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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="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4. LOGICAL CONNECTIONS
</SectionTitle>
    <Paragraph position="0"> The negation of a phase-set P1 is defined as follows:</Paragraph>
    <Paragraph position="2"> Note that (~R&gt; and N(R&gt; may be different because of non-equivalent phaseperspectives for ~R and R! For each two-place functor &amp;quot; u &amp;quot; (e. g. &amp;quot;Q&amp;quot; = &amp;quot;v&amp;quot;) we aegina PI a P2' if the sets PI and P2 are equal:</Paragraph>
    <Paragraph position="4"> the corresponding truth function (e. g. vel for &amp;quot; w,,). Obviously for every phase-operator P-O the expression P'O(PI'P2) &amp;quot;~ PI represents both a phase-set and a clear &amp;quot;tautology&amp;quot; - in other words - a phase-set that is &amp;quot;always true&amp;quot;, if PI is complete. Therefore, alt(P-0(PI,P2)-~P1 ) = U deg (where U deg is the phase-set that contains the time axis U as the only interval with the q-value T) reflects the double nature of the aforesaid implication.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5. TRUTH CONDITIONS
</SectionTitle>
    <Paragraph position="0"> The last considerations lead immedeately to the following definitions.</Paragraph>
    <Paragraph position="1"> The whole formalism requires two types of truth conditions, namely</Paragraph>
    <Paragraph position="3"> They have different status: (L) is used, if the phase-set P is considered as a temporal representation of something that is valid, independently of time. (M) is applied~if P is considered as something that represents a certain &amp;quot;time&amp;quot; (expressed by the phases of P). Because of the second possibility, alt appears not only in truth conditions, but it may constitute arguments in phase-operators etc., too. This will be shown in the examples below.</Paragraph>
    <Paragraph position="4"> Obviously one has for arbitrary</Paragraph>
    <Paragraph position="6"> By regarding the time axis U as a basic notion one has to take the trouble to consider the topology of U, and gets difficulties with closed and - 200 and open sets, environments etc.. This may be avoided by taking an axiomatic viewpoint: For all operations, relations etc. one formulates the essential properties needed and uses them without direct connection to the time axis. In this way U becomes a part of a model of the whole formalism. This is independent of the fact, that in definitions and explanations U may appear for making clear what is meant.</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
7 * TEMPORAL ADVERBI ALS
</SectionTitle>
    <Paragraph position="0"> In section 2. we have outlined, how propositions R are substituted by phase-sets (R&gt;. The same has to be done for temporal adverbials. First we consider definite adverbials: (tuesday&gt; is a phase-set P, where P~ is the set of all days (as spans p covering together the whole time axis U), and exactly the Tuesdays have the value q(p) = T. For (day&gt;~he set pm is the same, but it is q(p) = T for all p G Pm. (evening&gt; has as intervals suitable subintervals of the days with q(p) = T, whereas the remaining parts of the days form phases with q(p) = F in (evening&gt; . Analogously ( e~&gt; contains all years as spans p with q(p) = T, whereas (1986&gt; has the same spans, but exactly one with q(p) = T.</Paragraph>
    <Paragraph position="1"> Now we combine temporal adverbials with propositions. An e~ct representation would require that we list all possible structures of phrases, clauses etc. that express a certain combination. We use instead of this &amp;quot;standard paraphrases&amp;quot; as &amp;quot;a~ least on Tuesdays R&amp;quot;. If R is a certain proposition, e. g.</Paragraph>
    <Paragraph position="2"> R = John works in the library , then this paraphrase stands (as a remedy) for</Paragraph>
    <Paragraph position="4"> (1986 is a year, throughout which it is always true, tha~ every Tuesda~ is a day, when R occurs.) The second argument of PER is a phase-set defined by an air-operation. This phase-set has as T-phases exactly those maximal periods during which (8) holds.</Paragraph>
    <Paragraph position="5"> , PER((~ear~,...) selects the years that are covered by such a period, and the whole expression says that 1986 is such a year (and nothing about other years).</Paragraph>
    <Paragraph position="6"> The time of speech L is formally represented by a phase-set L deg with three phases, namely L itself with q(L) = T, and the two remaining infinite intervals with the q-value F. Then one may</Paragraph>
    <Paragraph position="8"> using the phase-operator ORD (cf. (I)) one introduces (yesterday) etc., and similarily (this year&gt; etc..</Paragraph>
    <Paragraph position="9">  (11) (in this year three times R</Paragraph>
    <Paragraph position="11"> In (11) a yes-no-decision is expressed (there are three T-phases of (R)in this year), but in (12) a &amp;quot;time&amp;quot; is defined, namely the three T-phases of (R&gt; in this year. Therefore~the truth conditions are different. The expression in (12) may appear as an argument in other expressions again.</Paragraph>
    <Paragraph position="12"> Now we apply the operation &amp;quot;choice&amp;quot;:  (13) (at most on Tuesdays three times R)</Paragraph>
    <Paragraph position="14"> OCC((R),x(day&gt;) determines the T-phases of (R) on a single day, KAR(...,3) keeps them iff there are exactly three (otherwise they become F-phases, cf. (I)), OCC(x(day},...) assigns to the single day the value Tiff the T-phases of (R) on this day have been preserved. Therefore, ~OCC(...,...) is a T-F-distribution over all days if x runs over all days, and the whole expression says</Paragraph>
    <Paragraph position="16"> These examples demonstrate the application of logical functors.</Paragraph>
    <Paragraph position="17"> As one oan see, the e~pressions render it possible to formulate even rather complex temporal relations in a comprehensible manner without much redundancy, the necessary arguments appear only once (or twice for certain quantifications as e. g. (tuesday) and (da~ in (8)). In order to handle durations, one needs another phase-operator EXT that is quite similar to KAR and ORD.</Paragraph>
    <Paragraph position="18"> The argument R stands either for &amp;quot;bare&amp;quot; propositions (without any temporal component) or for propositions with some temporal components. In the latter case the corresponding expression has to be substituted for (R):</Paragraph>
    <Paragraph position="20"> Similarily one obtains (qO) from (8).</Paragraph>
    <Paragraph position="21"> The truth condition in (8) causes that alt(...) occurs as argument in (qO).</Paragraph>
    <Paragraph position="22"> The sign &amp;quot;=&amp;quot; in the examples means that the left side is defined by the right side, the left side is stripped of one (or more) temporal components. In this sense (6), (8) and (9) are rules, (7) and (I0) include two rules in each case.</Paragraph>
    <Paragraph position="23"> The full and exact form of such rules requires more than the standard paraphrases, namely corresponding (syntactic) str~ctures on their left side.</Paragraph>
    <Paragraph position="24"> - 202 -</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
8. TENSES
</SectionTitle>
    <Paragraph position="0"> Till now nothing has been said about tenses. It is indeed possible to represent tenses in the formalism that we have outlined. But it is impossible to introduce &amp;quot;universal&amp;quot; rules for tenses. Even between closely related languages like English and German there are essential differences. So it does not make sense to explain here the details for the German tenses (of. Kunze 1987).</Paragraph>
    <Paragraph position="1"> The main points in describing tenses are these: At first one needs a distinction between &amp;quot;tense meanings&amp;quot; and &amp;quot;tense forms&amp;quot; (e. g. a Present-Perfectform may be used as Future Perfect). After that one has to introduce special conditions for special tense meanings (e. g. for perfect tenses in German and English, for the aorist in other languages). Further a characterization of tense meanings by a scheme like Reichenbach's is necessary, including the introduction of the time of speech L deg.</Paragraph>
    <Paragraph position="2"> On this basis rules for tense-assignment may be formulated expressing whioh tenses (= meanings) a phase xP or a phase-set P can be assigned to. From the formal point of view tenses then look like very general adverbials, and it is rather easy to explain how tenses and adverbials fit together. Tenseassignments create new expressions in addition to those used above. It is important that the position of the phases of (R&gt; does not depend on the tense R is used with: The tense selects some of these phases by phase-operators. So alt(NEX(xP,Ldeg)) * ~U deg is the basic condition for the actual Present (of. (G)).</Paragraph>
  </Section>
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