Temporal Relations in Texts and Time Logical Interferences

A calculus is presented which allows an efficient treatment of the following components: Tenses, temporal conjunctions, temporal adverbials (of "definite" type), temporal quantifications and phases. The phases are a means for structuring the set of time-points t where a certain proposition is valid. For one proposition there may exist several "phase"-perspectives. The calculus has integrative properties, i. e. all five components are represented by the same formal means. This renders possible a rather easy combination of all informations and conditions coming from the aforesaid components. I. Prerequisits We assume that propositions are replaced by phase-sets: A proposition R is something which is true or false at each point t of the time axis U: Value(R,t) = T(rue) or F(alse). A phase is an interval (span or moment) p on the time axis, which a truth value (denoted by q(p)) is assigned to: q(p) = T: p is an affirmative (main-)phase. q(p) = F: p is a declining (secondary) phase. A phase-set P is a pair [P~,q]: P~ is a set of intervals p and q is an evaluation function assigning a truth value to each p6 P~. The substitution of propositions R by phase-sets P is not unequivocal, but also not arbitrary. Some necessary conditions for the relationship between R and its "surrogate" P have been introduced and discussed elsewhere (Kunze 1986). One essential point is that the simple "moment logic" becomes an "interval logic". This is also connected with questions as expressed by the different definitions of HOLD, OCCUR and OCCURING in Allen 1984. Another fact connected with phases is the unsymmetry in the case of a negation: (I) The museum is open today. + The museum is open all day today. (2) The museum is closed today. = The museum is closed all day today. The proposition R is supposed to be fixed and given. P is considered as variable and provides a formal counterpart of different phase-perspectives for a certain proposition. The German sentence (3) Thomas raucht. has at least two of them (and consequently two meanings): "Thomas is a smoker" and "Thomas is smoking". Furthermore the use of phases enables us to consider some parts of T(R) as unimportant, accidental or exceptional. These parts form declining phases of R. The affirmative phases of R need not be disjunct, and they need not be contained in T(R). It is also possible to introduce nested phases, so that rather complicated courses may be rcpresented. 2. Some formal definitions • = p~ Let PI [ i' qi ] (i : 1,2) be two phasesets with P~ : P~. Then PI and P2 may be connected by means of sentential logic: For any functor "o" (e. g. "... and ..." and "if , then . ") . . . . . . . one defines P~ o P2 D~f [P~, ql o q21 with P~ : p~ : p~. Phase-operators connect arbitrary phasesets. As an example we take the phaseoperator OCC: [P~q] = P : OCC(PI,P 2) means P~ = PI and I T, if q1(p) = T and there is a p26P2 q(p)= with q2(P2 ) = T and p ~ P2 % @' F otherwise. " " " ~ P2" If one replaces pnp2 # ~ by p = , one gets the definition of P = PER(PI,P2) . P = OCC(PI,P 2) means "P2 happens during PI"' P = PER(PI'P2) "P2 happens throughout PI" The important point is that these relationships between PI and P2 are not represented by a Yes-No-decision, but again by a phaseset P: OCC(PI,P2) selects from the T-phases of the first argument those p for which the characteristic condition (= there is a P2