Matters

Model-Theoretic Semantics for the Web: "The word 'model' is not always defined the same way by logicians who study model theory, though the variability in these definitions is rather slight. Consider the following two definitions, the first taken from an introductory article on model theory, and the second from a textbook on mathematical logic.

Definition 1: 'Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that S comes to express a true or false statement, we are said to interpret S, and the added information is called an interpretation of S. If the interpretation I happens to make S state something true, we say that I is a model of S, or that I satisfies S, in symbols I |= S '

Definition 2: 'Now, using induction on formulas f, we give a definition of the relation I is a model of f, where I is an arbitrary interpretation. If I is a model of f we also say that I satisfies f or that f holds in I, and we write I |= f ... [the long, inductive definition follows]'

Definition 3: "Intuitively a model is a situation. That is, it is a semantic entity: it contains the kinds of things we want to talk about. Thus a model for a given vocabulary gives us two pieces of information. First, it tells us which collection of entities we are talking about; this collection is usually called the domain. Second, for each symbol in the vocabulary, it gives us an appropriate semantic entity, built from the items in D [the domain]. This task is carried out by a function F which specifies, for each symbol in the vocabulary, an appropriate semantic value; we call such functions interpretation functions. Thus, in set theoretic terms, a model M is an ordered pair (D, F) consisting of a domain D and an interpretation function F specifying semantic values in D"