Logical and Mathematical Semantics

For quite some time specialists in symbolic logic and set theory have been working with various issues which are highly significant to the work of semantically oriented linguists. Recent work in semantic theory has borrowed heavily from these two fields.

Logicians have been very much interested in denotation; in contrasting, for instance, a vague concept like “gryphon” which has no concrete referent in the real world with a concept like “horse” which has a definite referent. They make the distinction between intensional meaning, which is the semantic qualities of a word, and extensional meaning, which is the way a word extends or relates to the real world. They are also interested in whether an expression represents an accidental or purposeful action. And one of their most important interests concerns the analyticity of a statement. If the truth of a statement can be determined wholly by knowledge of the meanings of its parts, then it is said to be analytic. A sentence which does not have this quality is said to be synthetic. A synthetic sentence cannot be shown to be either logically true (tautological), or logically false (contradictory) by its internal structure. A contradiction is a statement in which two or more parts of the internal structure are incompatible with each other. For example, the sentence, Kings, presidents, magistrates, and dictators are rulers, is tautological because if you know the meanings of the words kings, presidents, magistrates, dictators, and rulers, you can correctly judge the sentence to be true. By the same process if you know the meanings of the words peasants and rulers, you can correctly judge the sentence, Peasants are rulers, to be contradictory. Both of these decisions were arrived at through studying the internal structure of the sentences. But with the synthetic sentence, Elmer is a ruler, there is nothing in the sentence itself which will enable us to judge its truthfulness or falseness. Instead we have to know something about the real world and about Elmer.

Logicians have also been very interested in sentence analysis. Consider, for example, the sentence John hit Mary. In this sentence hit is described as a two-place predicate because it requires two nouns; a subject and an object. In terms of logical predicates, this sentence could be analyzed as follows: hitxy, humanx, malex, namexz, Johnz, humany, femaley, namew, Maryw. And this would be read as follows: “x hit y, and x is human, and x is male, and x has the name z, and z is John, and у is human, and у is female, and у has the name w, and w is Mary”. This type of analysis is similar to recent proposals by some linguists in which such logical predicates as human, and male, are thought of as semantic features. It is also similar to some recent analyses in that it considers the predicate as the central word in the sentence.

Logicians are also working on identifying words as members of particular sets or categories, and they are trying to measure the effects of quantifiers such as all, no, and there exists; conjunctions such as and and or, casual relationship words such as if...then, and if and only if, and mathematical terms such as equals, is greater than, and is greater than or equal to. Such concepts as reflexivity, transitivity, and reciprocity are also being worked on. For example, two words are said to be reflexive if one refers back to the other, as in I cut myself. A transitive expression in the mathematical sense is one which extends a quality, for example such expressions as tall, old, fat, etc. in that if A is taller than B, and В is taller than C, then A is automatically taller than C. In contrast, the expression friend is not transitive. A could be a friend of B, and В a friend of С without A being a friend of C. Two expressions are reciprocal if their positions can be reversed in reference to the predicate. For example, if A is a neighbor of В, В is a neighbor of A. In the sentence, John married Mary, marry is a reciprocal predicate, but in the sentence, The priest married them, marry is not reciprocal.

People working in logical and mathematical semantics have devised an elaborate shorthand for dealing with the various concepts they are interested in. Some good books in which these concepts have been developed are Machine Translation of Languages edited by William N. Locke and A. Donald Booth (1955), Handbook of Mathematical Psychology edited by R. Duncan Luce, Robert R. Bush, and Eugene Galanter (1963), and Mathematical Linguistics in Eastern Europe by Ferenc Kiefer (1968).

Books on logical semantics include Word and Object by Willard Van Orman Quine (1960), Signification and Significance by Charles Morris (1964), The Language of Logic by Morton Schagin (1968), and Towards a Semantic Description of English by Geoffrey N. Leech (1970).