Alasdair Urquhart

The Theory of Processes

Some relatively unproblematic examples of processes are: the baking of a cake, the drying up of a pool of water, the flowering of a cherry tree, the reciting of Hiawatha, and a performance of Beethoven’s 9th. The consideration of examples of this sort leads to the recognition that a process embodies a temporally sequential, coordinated series of stages linked together in a cohesive unit. The stages at issue here can be viewed as the transient states within an ongoing system of changes, or rather of state-types since they, as well as the entire process at issue, must be repeatable, in principle at any rate. Thus while John’s growing up from his babyhood to his adulthood is indeed to be viewed as process, this is so because it is a concrete instance of the generic phenomenon of a boy’s growing up from infancy to adulthood. In accordance with this line of thought a process may thus be defined as: A programmed sequence (temporal sequence) of repeatable state-types. A process, in short, is a generic history.

The Background of Temporal Logic

The theory of temporal logic is an integral concern of philosophical inquiry, and questions of the nature of time and of temporal concepts have preoccupied philosophers since the inauguration of the subject. Kant wrote:

Tense Logical Characterizability and Definability

We must consider in some detail the fundamental question: How much of temporal logic can be expressed in terms of tenses alone? We have already seen that some quite simple properties of temporal ordering cannot be so expressed, while certain apparently very complex properties — such as continuity — can be expressed in this way. To make a start at a general theory of tense-logical expressibility, we can adopt some of the outlook of classical model theory by trying to link up formulas with the tense-structures that they characterize.

The Logic of World States

We have already had occasion to consider whether certain conditions on the U-relation are or are not expressible in terms of tense operators alone. For example, we have seen in the context of the completeness proof for K t , that irreflexivity of U is not so expressible1. We shall now present some further results along these lines, using a calculus of “world states” which gives some hope that the R-calculi can be developed within pure tense calculi.

The System K t of Minimal Tense Logic

We come now to the problem of devising a “tense logic” — that is, a set of rules governing the mutual logical interrelationships of the tense operators as defined within the framework of a (R/U)-calculus. Let us begin here with the question of an irreducibly basic or minimal tense logic; “minimal” in the sense of involving no assumptions whatsoever regarding the nature of the U-relation. This last condition is imposed to assure “minimality” in the sense of topological neutrality — i. e., the lack of any specific assumptions about the structure of time.

Linear Time: The System K l and Its Variants

The standard picture of time is that of a linear series. This appears not only in ordinary conceptions but is assumed in a great portion of physics. The absolute time of Newtonian physics is a one-dimensional linear continuum; even in relativistic physics the ordering of “local” time series is linear. In terms of the relation U of temporal precedence, this means that the temporal order in such conceptions satisfies the requirement both of transitivity:

Branching Time: The System K b

(Utt\prime \& Ut\prime t\prime \prime ) \supset Utt\prime \prime

The “Master Argument” of Diodorus and Temporal Determinism

and connectedness: