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The general problem of the relationship between instant-based and interval-based temporal semantics is studied. The paper is in two parts. In the first part we specified conditions for the mutual definability of instant and interval temporal structures. In this second part we extend this 'area of agreement' for temporal semantics proper and consider some natural 'logics of change' generated by this correspondence. In the first part of our paper we considered the possibilities of mutual definability for instant-based and interval-based temporal ontologies. Here we will try to extend this 'agreement area' on temporal semantics. We will consider various instant and interval valuations and will try to determine the conditions for their definability in terms of each other. It will be shown that such definable valuations give rise to natural logics, which could be regarded as logics of change. 1 Technical preliminaries We will be working in the frameworks of open dense linear instant-interval structures. Valuations on these structures will be represented as valuation predicates, that is, [/?],will denote a two-place valuation predicate V(/?,/) with respect to propositions and times (either instants or intervals). We will also be using the following notions: Definition 1 (i) A point α is a boundary point of an interval t (in notation, a + t) iff there is another point β such that t is bounded by these points. (ii) A point a is an internal point of an interval t (in notation, a. <• t) iff it belongs to t but is not a boundary point of it. Received September 9, 1988 TEMPORAL SEMANTICS: II 581 In order to give a smooth description of the necessary notions we will introduce a number of auxiliary temporal operators, which will simplify the presentation of subsequent results. Not all of them are independent, some even coincide (cf. Up and U/? below), but this is not essential for our purposes. But it is worth noting that they are sufficient for our descriptions, and this reflects the level of complexity required. Interval operators: [Tp]t ^ ( 3 s ) ( / < 5 & [p]s) [SP]t ^ ( 3 s ) ( S < / & [p]s) [Lp]t = (3a){a<t&(p)a) [Vp]t E ( 3 α ) ( α < t&(p)a) [l+p]t=(la)(a + t&(p>a) [lp]t = (a)(a<t^(3s)(a<s<t&[p]s)) [Γp]t = (α)(α <• t-+(ls)(a <• s < t & [p]s)) [l+p]t = (a)(a + t^(ls)(a + s*t& [p]s)). Instant operators: <Ap)a = (3t)(a<t& [p]t) <A°p)a = ( 3 ί ) ( α <• t& [p]t) (A+p)a = (lt)(a + t& [p]t) <U/?>« =(t)(a<t-*(3s)(a<s*t&[p]s)) < U » α s ( / ) ( α < t-+{3s)(a<' s<t& ίp]s)) < U » α a (0(α + ί->(3J)(α + 5 ^ ί & [p]s)) As can be seen, the operators L, A, and U depend on two valuations and hence their meaning is determined only if the destination' valuation is known, while the source valuation is not important. 2 Interval valuations We will begin with primary interval valuations, which we will call at-valuations (cf. Vlach [12]). However vague, the intuitive principle behind them is that an evaluation is made directly with respect to a given interval (and not, for example, by virtue of some subinterval of this interval). For such valuations we may expect that the distribution of truth values for a given proposition must depend mainly on this proposition and not on the properties of the valuation itself. Below we will describe the main types of interval-evaluated propositions (cf. Burgess [6]). Note that we give, in fact, pairs of dual properties. This duality will play an essential role in the following. Definition 2 (i) A proposition p is called persistent if the formula (Ύp -* p) is universally valid, (ii) A proposition is called negatively persistent if the formula (p -> ~ T~p) is universally valid. It is clear that a proposition is negatively persistent if and only if its (classical) negation is persistent and vice versa. It is easy to show that (negatively) persistent propositions are representable as having the form Ύp (~ T~p) or — S~p (Sp) for some proposition p. As an example of persistent propositions we will mention aspectual types of activities and states. 582 ALEXANDER BOCHMAN Definition 3 (i) A proposition /? is intervally convex if the formula (Ύp & Sp -» /?) is universally valid, (ii) A proposition p is negatively intervally convex if the formula (p -> ~ T~p v ~S~/?) is universally valid. Intervally convex propositions are representable as having the form Ύp & Sp and thus are definable as conjunctions of persistent and negatively persistent propositions. In the same way negatively convex propositions are definable as disjunctions of persistent and dual persistent propositions. A proposition is convex and negatively convex if and only if it is persistent or negatively persistent. Note that interval convexity was proposed in fact in Taylor [10] as a characteristic property of activities. Definition 4 (i) A proposition p is cumulative if Vp-+p is universally valid, (ii) A proposition/? is negatively cumulative if p-> ~Y~p is universally valid (that is, if its negation is cumulative). (Negatively) cumulative propositions are representable as having the form Γp (~Γ~/?). Note also that persistent propositions are obviously negatively cumulative, while negatively persistent ones are cumulative. It can be shown that propositions of all elementary aspectual types are in fact cumulative in at-valuations. Indeed, for activities, states, and generics this is obvious, while for achievements and accomplishments it is trivially true, because the former are false at intervals, whereas intervals at which an accomplishment is true are not overlapped. Definition 5 (i) A proposition p is strongly cumulative if ~ S~Sp ->p is universally valid, (ii) A proposition/? is strongly negatively cumulative if p -> S~S~p is universally valid. Among aspectual types only states and generics are strongly cumulative, while activities are only simply cumulative, because two abutting temporal intervals of some activity with an instantaneous interruption between them (for example, an instantaneous stopping in motion) do not form an interval of an (uninterrupted) activity. It must be noted here that so-called 'mass analogy' for aspectual types corresponds to strong cumulativity rather than to simple cumulativity. We will say that a proposition is (negatively) homogeneous if it is (negatively) persistent and (negatively) cumulative, and that it is strongly (negatively) homogeneous if it is (negatively) persistent and strongly (negatively) cumulative. It is easy to show that p is homogeneous iff ~L°~A°/? <+p is valid and negatively homogeneous iff p <-> L°~A°~/? is valid. A proposition p is strongly homogeneous iff/? <-• ~S~S/? is valid, while /? is strongly negatively homogeneous iff/? <-> S~S~p is valid. In addition to at-valuations there are two other valuations common in temporal semantics for natural languages (cf. Vlach [12]). For-valuations are persistent for all propositions, while in-valuations are negatively persistent for all propositions. It is clear that the properties of persistence and negative persistence TEMPORAL SEMANTICS: II 583 here characterize the valuations themselves, rather than the corresponding propositions. Note that any for-valuation could be represented as (ls)(t <s& V(/?,s )) or as (s)(s < t -* Y(p,s)) for some valuation V, whereas a negatively persistent valuation is representable as (3s)(s <t& V(p,s)) or as (s)(t < s-+ V(p,s)). What is important for the following is that since all aspectual types are cumulative in at-valuations, in the for-valuation, defined as (ls)(t < s & V(/?,s)), where V is an at-valuation, they all will be homogeneous. 3 Instant valuations There are also plausible classifications for instantevaluated propositions. Definition 6 (i) A proposition^ is open iff the following formula is universally valid: p <-> A°~L°~p. (ii) A proposition p is closed iff the formula p <-• ~ A°~L°p is universally valid. According to Aristotle, propositions expressing motion are in fact open, while states are in general closed. Definition 7 (i) A proposition p is regular open iff the formula p <-> A° S ~ L°p is universally valid. (ii) A proposition p is regular closed iff the formula p <-> ~A°~S~L°~/? is universally valid. As will be shown, the above four types of propositions strongly correspond to interval types of (negative) homogeneity and strong homogeneity. 4 Finiteness requirements There are many reasons to think that empirically definable propositions and valuations cannot"change their minds" too often, or, to be more exact, corresponding truth values couldn't change an infinite number of times in a finite interval. We now give a formal description of this requirement for instant-evaluated propositions: Definition 8 An instant-evaluated proposition p v& finite iff the following formula is universally valid: (1) ~U~(~L°~/?v~L7>). For compact intervals this means that they may be divided on a finite number of subintervals, such that for any of them p is either true throughout or false throughout. Another equivalent to condition (1) is that the truth-set of p has a discrete boundary in the topological sense. We will say that a valuation is finite if all evaluated propositions are finite. The finiteness requirement (1) implies the universal interval validity of the following formula: (2) S ( ~ L ° ~ p v ~ L » . This latter condition corresponds to von Wright's requirement about the possibility of dissecting any interval on subintervals in which p is either true throughout or false throughout. 584 ALEXANDER BOCHMAN Now we will give corresponding finiteness requirements for interval-evaluated propositions. These requirements are less straightforward, but we may give for them a natural indirect description. For any interval valuation V(p,t) we may define the following 'two-indexed' instant valuation: V'(p,α,j8) s (3f)(α + t & β + t & V(p,t)). Note that if a Φ β then V'(p,a,β) ++ V(/?,[α0]). Now, for any fixed β, V'(p,a,β) is an ordinary instant valuation, and it is natural to require that all these valuations be finite. But

Concerted instant-interval temporal semantics. II. Temporal valuations and logics of change.