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<Paper uid="C88-2086">
  <Title>Solving Some Persistent Presupposition tProblems</Title>
  <Section position="3" start_page="0" end_page="422" type="metho">
    <SectionTitle>
2 General Background
</SectionTitle>
    <Paragraph position="0"> There has been a long history of attempts to define methods that would produce the presuppositions of a sentence. The default logic approach that is highlighted here follows the general framework set out in /Gazdar 1979/. One feature of this framework is that the speaker is governed by Grice's Principle of Cooperative Conversation.</Paragraph>
    <Paragraph position="1"> Assuming these general guidelines allows a competence model of the hearer's interpretation to generate the appropriate presuppositions of sentences with the forms 'a or b' and 'if a then b'. Details of this process is given later.</Paragraph>
    <Section position="1" start_page="0" end_page="420" type="sub_section">
      <SectionTitle>
2.1 Linguistic Presuppositions
</SectionTitle>
      <Paragraph position="0"> Being implied by a natural language sentence and the natural (or preferred) interpretation of its simple negation is the primary quality that qualifies an inference as a presupposition. This evaluation *This ~esearch was partially supported by NSERC grants A7642 (to It. Reiter) and A3039 (to P. C. Gilmore).</Paragraph>
      <Paragraph position="1">  of inferences is called the negation test. Presuppositions are generated from lexical and syntactic contexts. Those contexts which pass the negation test can be termed presuppositional environments. Sentences (1)-(2) demonstrate some prototypical examples of presuppositions produced by the presuppositional enviromnents, factive verbs and definitions of words. In each of these examples tim truth of the affirmative a-sentence always implies the truth of the c-sentence, and the truth of the negative b-sentence normally implies the truth of the c-sentence.</Paragraph>
      <Paragraph position="2">  (1) a.</Paragraph>
      <Paragraph position="3"> b.</Paragraph>
      <Paragraph position="4"> C.</Paragraph>
      <Paragraph position="5"> Mary is surprised that Fred left.</Paragraph>
      <Paragraph position="6"> Mary is not surprised that Fred left.</Paragraph>
      <Paragraph position="7"> Fred left.</Paragraph>
      <Paragraph position="8"> (2) a. My cousin is a bachelor.</Paragraph>
      <Paragraph position="9"> b. My cousin is not a bachelor.</Paragraph>
      <Paragraph position="10"> c. My cousin is a male adult.</Paragraph>
    </Section>
    <Section position="2" start_page="420" end_page="420" type="sub_section">
      <SectionTitle>
2.2 Projection Rule Procedures
</SectionTitle>
      <Paragraph position="0"> The procedures for deriving presuppositions of complex sentences prior to/Mercer 1987, 1988/that have received most attention are the ones based on the projection rule. These include/Karttunen 1973, 1974/~ /Karttuuen and Peters 1975~ \]979/~ /Gazdar 1979/, and /Soames 1979~ 1982/. The details of these theories are not important.</Paragraph>
      <Paragraph position="1"> What is of importance is the linguistic basis for these theories.</Paragraph>
      <Paragraph position="2"> Crucial to any theory, of natural language presuppositions is the concept of a presuppositional environment. These lexical or syntactic enviromnents generate inferences, which are called presuppositions, whether they are in the scope of a negation or not. In addition to tl~e concept of presuppositional environments~ what is common to all the linguistic theories is the notion of a projection rule which projects the generated inferences as presuppositions of the sentence. The naive projection rule proposed in /Langendoen and Savin 1971/takes all the presuppositions from all the presuppositional environments conrained in the sentence and projects them as presuppositions of the sentence.</Paragraph>
      <Paragraph position="3"> Although the modifications to this simplistic rule differ (see/Karttunes 1973, 1974/, /Karttunen and Peters 1975, 1979/, /Gazdar 1979/, and \]Soames 1979, 1982/), a common theme is that presuppositions are connected with surface phenomena. Although the methods differ in the importance that the semantic representation plays in the derivation of the presuppositions, without exception the potential presuppositions that are candidates for the (modified) projection rule are generated because the presuppositional environment exists in the surface form of the sentence;</Paragraph>
    </Section>
    <Section position="3" start_page="420" end_page="420" type="sub_section">
      <SectionTitle>
2.3 A Default Logic Approach
</SectionTitle>
      <Paragraph position="0"> The approach presented in /Mercer and Reiter 1982/ and /Mercer 1987~ 1988/has a number of distinguishing features.</Paragraph>
      <Paragraph position="1">  1. The method is based on inferencing in a logical system, although the logic is not a classical one.</Paragraph>
      <Paragraph position="2"> 2. The me~hod uses semantic representations of the naturM la.nguage sentence. Iu the case of 'if a then b' the semantic representation that is nsed directly is a derived representation (a D b can be derived from a &gt; b, where &gt; is StalnMC/er's connective for 3. All presuppositional environments that generate presuppositions  must be within the scope of a negatiou eil:lter in the represen.. taUon el' the sentence or some logical for:m derived from tlfis representation.</Paragraph>
      <Paragraph position="3"> ilow the method interact:~ with sentential adverbs is the main theme of this paper. The definition of presupposition and the working of the inference procedure in /Mercer 1987, 198'8/ solves the seeping problenrs caused by the interaction of negation and other environments. In the discussion of sententiM adverbs it will be shown that the normal ~mntence-seope for negation is circunwented in certain circamst;utc(:s. This circumvention of the normal rule explains the presuppositi, utal behaviour of the sentential adverb environment.</Paragraph>
    </Section>
    <Section position="4" start_page="420" end_page="420" type="sub_section">
      <SectionTitle>
2.3.1 Log.eal tEepresentatlon of
Pre.quppoMtions rising Default
R.ulos
</SectionTitle>
      <Paragraph position="0"> A no,'mal default ~*ule is a rule of inference denoted fl(Y) where a(Y) mM fl(x') are a.1\] first order formulae whose fl'ee variables are among those of .~ = xl,...,x,,. Intnitively, a default rule can be interpreted as: For all individuals xi .... ,xm, if the prerequisite t~(Y) is belie,rod I and if fl(,~') is consistent with what is believed, then the consequent fl(.~deg) nray be conjectured. A ~lormal default theory is a set of l\[rst order lbrnrulae together with a set of normal defaults. A fized point of a normal default theory is the deductive closure of tire set eomprised of the first order fornrulac and some maximal set of eonsequents that are consistent with the fixed point.</Paragraph>
      <Paragraph position="1"> The CONSIs'QUENTS'{D} is the set of all conseqaents of the default rules in the default thnory.</Paragraph>
      <Paragraph position="2"> For the purposes of this paper, I will change slightly the interpretation of the default rule to mean: if the speMC/er says 'a(~')' and fl(Y) is consistent with the heater's knowledge base, KBH, then the hearer can conjecture fl(,~;). it is not M)solutely clear what the verb says means (u' how it should be represented. For the purposes of this 1raper I only require those notions first presented in/Grice 1975/under the title Principle of Cooperative Conversation and formMized in /Gazdm&amp;quot; 1.979/. Under Ga:zdar's interpretation of Grice's maxims the speaker is e(,mmitted to the truth of u, the sentence that he utters.</Paragraph>
      <Paragraph position="3"> Therefore the speaker knows u. The conversational approach that I take views the contrihntion of a speaker's ntterance u as the addition of Ksu to 1(t111 along with other conversational intbrmation wldch is detailed in section 2.3.2. The meaning of the utterance is then a function of the inierencing process on I~Bl-1 U {I&lt;su }.</Paragraph>
      <Paragraph position="4"> The default rules require some extra informa*:ion to guard against misuse of the default rules. This information is a conjunct in the prerequisite of the default rule. Except for this technical aspect this extra intbrmation plays no role. Since it creates long default rules, i have tell it out of all the examples. For further details see/Mercer /.987/.</Paragraph>
      <Paragraph position="5"> Whenever the discussion concerns the default logic approach, I will assnme that the speaker's utterance has undergone the first phase of the interpretation process which generates a semantic representation (logical form) of the sentence uttered. This ,;entantic representation The verb believe sholdd be t~kea to lm?&amp;lt first order dctiw~ble or conjectured from the default theory.</Paragraph>
      <Paragraph position="6"> will be a well-formed sentence in a first order $4 modal language containing a countably intinite set of predicate syntbols, constant symbols, and variable symbols, plus the logical symbols A, V, D, -, Ks, ~nd Ps. The last two symbols, called modal operators, are to be interpreted as 'the speaker knows that' and 'for all the speaker knows, it is possible that', respectively. Although there is no general method known to generate this representation, some generM rules (:an be followed. Any sentence with an explicit negation is translated into the widely seeped negation of its affirmative counterpart. Any compound sentence is mapped clause by clause into a logical form, each clause being treated as a sentence.</Paragraph>
    </Section>
    <Section position="5" start_page="420" end_page="420" type="sub_section">
      <SectionTitle>
2.3.2 Deriving Presuppositions in
Complex Sentences
</SectionTitle>
      <Paragraph position="0"> The concept discussed herein -- using default logic to derive presuppositions -- is strongly influenced by Gazdar's method. I will present the representation of presuppositions in following sections with little explanation. For a complete discussion of how presul)postions are represented by default rules in a default theory together with how default logic proof theory captures Gazdar's idea of presuppositions being consistent with a context see/Mercer and Reiter 1982/or /Mercer 1987/. Another influence is the use of clausal implicatnres in connection with deriving presuppositions fl'om complex sentences, tu the default logic approach the clausal implicatures are used to control the division of the original theory into its first order cases.</Paragraph>
      <Paragraph position="1"> The clausal implicatures are derived fi'om the natural language sentence according to Gaz, dar's formal treatment of Grice's converss tional principles (/Griee 1975/). The sentence uttered by a speaker commits the speaker not only to the truth of the sentelme but also to the possibility of its clauses (its parts). So in tlm case of the speaker uttering 'A or B' or 'if A then B', unless tlmre is background knowledge or there are linguistic reasons to prevent it, the speaker b; committed to PsA, Ps-,A, PsB, aud Ps,'~B. These implicatures will provide the means to restrict the division of the theory representing the utterance into its cases.</Paragraph>
      <Paragraph position="2"> Becmtse default logic proof theory does )tot display any analogue to the law of the excluded middle (the antecedents of tim default: rules nrust be provable and there is no equivalent to the deduction theorem) and because presuppositions do arise from the clauses of complex sentences, some form of analysis by eases is required. Since a statement is provable in a case anMysis only if it is provable in all cases, the choice of cases is critical. As in the case of a tirst order theory, too few cases would allow inappropriate statenmuts to Ire proved. In addition because of the non-monotonic nature of default logic, having too many casts could prevent al)propriate statements being proved.</Paragraph>
      <Paragraph position="3"> In general the choice of cases must reflect two principles. Since the case analysis is a proof theoretic analogue of the model theoretic law of the excluded middle, each ease must completely determine the truth values of each of the disjunets found in the statement to which case analysis is being applied. Also, since the case analysis is justified solely on linguistic grounds (see /Mercer 1987/ for further discussion), the cases must reflect this linguistic situation. 3'o justify a. case, the possibility of the statement that distinguishes the case must 1re provable tY=om the original default theory. Since none of the modM statements take part in the proofs, they are left out of the cases. An example should clarify these ideas.</Paragraph>
    </Section>
    <Section position="6" start_page="420" end_page="421" type="sub_section">
      <SectionTitle>
Example
</SectionTitle>
      <Paragraph position="0"> Suppose the sentence 'A or B' is uttered. '\]?he default theory repre.senting this utterance would be</Paragraph>
      <Paragraph position="2"> where ax,..., a, represent the appropriate first order statements and $1,. *., ~n represent the appropriate default rules.</Paragraph>
      <Paragraph position="3"> Since A A-~B and -~A A B completely determine (that is, determine the truth values of both) A and B, and since the statements Ps(A ^ -~B) and Ps('~A A B) can be derived, A ^ -~B and -~A ^ B distinguish the two cases. Note that although PsA, Ps'~A, PsB, Ps'~B are all derivable, none of A, -~A, B, -~B are candidates for distinguishing a case because, individually, none of them completely determine the truth values of both A and B.</Paragraph>
      <Paragraph position="4"> Ilence the two cases of the original theory, T, are</Paragraph>
      <Paragraph position="6"> The simple negated sentence, an example of which is presented in section 2.3.1, is just a special instance of the case analysis procedure.</Paragraph>
      <Paragraph position="7"> In the simple negated sentence, -~X (which is represented as Ks-~X), the possibility of the only case (distinguished by -~X) can be proved using the utterance and the theorem ~- Ks~X F- Ps-~X.</Paragraph>
      <Paragraph position="8">  Definition 1 A sentence ex is a presupposition of an utterance u, represented by the default theories Auc~.~t ..... Aua .... 2, if and only /f A,c~ , ~-A a for all i and a e Th(CONSEQUENTS{D}), but Au ~/ o~ and Au \[/A .,~3.</Paragraph>
      <Paragraph position="9">  This definition can be loosely paraphrased as: ifa is in the logical closure of the default consequents and is provable from the utterance, and all proofs require the invocation of a default rule and in the case of multiple extension default theories, a is in all extensions, then a is a presupposition of the utterance.</Paragraph>
    </Section>
    <Section position="7" start_page="421" end_page="422" type="sub_section">
      <SectionTitle>
2.4 Important Differences
</SectionTitle>
      <Paragraph position="0"> The previous approaches which have been mentioned above rely on two ideas. Firstly, presuppositions are generated from positive and negative presuppositional environments, if these environments occur in the surface sentence. Secondly, a number of different methods, collectively called projection methods, are used to screen out those potential presuppositions which are not to be projected. A brief description of Soames' method is given in section 4.1.</Paragraph>
      <Paragraph position="1"> The default logic theory described in detail in/Mercer 1987, 1988/ approaches the problem of presupposition-generation from the level of logical representation. Presuppositions are generated from the logical representation if negated presuppositional environments occur in the logical representation of the natural language sentence or some logical form which can be derived from this representation. Malay of the results that the modified projection methods achieve are just proof theoretic results in the default logic approach to natural language ~For purposes of this definition, the only defaults in each Auco,o i a~e the presupposition generating defaults. In reality the default theory would contain many other kinds of defaults. The definition would have to be changed so that the proof of c~, requires the invocation of a presupposition generating default, and that a E Th(CONSEQUENTS{D'}), where D' is the set of presupposition generating defaults ~All of the examples presented in this paper deal with default theories having single extensions. In those theories which have multiple extensions, some way of stating that a presupposition is in all extensions is required. Since extensions of normal default theories are orthogonal, if An has multiple extensions then there exists a sentence fl such that Au I-A fl and Au V-a ~/?. I will call this situation being split along the fl-dimensiom If the extensions do not split along the ~-dimension then either ~ is in all extensions or a is in no extension. So if Au f-a ~ (which means that at least one extension contains ~) and Au V/a ~a (which means that no extension contains ~a, which means that the extensions do not split on the a-dimension) then ~ is in all extensions.</Paragraph>
      <Paragraph position="2">  presuppositions. In addition, once the logical representation of sentential adverbs is presented, it will be shown that the solution to the problem of presuppositions derived from sentential adverbs is again obtained in the default logic approach without any modifications.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="422" end_page="422" type="metho">
    <SectionTitle>
3 Sentential Adverbs
</SectionTitle>
    <Paragraph position="0"> The two sentential adverbs that will be presented are those found in the examples given in/Soames 1982/: 'too' and 'again'. Becanse one of the defining properties of a presuppositional environment is indio caring positive to the negation test 4, I will first look at each when there is a negation present. The interesting property displayed by sentential adverbs is that in addition to any interaction between negation and the underlying form, there is also an interaction between negation and the adverb. This interaction can be captured in two different logical representations.</Paragraph>
    <Paragraph position="1"> The sentential adverbs have the added complication that they can take any part of the sentence as their focus of the adverb. The focus of the adverb will be capitalized. Although the verb of the sentence can be focussed, a presentation of this particular focus would require an event-based representation. I do not discuss this focus in the following. However, it, too, would behave analogously.</Paragraph>
    <Section position="1" start_page="422" end_page="422" type="sub_section">
      <SectionTitle>
3.1 Too
</SectionTitle>
      <Paragraph position="0"> The representations of 'kick too' are shown in (3) and (4). These two representations convey the different foci of the adverb, 'too', the subject and the object of 'kick', respectively. I will be only interested in the representation which focuses on the subject, that is (3). The explanation for presuppositions that arise from the adverb focussing on the object is similar to the discussion presented below.</Paragraph>
      <Paragraph position="2"> Sentential adverbs have a most peculiar attribute when they interact with natural language negation. The adverb can be either inside or outside the scope of the negation. Sentences (5) and (6) point out the two possible interpretations in the case of 'too'. One particularly interesting phenomenon is that all of the possible scopes of the negation and the adverb may not occur in surface form. For instance,  (6) would normally be uttered as 'BILL didn't kick the ball, either.'. I will use the incorrect surface form in the examples, however. The italicized portions of the sentences indicate the portion which is in the scope of 'too'. (5) is to be interpreted as: Although someone else kicked the ball, Bill didn't. (6) is to be interpreted as: Both Bill and someone else did not kick the ball.</Paragraph>
      <Paragraph position="3"> (5) BILL didn't kick the ball, too.</Paragraph>
      <Paragraph position="4"> (6) BILL didn't kick the ball, too.</Paragraph>
      <Paragraph position="5">  The representations for the unnegated 'BILL kicked the ball, too.' and the sentences (5) and (6) are shown in (7)-(9), respectively. As proposed in /Kempson 1975/, /Wilson 1975/, and implemented in /Mercer 1987/, the representation of the simple negation of the sentence 'BILL kicked the ball, too.' is just the wide-scoped negation as shown in (8). I have shown the right-hand side equivalents of the appropriate representations so that I can contrast the two different uegatlons.</Paragraph>
      <Paragraph position="6">  (7) KICK(Bill, ball) A 3x.KICK(x, ball) A x ~ Bill 4A positive indication to the negation test means that a sentence, 8, containing the purported presuppositionai environment and the preferred interpretation of not S both have the same inferences arising from the environment in question. (8) -,\[KICIt( Bill, ball) A -lx.KIClf(x, ball) A x C/ Bill\] (9) -~KIG'l( ( Bill, ball) A Hx...~KICK (x, ball) A x ~ Bill What is important for tile presuppositional analysis is that only (8) can be a candidate for the negation test. One of the prerequisites of this test is that the supposed presuppositional environment is  within the scope of the logical negation in the logical representation of the sentence. The logical representation of (9) does not meet this requirement.</Paragraph>
    </Section>
    <Section position="2" start_page="422" end_page="422" type="sub_section">
      <SectionTitle>
3.2 Agai a
</SectionTitle>
      <Paragraph position="0"> The situation for the sententia |adverb, 'again', is somewhat similar to that described above for 'too'. The adverb can be inside or outside the scope of the negation. Accordingty~ the adverbs found in (10) and (11) are the presupl)ositional and non-presuppositional enviromnents with respect to the positive sentence 'Fred called again,'. (10) is to be interpreted as: At some time in the past Fred called and during some interwl of time which is important to the context in which the sentence is uttered, Fred didn't call. (11) is to be the following interpretation: At some time in the past Fred didn't call and during some interval of time which is important to the context in which tile</Paragraph>
      <Paragraph position="2"> The representations for the unuegated 'FRED called again.' and the sentences (1(t) and (11) are shown in (13)-(15), respectively. As in the case of 'too', the representation of the simple negation of the sentence 'Fb~ED called again.' is just the wide-scoped negation as shown in (1+1). i have shown the right~hand side equiwdeuts of the appropriate representations so that I can contrast the two difihrent  negations.</Paragraph>
      <Paragraph position="3"> (13) CALL(Fred, you, t) A :3tj .CALL(16~d, you, h) A tl &lt; t (1.4) -,\[ CAL t,(1,Yed, you, t) A i~tt. CAI, L( Ft~d, you, 11) A t 1 &lt; t\] (15) -,CALL( I;Yed, you, t) h ~tl.'-,CALL( Fred, you, h) A tl &lt; t  As in the case for 'too', the only representation of 'again' that sanctions the use of presuppositioual machinery is (14) s.</Paragraph>
    </Section>
  </Section>
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