20th WCP: Interval Description of Change

The propositions about the transition states are the proper subject of temporal logic. Usually the temporal logic calculi presuppose the concept of a time moment as a starting abstraction. An evaluation of the formulae is carried out relatively to the moments which are odered by an earlier - later relation. A change of truth value of a formula testifies to a change of a state of affairs. The change of truth value is considered here as instantaneous.

The first approach to the formal explication of change by that means is von Wright's formalism [9]. The values of the formulae are determined relatively to temporal structures with a discrete order of moments. An event is considered in this formalism as a pair of states of affairs, i.e. as a transition state. It is represented by the formula ATB, where A and B designate the propositions about the states, T is the binary copula "and next".

However, it is duration that is the most important feature of time. A series of the problems arise when we have to deal with objects which do not possess a temporal extent. Recent investigations show that moments are inapplicable to the study of the phenomenal continuums: for many kinds of the quality changes, any subdivision of existence time of an object does not separate clearly the state before change from the state after it. It is not possible to determine the last moment of a prior state and the first moment of a posterior one. So, the predicates of natural language are not valued relatively to moment of time [3, 414; 7, 49]. The beginnings and the ends of the many states and processes are placed somewhere in the intervals of time, not at the fixed points. In virtue of vagueness of the state boundaries propositions describing the state before and after change are both true or their truth status is not determined. In von Wright's semantics the formula A is a semantic consequence of the formula ATB, while the formula B is not. In virtue of discreteness of temporal structure there is no intermediary moment between the last moment of A's truth and the first moment of B's truth. So, in this approach the moment of the transition coincides with the last moment of A's truth or with the first moment of B's truth. On the other hand, if the order of moments is dense, there is always a moment between A's and B's truth. Than the formula ATB is true at the moment, which is different from the truth of A and truth of B [4; 190, 192 ].

The first axiomatic theory of the intervals of time was proposed by C.L. Hamblin [3]. An irreflexive, antisymmetric and transitive "later then" relation was taken as a primitive constant. The attempt to justify that this relation is more than simply an order relation has led to another variant of the interval theories. Needham's aim was to build a theory of linear order in such a manner that the direction and the distinctions of future and past would be inexpressible [7].

J.N. Woodger has built an original axiomatic theory with an elementary relation "to be a part" both in spatial and temporal senses [8]. In terms of this relation and of relation of time precedence the concepts of the sum of parts, of moment, of organised units (such as a cell, an organism), of instant part of an organised unity were defined. These means have allowed to formalise the biological relations, which arise from model of cell division. A special kind of the continuous change, branching continuum has been formally expressed.

In order to study physical space-time J.-M. Laforge used as an initial a concept "to be a fragment of a physical aggregate" (F) and the concept of the territory of a physical compound (T): " The territory is what completely overlap a class of the objects having common mathematical or other well defined property" [5, 35]. A transformation of a physical compound, relised in the breaking up and subsequent joining up of the parts, can break a topological structure of a physical compound, but does not break an invariance of the relations F and T.

In terms of F and T the pseudo-topology of a physical compound is built. The concept of boundary, dividing set of points , is replaced by the concept of pseudo-boundary. The pseudo-boundary of a fragment A of a certain physical compound is a class of fragments intersecting A and its complement. An object which belongs to the pseudo-boundary of the fragment A has the properties of the objects from the fragment A and from its complement.

The temporal structures in two theories mentioned above contain the moments. In addition to the notion of truthfulness defined relatively to the moments, E.Lemmon has introduced the notion of truthfulness relatively to the intervals. The basic relation of this logic of space-time zones is four-dimentional relation "part-whole". The usage of a space-time structure allows justify the distinctions between the simple and continuous tenses of verbs, between perfective and imperfective aspects of verbs, and to construct a logic of certain tense adverbs [6, 101]. A next step in the creation of the logic of space-time zones is a temporal interval logic.

I will discus a description of change in the framework of one branch of temporal logic, namely tense interval logic. The goal is to present an approach to the construction of a tense paraconsistent logic in addition to the proposals of N.C.A. da Costa and S. French [2]. In accordance with Hamblin's intention, I consider change as that occupying an interval of time which has fuzzy boundaries. The description of change consists in the the description of those states overlapping the interval of change. As a result the description contains inconsistent statements.

These intuitions are embodied in the semantics of interval paraconsistent logic. It is defined relative to a model I of the form < T, Í , ½ ½ >, where T - class of time intervals; Í - mereological subinterval relation; ½ ½ : P ´ T ® {0,1} is a value function for propositional variable. The function ½ ½is extended inductively to supply values for the set of the formulas of the language of interval paraconsistent logic.

Definition 1. The truth condition for the formulas of the language of interval paraconsistent logic are:

Validity is defined in the following manner:

Definition 2 . A formula A of the language of interval paraconsistent logic is valid iff for any model I ½A½t = 1 for every t Î T.

By addition of the earlier-later relation R the model I is turned into a tense interval paraconsistent model TI : < T, Í , R , ½ ½ >. The truth conditions T1- F4 are supplemented by the conditions for the formula with tense operators "it will always be the case, that..."(G), "it has always been the case, that..."(H), "it will be the case, that..."(F), "it has been the case, that..."(P)

and by the special conditions for the operators "it always be the case, that..."(L), and "it sometimes be the case, that..."(M)

We can use this semantics for the study of many-valued logic. D. Bochvar has offered a calculus with internal and external connectives[1]. The truth matrices for some of them are:

where ØA is intrinsic negation of A, +A is extrinsic assertion of A ("A is true"), Ç is an intrinsic conjunction, * is the value "meaningless". The connectives Ø, + and Ç make a functionally complete system of connectives.

Theorem.

Bochvar's matrices are recoverable by the truth conditions of interval paraconsistent semantics.

Proof. The matrix I is represented by conditions:

The matrix II is represented by conditions

Thus the interval interpretation of Bochvar's calculus is derived.