates Miss Moneypenny; James Bond is under the desk). Predicates are represented by upper-case letters of the alphabet, like A, B, C and so on. When constants and predicates are combined, this results in a formula. For example, the sentence in (14a) can be expressed by the formula in (14b), where upper-case S represents the predicate sings and lower-case f represents the constant Fred. By convention, the predicate occurs first in the predicate calculus formula, so the ‘translation’ does not reflect the word order of English.
Example (15) illustrates a formula in which a two-place predicate combines with two constants. The relative order of the constants is important, because this reflects the difference in meaning contributed by the syntactic structure: like the natural language sentence in (15a), the formula in (15b) says that Jane loves Tom, not that Tom loves Jane.
In sentences like Jane loves Tom and Tom loves Jane, which consist of two or more conjoined clauses and thus express two or more propositions, the clauses are connected by natural language connectives like and, or, but and so on. In sentences like Jane does not love Tom or Jane loves Tom but not Bill, the negation word not is an operator, an expression that takes scope over some part of the sentence and affects its meaning. Natural language expressions like all, every and some are also operators. These are quantifiers and take scope over some part of the sentence by quantifying it (for example, the sentences Every police man witnessed some crimes and Some policemen witnessed every crime each express a different proposition due to the positions of the quantifiers, despite the fact that they contain the same predicates and constants). Connectives and operators are represented by the logical symbols in Table 13.1, where the column ‘syntax’ shows how these symbols can be combined with other units.
Example (16) shows how the sentence in (16a) is translated into a predicate calculus formula (16b). The expression in (16c) shows how the predicate calculus can be ‘read’. In this example, x represents a variable. This is an expression that, like a constant, relates to an entity or group of entities (hence the lower-case symbol); unlike a constant, a variable does not indicate a specific entity. The lower-case letters x, y and z are reserved for variables.
