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<Paper uid="E89-1027">
  <Title>pect and Events within the setting of an Improved Tense Logic. In: Studies in Formal Semantics (North-Holland</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2. PHASE-SETS AND PROPOSITIONS
</SectionTitle>
    <Paragraph position="0"> A phase p is an interval (either unbounded or a span or a moment) which a truth value (denoted by q(p)) is assigned to: q(p) = T : p is considered as an affirmative phase.</Paragraph>
    <Paragraph position="1"> q(p) * F : p is considered as a denying phase.</Paragraph>
    <Paragraph position="2"> The intervals are subsets of the time axis U (and never empty!).</Paragraph>
    <Paragraph position="3"> A phase-set P is a pair tP',q3, where Pa is a set of intervals, and q (the evalnation function) assigns a truth value to each pe Pa. P has to fulfil the following consistency demand: (A) For all p,,p&amp;quot;aP&amp;quot; holds: If p'n p&amp;quot; @ ~, then q(p') * q(P&amp;quot;). A phase-set P is called complete, iff the union of all phases of P covers U. Propositions R are replaced by complete phase-sets that express the &amp;quot;structured&amp;quot; validity of R on the time axis U. Such a phase-set, denoted by (R&gt;, has to be understood as a possible temporal perspective of R. There are propositions that differ from each other in this perspective only: Por (I) John sleeps in the dinin~ room~ one has several such perspectives: He is sleeping there, he sleeps there because the bedroom is painted (for some days), he sleeps always there. SO the phases of (R&gt; are quite different, even with clear syntactic consequences for the underlying verb, The local adverbial may not be omitted in the second and third easel ~ I skip here completely the following  problems: - A more sophisticated application of nested phase-sets for the representation of discontinuous phases in (R&gt;; - the motivation of phases (e. g. according to Vendler (1967)) and their adequacy. null 3. PHASE-0PERATORS  A phase-operator is a mapping with phase-sets as arguments and values.</Paragraph>
    <Paragraph position="4"> There are phase-operators with one and with two arguments. A two-place phase-operator P-O(PI,P 2) is characterized by the following properties: (B) If P = P-O(PI,P2), then P&amp;quot; * P~, i. e. the set of intervals of the resulting phase-set is the same as of the first argument; (C) For each phase-operator there is a characteristic condition that says how q(p) is defined by q1(p) and P2 for all p PS P~. This condition implies always that q(p) = F follows from q1(p) = F.</Paragraph>
    <Paragraph position="5"> SO the effect of applying P-O(PI,P 2) is that some T-phases of PI change their truth value, new phases are not created. The characteristic conditions are based on %wo-place relations between intervals. Let rel( , ) be such a relation. Then we define (by means of tel) q(p) according to the following scheme:</Paragraph>
    <Paragraph position="7"> We will use three phase-opera~ors and define their @v~uation functions in the following way by (D)s  (E) P = 0CC(P1,P2): rel(P2,p) is the relation &amp;quot;P2 and p overlap&amp;quot;, i. e. P2nP ~ ~. (F) P = PER(P1,P2)s rel(P2,p) is the relation &amp;quot;P2 contains p&amp;quot;, i. e. P2 ~ p&amp;quot; (G) P = NEX(PI,P2)s rel(P2,p) is the relation &amp;quot;P2 and p are not seperated from  each other&amp;quot;, i. e. P2uP is an interval.</Paragraph>
    <Paragraph position="8"> As an illustration we consider some examples. Needless to say~ that their exact represention requires further formal equipment we have not introduced yet. Typical cases for 0CC and PER ares  (2) yesterda~ was bad weather. Overlapping of (yesterday&gt; and a T-phase of (bad weather&gt;.</Paragraph>
    <Paragraph position="9"> (3) John worked the whole evening.  A T-phase of ( evening&gt; is contained in a T-phase of (John works&gt;. (for (evening&gt;, (yesterday&gt; cf. 7.) There is only a slight difference between the characteristic conditions for 0CC and NEX: NEX admits additionally only b~EETS(P2,p) and ~LEETS(p,p2) in the sense of Allen (1984). Later ! will motivate that NEX is the appropriate phase-operator for the conjunction when. Therefore, sentences of the form (~) R1, when R 2. (cf. (N), (0)) will be represented by an expression that contains NEX((R2&gt; , (Rfl&gt;) as core. The interpretation is thaC nothing happens between a certain T-phase of (RI&gt; an~ a certain T-phase of (R2~ (if they do not overlap).</Paragraph>
    <Paragraph position="10"> The next operation we are going to define is a one-place phase-operator with indeterminate character. It may be called &amp;quot;choice&amp;quot; or &amp;quot;singling out&amp;quot; and will be denoted by xP~, where P1 = \[~,qd\] is again an arbitrary phase-sets</Paragraph>
    <Paragraph position="12"> If we need different choices, we write xP1' YPI' zP2' ..., using the first sign as an index in the mathematical sense.</Paragraph>
    <Paragraph position="13"> Moreover~we define one-place phase-operators with parameters:</Paragraph>
    <Paragraph position="15"> integers g: ORD(PI,g) assigns the value T exactly to the g-th T-phase of PI' if there is one, with certain arrangements for g (e. g. how to express &amp;quot;t~e last but second&amp;quot; etc.) Finally we define the &amp;quot;alternation&amp;quot; alt(P I) of an arbitrary phase-set P1 = \[l~1,ql\]. By alternation new phases may be create~s alt(P1) contains exactly those phases which one gets by joining all phases of P1 that are not seperated ~o - 199 from each other and have the same value ql(Pl). So the intervals of alt(P I) are unions of intervals of PI' the q-values are the common ql-values of their parts (of. (A)). It is always alt(alt(P1)) = alt(P1) , and alt(P I) is complete, if PI is complete. Going from left to right on the time axis U, one has an alternating succession of phases in alt(P1) with respect to the q-values.</Paragraph>
    <Paragraph position="16"> alt(P I) is the &amp;quot;maximal levelling&amp;quot; of the phase-set PI&amp;quot;</Paragraph>
  </Section>
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