In those ambiguous or catastrophic situations where the evolution of phenomena seems ill determined, the observer will try to remove the indeterminacy and thus to predict the future by the use of local models. The single idea of a spatiotemporal object already implies the idea of a model (this is discussed in Chapter 2). From this point of view, we say that a system of forms in evolution constitutes a formalizable process if there is a formal system P (in the sense of formal logic) satisfying the following conditions:
1. Each state A of the phenomenological process under consideration can be parameterized by a set of propositions a of the formal system P.
2. If, in the course of time, state A is transformed into state B, then B can be parameterized by a set b of P such that b can be deduced from a in P.
In other words, there is a bijective map h from some or all of the propositions of P onto the set of forms appearing globally in the process, and the inverse of this map transforms temporal into logical succession.
Such a model is not necessarily deterministic, for a set a of premises of P can, in general, imply a large number of formally different conclusions, and so the model is not entirely satisfactory for, being indeterministic, it does not always allow prediction. Atomic models, in which all forms of the process under consideration arise by aggregation or superposition of elementary indestructible forms, called atoms, and all change is a change in the arrangements of these atoms, are to some extent models of this type. But the theory will not be satisfactory unless it allows for prediction, and for this it is necessary, in general, to construct a new theory, usually quantitative, which will be the thermodynamical theory governing the arrangement of these particles.
All models divide naturally in this way into two a priori distinct parts: one kinematic, whose aim is to parameterize the forms or the states of the process under consideration, and the other dynamic, describing the evolution in time of these forms. In the case envisaged above of a formalizable process, the kinematic part is given by the formal system P together with the bijective map h of P onto the forms of the process. The dynamic part, if it is known, will be given by the transition probabilities between a state A parameterized by a set of propositions a and a state B parameterized by b, a consequence of a. In this way the kinematical theory, if formalizable, implies a restriction on the dynamical theory, because the transition probability between states A and C must be zero if 1/h(C) is not a consequence of a = 1/h(A) in P. It is altogether exceptional for a natural process to have a global formalization; as we shall see later, it is a well-known experience that initial symmetries may break in some natural processes.(note). As a result we cannot hope for a global formalization, but local formalizations are possible and permit us to talk of cause and effect. We can say that the phenomena of the process are effectively explained only when P is effectively a system of formal logic; in most known cases P has a less rigid structure, with only a preordering in the place of logical implication. Dropping the natural restriction that P contains a countable number of elements (parameterized by symbols, letters, etc.), we obtain quantitative or continuous models.