Tarskian Semantics, or No Notation Without Denotation!

Tarskian semantics is called "Tarskian" for historical reasons (Tarski, 1936). A more descriptive name would be "systematic denotational semantics," or SD for short. The method is called "denotational" because it specified the meanings of a notation in terms of what its expressions denote. The method is called "systematic" in hopes that the rules that assign meaning are precise enough to support statements and occasionally proofs of interesting properties of the notation. In a typical predicate calculus, we assign to primitive symbols denotations which consist of objects, functions, or predicates. Then the meanings of more complex expressions are defined by rules which define their meanings in terms of the meanings of their parts. For sentences in such a language, this amounts to specifying the conditions which make any given sentence true. That is, the meaning of a sentence is a specification of what would make it denote T and what would make it denote NIL. This specification may thus be thought of as a generalization of an ordinary LISP predicate definition. For example, we may assign to the predicate symbol PTRANS a predicate which is true only if its first argument has ever caused its second argument to be physically transferred from its third argument to its fourth argument. Then the denotation of (ACTOR x ¢=~ PTRANS OBJ y FROM u TO v) should be T just if the denotation of x has ever transferred the denotation of y from the denotation of u to the denotation of v. (This is a long-winded way of writing a typical semantic rule, which maps the syntax of an expression into a denotation systematically. Syntax is not an issue in this paper, but notice that the use of SD does not commit us to any syntax in particular, so long as it is precise.) It is clear that the first argument of the denotation of PTRANS should be an animate agent; its second, a physical object; its third and fourth, places. If we wish to be precise, we must somehow forbid incongruous types to appear in these places, or go on to specify what the denotation (ACTOR x ¢e~ PTRANS OBJ y FROM u TO v) is when x, y, u, or v is incongruous. So far this may seem very fluffy stuff. What have we gained by (apparently) just repeating in the semantic domain what is fairly obvious in the first place? Mainly …