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<Paper uid="C94-2181">
  <Title>CONSEQUENCE RELATIONS IN DRT</Title>
  <Section position="4" start_page="0" end_page="1114" type="metho">
    <SectionTitle>
2. Overview of DRT
</SectionTitle>
    <Paragraph position="0"> In this section, we give a brief introduction to DRT. For a detailed exposition, the reader should consult Asher (1993). The basic idea of DRT is to formalize a dynamic representation of partial interpretations by means of classical models using a construction algorithm of discourse representation structures (DRSs).</Paragraph>
    <Paragraph position="1"> Observe that DRSs can be regarded as such abstract objects as partial models, mental representations, or (partial) possible worlds. But,  such identifications do not seem essential to the significance of DRT.</Paragraph>
    <Paragraph position="2"> '\]'he language of DRT is called Discourse Representation Language (DRL), which is like a standard quantifier-free first-order language except discourse referents and conditions. The logical symbols of DRL include =: (identity), --~ (comlitional), V (disjunction) and ~ (negation). A discourse representation (DR) K is expressed as a pair (UK, ConE), where UE is a set of discourse re\]erents, and Conic is a set of conditions. Each condition is either atomic or complex. Complex conditions are of the form: K1 :~ K2, KI V K2 or ~K1, where both K1 and K2 are Dl~s.</Paragraph>
    <Paragraph position="3"> A discourse representation structure (DRS) is a partially ordered set of DRs, which can be constructed by means of DRS construction rules whose application reflects the syntactic composition of the sentences in the discourse.</Paragraph>
    <Paragraph position="4"> When each DR of a DRS is maximal, the DRS is called a complete DRS. Intuitively speaking, each stage in the construction algorithm can be viewed as a partial possible worlds, in which more information resulting from the processing of a further bit of the discourse changes it into a more precise description of the world.</Paragraph>
    <Paragraph position="5"> A model for DRL is an ordered pair (DM, FM), where DM is the domain of M and FM is an interpretation function of constants and predicates. An embedding \]'unction for a DR K in a model M is a mapping from discourse referents in UK into the domain of M. An extension of an embedding flmction f for K in M to an embedding function g for K' in M is defined as g: (Dora(f) U UE, ) --~ DM. We write f C K g to mean that g extends an embedding function f to an embedding of K'. The notaion M ~-t,K C abbreviates that M satisfies C under ffor K. A proper embedding of K in M is an embeddhtg flmetion such that f ~K g and for any condition C in K, M ~g,E C. The notions of proper embedding and satisfaction can be extended to general cases by slmnltaneous recursion; see Asher (1993). A DR K is shown to be true in a model M iff there is a proper embedding of K in M. A DR K implies a DR K' iff every model in which K is true is also a model in which K' is true. This definition induces a consequence relation in DRT, but we have no reason to consider it as the only plausible for DRT. In fact, it is our job in tMs paper to seek alternate definitions.</Paragraph>
  </Section>
  <Section position="5" start_page="1114" end_page="1115" type="metho">
    <SectionTitle>
3. Consequence Relations and Sequent
</SectionTitle>
    <Paragraph position="0"> C alcull A partial semantics for classical logic is implicit in the so-called Beth tableaux. This insight can be generalized to study consequence relations in terms of Gentzen calculi. The first important work in this direction has been done by van Benthem (1986, 1988). We here try to apply this technique to DRT. Since our approach can replace the base logic of DRT by other interesting logics, we obtain alternative versions of DttT.</Paragraph>
    <Paragraph position="1"> Recall the basic tenet of Beth tableaux.</Paragraph>
    <Paragraph position="2"> Namely, Beth tableaux (also semantic tableaux) prove X --~ Y by constructing a counterexampie of X K: ~Y. In fact, Beth tableaux induce partial semantics in the sense that there may be counterexamples even if a branch remains open. Let X and Y be sets of formulas, and A and B be formulas. And we write X b Y to mean that Y is provable from X. Van Benthere's partial semantics for classical logic can be axiomatized by the Gentzen calculus, which has the axiom of the form: X, A P A, Y and the following sequent rules:</Paragraph>
    <Paragraph position="4"> Van Benthem's formulation can be extended for partial logics. Because such an extension  uses the notion of partial valuations, it is not difficult to recast the tzeatment for DRT. Let V be a partial valuation assigning 0, 1 to some atomic formula p. Now, we set V(p) = 1 for p on the left-hand side and V(p) = 0 for p on the right-hand side in an open branch of Beth tableaux. This construction can be easily accommodated to sequent calculi. Then, we can define the following two consequence relations:</Paragraph>
    <Paragraph position="6"> where Pre and Cons stand for premises (antecedent) and conclusion (succedent) of a seqnent, respectively. In a classical setting, (C1) and (C2) coincide. It is not, however, the case for partial logics.</Paragraph>
    <Paragraph position="7"> The Gentzen calculus G1 for C1 is obtainable from the above system without right (~)rule by introducing the following rules:</Paragraph>
    <Paragraph position="9"> Van Benthem (1986) showed that G1 is a Gentzen type axiomatization of C1. To guarantee a cut.free formulation, we need to modify van Benthem's original system. We denote by GC1 the sequent calculus for GC1, which contains the axioms of the form: (A1) A }- A and (A2) A, --~A ~-, with the right and left rules for (&amp;), (V), (~), (~ &amp;) and (~ V) together with (Weakening) and (Cut). It is shown that GC1 is equivalent to G1 without any difficulty. As a consequence, we have: Theorem 1 C1 can be axiomatized by GC1.</Paragraph>
    <Paragraph position="10"> The Gentzen system GC2 for C2 can be obtained from (GC1) by adding the next axiom:</Paragraph>
    <Paragraph position="12"> There are alternative ways to define consequence relations by means of sequent calculi. For example, it is possible to give the following alternate definitions.</Paragraph>
    <Paragraph position="14"> The new definition obviously induces inconsistent valuations. The Gentzen system GC3 is obtainable from GC1 by replacing (A2) by the following new axiom:</Paragraph>
    <Paragraph position="16"/>
  </Section>
  <Section position="6" start_page="1115" end_page="1116" type="metho">
    <SectionTitle>
4. Relation to Partial Logics
</SectionTitle>
    <Paragraph position="0"> In this section, we compare the proposed Gentzen systems with some existing partial logics, in particular, three-valued and four-valued logics in the literature; see Urquhart (1986). To make connections to partial logics clear, we extend DRL with weak negation &amp;quot;--&amp;quot; to express the lack of truth rather than verification of falsity in discourses. We denote the extended language by EDRL. In the presence of two kinds of negation, we can also define two kinds of implication as material implications. We need the next rules for weak negation:</Paragraph>
    <Paragraph position="2"> In fact, these rules provide a new consequence reation of EDRL denoted by ~EDRL. Our first result is concerned with the relationship of GC1 and Kleene's (1952) strong three-valued logic KL, namely Theorem 4 The consequence relations of GC1 and KL are equivalent.</Paragraph>
    <Paragraph position="3"> From this theorem, EDRL can be identified with the extended Kleene logic EKL. Let A -~,, B be an abbreviation of ~A V B. Then, we can also interpret Lukasiewicz's three-valued  logic L3. In fact, the Lukasiewicz huplication D can be defined as follows: A D B =a~t (A -~0 B) &amp; (~B-*,~ ~A) which implies t=EKL h D B iff A ~:EKL B and ~B ~EKL ~i.</Paragraph>
    <Paragraph position="4"> This is closely related to the consequence relation C3.</Paragraph>
    <Paragraph position="5"> Theorem 5 AFc, a B iff ~EKL A D B.</Paragraph>
    <Paragraph position="6"> If we drop (A2) from GC1, we have the sequent calculus GCI-, which is shown to be equivalent to Belnap's (1977) four-valued logic BEL.</Paragraph>
    <Paragraph position="8"> The four-valued logic BEL can handle both incomplete and inconsistent information. We believe that four-vaNed semantics is plausible as a basis for representational semantics like DRT, which should torelate inconsistent information in discourses. In view of these results, we can develop some versions of DRT which may correspond to current three-valued and four-vahed logics; see Akama (1994).</Paragraph>
  </Section>
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