Declarative and procedural aspects of theories and models

The present account of the relation between theories, models and reality is, of course, highly simplified from the point of view of scientific methodology and the philosophy of science. But it provides a useful system of coordinates for locating the different kinds of work done in Computational Linguistics.

Theory:

A theory is a set of theorems, some of which are axiomatic and others derived. A theorem is a sentence in a formal language defined in terms of a formal syntax, related to other sentences by rules of inference, and interpreted as true or false, sound or complete, with reference to a model.

Model:

From a formal point of view, a model is a structure with an interpretation function which provides a denotational semantics for a theory. From an empirical point of view, a model is an abstraction, idealisation, selection, stylisation of reality which provides a systematic starting point for scientific study of reality.

The theory-model relation is defined in mathematics, in the field of model theory. A given theory may have several different models, such as formal algebraic models, or operational models such as computer programs or working hardware devices.

The model-reality relation is defined in the descriptive sciences. A given model may relate to different theories, i.e. several theories may compete to provide the best description and explanation of the model.

An empirical model may be informal, systematised, or, in the extreme case formalised as an abstract structure.

A model, whether formal or informal, may have an operational semantics which shows how to derive conclusions from premises, either manually (as in writing derivations or proofs by hand) or automatically with a computer program.

The diagramme shows a number of the relations between theories and models.