STRUCTURES A ND D YNAMICS O F T HEORIES

The most natural way to formulate a scientific theory is to axiomatize it. Among the various possibilities of interpreting the phrase 'to axiomatize a theory', a particularly attractive one for logical studies consists in taking it as meaning 'to define a set theoretic predicate'. We shall therefore sometimes speak of the set theoretic predicate corresponding to the theory in question; e.g., the theory of groups is axiomatized by introducing the corresponding set theoretic predicate 'is a group'; quantum mechanics is axiomatized by introducing the set theoretic predicate 'is a quantum mechanics'. We shall not presuppose that the set theory itself is formalized. The set theoretic predicate used to axiomatize a theory will, therefore, always be an informal predicate. If the non-logical vocabulary of the theory contains only quantitative concepts, i.e. functions of various kinds, the set theoretic predicate corresponding to the theory describes a mathematical structure S. For the time being we will use only an intuitive concept of a theory. What matters for the moment, given a theory T, is only its mathematical structure S (T) and the extension of this predicate 'S' which we call the set Ms(r) of models of our theory. The empirical statements made with the help of our theory are sentences of the form