Document 185699

From: Proceedings, Fourth Bar Ilan Symposium on Foundations of Artificial Intelligence. Copyright © 1995, AAAI ( All rights reserved.
How To Derive
Michael Morreau
Philosophy Department & UMIACS,
University of Marylandat CollegePark,
College Park, MD20742-7615,
BelowI will analyze defensible inferences underlying the interpretation of conversational implicatures
and presuppositions; a nonmonotonicconsequencenotion will lead from premises including pragmatic
generalizations to conclusionsabout howthose sentencesare best interpreted..
Suppose you have said something in mypresence. Nowgiven our commonknowledgeof the norms of
language use and given reasonable assumptionsof mine about your dispositions and cognitive states
given all this, what amI entitled to supposeyou have conveyed?Andwhat kind of reasoning allows meto
derive what is conveyedfrom the fact of your utterances? Theseare the questions to which,borrowingfrom
the philosophyof languageand the Artificial Intelligence literature of nonmonotonic
reasoning, I will offer
an incomplete answer.
Utterances conveymeaningsbeyondtheir conventional content. I will use the term conveyedmeaning
in a sense whichsubsumesthe conventional consequencesof utterances, their conversational implicatures
and their "accommodated"
presuppositions. Their metaphoricimplications and morecould be included too,
but I won’t get to that here; I will focus on what is conveyed through the "accommodation"of
presuppositions (in particular the existential presuppositions of definite descriptions) and on what
conveyed by conversational implicatures (in particular the quantity presuppositions of indicative
conditionals). I will look too at what happens when these two kinds of conveyedmeanings comeinto
The following examples, which will be analyzed in detail, illustrate someof these presuppositions,
implicatures and conflicts.
Supposefor reasons of your ownyou say out of the blue, "the cat is not on the mat." And,for the
meantime,you say nothing else. Thenyou convey, amongother things, that there is a cat, and that there
is a mat. The "accommodation"of the existential presuppositions of definite descriptions has been
supposedto play a part in this; here I will supposesimply that speakers of English knowthat uttering a
definite description or another "presuppositional trigger" typically has the effect of conveying the
presupposition, without going into howand whyit has this effect.
This is not a typical case, though, since you continue with the explanation: "there isn’t a cat." Now
your two utterances taken together will still conveythat there is a mat, but no longer that there is a cat.
This illustrates the defensibility of pragmaticreasoning. In the analysis sketchedhere defensibility emerges
from the nonmonotonicreasoning which, I claim, links premises including pragmatic generalizations and
the fact that certain sentenceshave beenuttered to conclusionsabout what they convey.
Suppose, to take another kind of example, you utter a conditional in the indicative mood:"if my
chequehas arrived then I’ll pay you backtoday". I will again be able to drawdefensible conclusionsabout
whatthis conveys:typically, that youare in a position to assert neither the consequent,"I’ll pay youback
today", nor the antecedent, "mychequehas arrived" (whichtogether with the conditional itself entails the
consequent).Suchinferences are standardly treated as conversationalimplicatures of Grice’s conversational
maximof quantity and that is howI will treat themhere, too.
¯ Researchsupportedin part by the NSFand ARL.
Interestingly, in cases wherethere is a conflict, the quantity implicatures of indicative conditionals
override the existential presuppositionsof definite descriptions occurring in them. If yousay "the cat is on
the mat if she’s in the house"there is no such conflict and your utterance conveysboth that there is a cat
and that youare unsurewhethershe is inside. If on the other handyousay "the cat is on the mat, if there is
a cat" there is a conflict betweenthe existential presupposition,that there is a cat, and the conversational
implicature, that you are unsure whetherthere is a cat, since youcan hardly he supposedto conveyboth of
these things. It seems that in cases like this the conversational implicamrewins out: you conveyonly
that youare unsure whetheror not {hereis a cat.
Theseare the only examplesI will consider here. In analyzing themI have tried to keep in mindmany,
manyother cases which a successful theory must account for too. Dealing with those other exampleswill
engage the cogs and wheels whichhere sometimesturn without doing muchuseful work.
WhatKnowledge Underlies Interpretation?
The derivation of conveyed meanings is supported by knowledge some of which is broadly speaking
linguistic, thoughmuchof it is not. In order to express {his knowledge
precisely it is helpful to introduce,
besides the natural languageof the utterances, two formal languages. I assumea natural languagef-N, a
formal languagef-LF (here a languageof first-order modallogic) in whichto write the logical formsof the
expressions of I.N and a language £ in which to formalize the pragmatics of LN. I assume £LF is a
sublanguageof L. To express what we knowabout the semantics and pragmatics of f.~ r. contains terms
whichrefer to expressions in those other two languages:for any English sentence o, "o" is an individual
term of L. Andfor every £.~ sentence q~, "q~" is an individual term of L, (Goedelnumberingis one wayof
introducingsuch terms.) Since £-tF is a sublanguageof f., this latter stipulation meansthat Lhasindividual
terms whichmustbe interpreted as (someof) its ownformulas; as far as I can see at the moment
{his does
not introduce paradoxes.
A possible-worlds modelfor L has in its domain~7 the sentences of ~ and those of LL~along with
somespeakers of f.~ and somethings for themto talk about. (Weneeda ca,., and a mat.) Talk about these
things is captured in relations on ~ and betweenpossible worlds whichare suitable for interpreting, in
addition to the vocabulary of ~ the following relation symbolsand modaloperators. For each speaker s
there is a monadicpredicate Utts; for any sentenceo of English, the intendedinterpretation of UUs"O"
is that
s has uttered (sometoken of) (Y. Thereis a binary relation If between(namesof) English sentence types
(namesof) their logical forms, lfCo","cp") expressesthat "¢p" is the logical formof "o". Thereis a binary
relation <i between (namesof) logical forms; "q~"<i"W"means, informally speaking, that an utterance
whoselogical form is "¥" wouldbe moreinformative than one with logical form "q~". For each speaker s
and sentence q} of ~ I assumemodaloperators cons and bels; cons(~) meansthat the speaker has conveyed
q}; belscP, that the speakerbelievescp. I assumealsoa modaloperatortel ; re/q~ is intendedto expressthat q}
lis relevant to the dialogue currently underway,
Finally I assumethat L contains a weakmodalconditional ¯ with which to express generalizations
about howspeakers normallyuse languageto conveymeanings.The intended interpretation of q~>~is that
if ¢p, then normally¥. This conditional is interpreted in the mannerOf Stainaker [1968] or Lewis[1973]
using the device of possible-worldsselection functions. Its umthconditions are completelystandard except
for the fact that the modalconstraint centering (Cheilas [1980] calls it rap) is not imposedon worldsselection functions. Suchweakconditionals have been used to express pragmaticgeneralizations before, by
Lascarides and Asher [1993] for example. What I will say about presuppositions and conversational
implicatures dovetails with and complements
their account of the interpretation of temporal and rhetorical
relations left implicit in texts.
I can nowformalize representative examplesof pragmaticgeneralizations and other information which
enables the derivation of conveyedmeanings.First, I assumethat the interpreter of £ (whichI will now
assumeto be English) knowsthe conventional content of its sentences. That is, I assumethe interpreter
knowscertain facts about the logical formsof English sentences. Representativeexamplesare
lOf coursetel oughtto be givenfurther analysis, perhapsin termsof counterfactualsaboutthe interpreter’s epistemicstate
and the goals of the dialogue. Roughlyspeaking,a proposition is relevant at a given point in a dialogue if the hearer does
not already believe it andif the goals of the dialoguewouldbe furthered wereit to be conveyed
to him. I can’t go into this
That such knowledgeshould be represented in the way I have just suggested is very simplistic in several
ways; for one thing, as linguists since Chomsky
have madeclear, the fact that we can understandsentences
we’ve never heard before makes in inconceivable that such knowledgeis not derived somehowfrom
underlying knowledgeof syntax. For another, anaphoric relations betweendifferent utterances makethe
notion that logical forms are formulasof my£tFexceedinglyimplausible; alternative representations which
would be better include Kamp’sDiscourse Representation Structures, presented in Kamp&Reyle
This having been said, these oversimplifications do not amountto objections to the approach I am
suggestinghere, so muchas directions in whichthere is every reason to believe it can be extendedin the
Concerningthe generalizations underlying interpretation, someof themare not linguistic at all but
concern the cognitive states of the participants in a dialogue. To begin with, our beliefs tend not to be
That we should knowany such thing is implausible and again an oversimplification is responsible. What
is required for the pragmaticreasoningformalized belowis that speakers typically do not simultaneously
entertain two beliefs of which they are aware, and which are obviously incompatible. Suchan assumption
is a lot moreplausible but it is indistinguishable fromthe generalization stated aboveif belief is modelled
in the simplistic wayI’ve chosenhere, as a (nonalethic) modaloperator along the lines of . Recentwork
in Artificial Intelligence on the notion of awarenesswill hopefully give rise to a plausible alternative to
Other generalizations are pragmatic insofar as they derive from the assumption that speakers are
governedby Gricean maximsof conversation (and a few others whichI will get to). A principle related
Grice’s maximof quality, firstly, requires that we conveyonly what we believe to be true. A speaker who
accepts this maximwill normally conveyonly propositions whichhe believes:
cons~P> bels~P.
Exceptionsto this generalization include lies and other utterances whichconveythings the speaker fails to
believe -- whichhe believes, in the case of lies, to be false. Not all such utterances are insincere though,
or intended to mislead; they include guesses, for example.In fact this principle is stronger than Grice’s
maximof quality, whichrequired only that speakersavoid utterances they take to be false.
Grice’s maximof relevancerequires us to convey9nly what we take to be relevant:
consq~ > belsrel ¢p.
Exceptionsto the rule that conveyedmeaningstypically are taken to be relevant include cases in whicha
speaker movesthe conversational goals. Thus for examplewhile waiting for a taxi one might interrupt a
trivial converationabout someor other topic by announcingthat the taxi has arrived. (This examplemakes
clear that the notion of relevancewhichI intend is a highly context-sensitiveone: of coursethe fact that the
taxi has arrived is relevant in a broadsense; that is whyit is not considered rude to interrupt with the
announcement.But in general it is not relevara to the ongoingconversation, whichmight as well be about
gardening.Noneof this context sensitivity is present in myanalysis, though, so here too there is scopefor
developmentof the notion of a conversational goal.)
Finally Grice’s maximof quantity requires speakers to be as informative as possible, within limits
imposedby quality and relevance. I formalize this by supposingthat a speaker normallywill prefer a less
informative to a more informative utterance only if he cannot assert the more informative thing, either
becausehe does not believe it, or becausehe does not think it relevant.
conscp ^bels("qPci’~) -, (bels¥ ^ belsrel ¥)
Otherpragmaticgeneralizations whichas far as I can see do not derive from broadly Griceanmaximsinclude
the law that utterances tend to conveytheir conventionalcontent:
Utts"Ct’ Mf(’~’O) > const
Exceptionsto this generalization include utterances whichare sarcastic, polite, ironic, metaphoricalor
Finally, the following principle expresses the notion that anything which typically holds whena
sentenceo is uttered (underany given conditions 7,,) is typically conveyedby such an utterance, provided
is relevant:
2~ is a binary connectiverepresenting the indicative conditional; THEa definite-description operator whichturns a monadic
predicate into an individual term.
I expect that somesuch principle is responsible for the sense of paradox engenderedby G.E. Moore’s
famousutterance: "the cat is on the mat, and I don’t believe it". By conventionalcontent and quality the
first part of this utterance, "the cat is on the mat", is normallyuttered in a context in whichthe speaker
believes that the cat is on the mat. Fromthis and the fact that "the cat is on the mat" has in fact been
uttered will follow, with signification, that the speaker in uttering these wordsnormallyconveysnot only
that the cat is on the mat, but also that he believes this to be so. Nowthis latter thing of course directly
contradictsthe conventionalcontentof the secondpart of the utterance, "...I don’t believeit [i.e. that the cat
is on the mat]".
In this illustration (whichcan he given a rigourons treatment once a notion of nonmonotonic
using > has been introduced), signification was called on to conveycertain propositions about the mental
state of the speaker. But I havestated the principle moregenerally than this; it asserts that any relevant
meaningq~ is conveyedby an utterance evenif it is associated with that utteranceL, say, by laws of nature,
and does not arise fromthe conventionalmeaningof the sentences uttered and pragmaticlaws. In fact I will
makeuse of this schemeonly with q~ instantiated as a sentence of the form bels¥ or of the form ~bels¥.
It seemsto methat this principle is plausible for other propositionsq~ too, though,especially propositions
of whichthe speaker can be expectedto be awareof, so I’ve stated the principle as generally as possible.
Whetherit is too permissivehas to be seen.
Finally I assumethat merely uttering certain lexical items, including "presuppositionai triggers"
contributes to conveyedmeaning. For just one example, the following generalization concerns definite
descriptionswhich,I suggest, tend to conveythat there are things fitting those descriptions:
uttsC...~e IZ..’) > cons3xII
Here H is a predicate expression of English but doubles as its formal representation. Similar
generalizations can he stated for other presuppositionaltriggers listed by Levinson[1983]: factive verbs,
implicative verbs, changeof state verbs, iteratives, verbs of judging, temporalclauses, cleft sentences,
comparisons,contrasts and all of the rest.
The pragmaticand other generalization schemesjust described makeup a backgroundtheory against which,
in the following, ,.he interpretation of a speaker’s utterances will proceed. Call the set of all of their
instantiations BT,and let a be someor other sentence of English. In the followingsection I will introduce
a nonmonotonic
consequencenotion by meansof whichit will be possible to derive, from BTand utterance
facts of the form Utts("~"), conclusions of the form consTO.Meanings9 such that cons9 can be derived
fromBTand Utts(’W’)are the meaningswhichare conveyedby the speaker’s uttering
Nonmonotonic Reasoning
NowI introduce a notion of nonmonotonic
inference whichenables the consequentsof these conditionals to
be detachedin cases wherethis does not introduceinconsistency.
Conditional logic has been used before as the basis for nonmonotonicreasoning by, amongothers,
Delgrande [1988], Asher&Morreau
[1991], and Boutifier [1992]. The form of inference which I will use
here is different from all of these; it does not face the problemwith "irrelevant" premiseswhichDelgrande
and Botilier ran up against and it is from a conceptual and technical point of view not as dismayingly
complicated as the notion of "commonsense
entailment" introduced by Asher and Morrean. In fact, as a
later theoremwill show, the inference notion amountsto a trivial fragmentof (a slight generalization of)
Reiter’s Default Logic(the defaults are understoodto he defined L, not on classical logic as they are in
Reiter’s original [1980]presentation.)
The informal idea is that premisesincluding BTand UttsC¢7")can he strengthenedby adding, as default
assumptions, as manyinstances of modusponens, q~>¥---* q~-~¥, as is possible without introducing
inconsistency. This last schemeI will abbreviate as q~>/---~¥. I will also consider examplesin whichthe
default assumptions are of the form (q~>¥ ^ ¥>X) "-~ (¢P>Z)- The effect of adding these default
assumptions is that defeasible forms of the argumentforms modusponens and hypothetical syllogism
becomeavailable (neither argumentschemeis logically valid):
Onebasic conceptis the focal vector, whichis just a vector of (sets of) default assumptionsrelative
which nonmonotonic
inference will be defined. It plays very muchthe same role here as Reiter’s sets of
defaults do, in DefaultLogic:
,~= <~1, ~2, ....~> is a focal vector just in case each gi is a set of sentences.
Afocal vector should be thought of as a stock of potential assumptions:assumptionswhichcan be addedto
what is knownin any given case where an utterance is to be interpreted. These assumptions comewith
different priorities: assumptionsin Gi havehigher priority than those in ffj, if i<j. In cases of conflict
betweenpotential assumptions,whatwill be assumedand whatwill not is determinedby this priority order,
since an assumptionof lower priority will never be madeif it conflicts with an assumption of higher
priority whichcould have been madein its place. Assumptionsin ~1 will take precedencein conflicts with
those in ~2, if3,...; assumptionsin G2will take precedencein conflicts with those in if3, ~4 .... and so on.
Below, .~ will be a vector <~1, g2, ... (k> of (sets of) instances modus ponens whose eff ect, when
assumed,is to enable conclusionsto be drawnabout what is conveyedin actual utterance situations.
An example illustrates assumption-makingin which higher-priority assumptions concerning the
conventional content of an utterance override lower-priority assumptionsbased on the speaker’s choice of
definite-description syntax. Considerthe followingtwo utterance situations:
(I) A speakersays "the cat is not on the mat"(and nothingelse).
(11) A speakersays "the cat does not exist" (andnothingelse).
In the first case the speaker conveysthat there is a cat, whichis not on the mat. In the secondcase the
speaker conveysthat there isn’t a cat. Let ~ be the vector <ffl,~2>, of which~1 contains just the following
five instances of modusponens (all can be gotten from BTin a way which will becomeclear as we go).
Addingthe assumptionsin .~ to the utterance facts as describedin (I) and 01) will, relative to BT,enable
interpreter to derive appropriate conclusions about what the speaker has conveyed in each of these
First, ~1 contains T>l...>--~bels.L Relative to a backgroundtheory including CONSISTENCY
to assumethis sentence is of course simplyto assume~bels.L, whichexpresses that the speaker’s beliefs arc
Second,gl contains the following two sentences:
cons(---~xcatx) >1...) bels(--~xcat x)
cons(3xcatx) >1...) bels(3xcat x).
Relative to a backgroundtheory including QUALITY,
and in a context in which the speaker has conveyed
that there is (not) a cat, to assumethese sentences is simply to assumethat the speaker believes what
has conveyed.
Finally, if! contains the following two instances of modusponens which, so to speak, cash out the
generalization CONVENTIONAL
The first is
Utts"the cat is not on the mat"^
/.if"the cat is not on the mat", "--on(THEcat,THEmat)"
This sentence expresses the following thing: if an utterance of "the cat is not on the mat" normally
conveysthe conventionalcontent of this sentence, namelythat the cat is not on the mat, and if in addition
this sentence has been uttered, then the speaker has conveyedthat the cat is not on the mat. The other
sentence in ~1 expresses the analogous thing about the sentence "the cat does not exist" and its
conventional content:
~2 is chosen to include sentences which cash out the generalization DEFINITE
in muchthe
sameway. (It is significant that they appear in ~2 instead of GI. This is howthe interpretation process
incorporates an empirical hypothesis whichI nowput to you: that assumptionsderiving from the speaker’s
choice of syntax have lower priority than assumptionsderiving from his choice of conventional content,
assumptionsabout the consistencyof the speaker’sbeliefs, and so on.) Thefirst sentencein g2 is,
notonlterr~’ >1.., cons3~2vtx.
It expressesthe followingthing: if an utterance of "the cat is not on the mat"normallyconveysthat there
is a cat, and if in addition this sentencehas in fact beenuttered, then the speakerhas in fact conveyedthat
there is a cat.
The second sentence in f2 does the samething for the sentence "the cat does not exist". This is an
interesting case because whenthis sentence is uttered its conventional content and the use of definitedescriptionsyntaxtend to pull the interpreter of the utterancein different directions:
uns’~he~ d~ ~e~sf’ >/_, consular
This sentenceexpressesthat if an utterance of "the cat does not exist" normallyconveysthat there is a cat,
and if in addition this sentencehas in fact beenuttered, then the speakerhas in fact conveyedthat there is a
cat. Wewill see that because this assumption has lower priority than the sentence dealing with the
conventionalcontent of "the cat does not exist", the effect of uttering this sentenceis simplyto conveythat
there is no caL
I amunhappythat the procedureI’ve just followedin deriving defaults fromBT(this procedurewill be
stated explicity below)should lead to the inclusion in our focal vector of an assumptionas implausible as
this one! Its presence doesn’t seem to do muchdamageinsofar as it does not lead us to derive
counterintuitive conveyed meanings, but it is a blight on myaccount. The problem seems to be that
generalizations should not be treated as I havetreated them, as schemesor universally quantified sentences.
It is true I think that a definite description normallyis uttered in contexts in whichthe existence is
conveyedof a uniquesalient thing satisfying the description. But this should not entail that, say, "the cat
does not exist" is normallyuttered in contexts in whichthe existence is conveyedof a uniquesalient thing
satisfying the description (a cat in this case). This question and its answerare discussed in a little more
detail in mypaper "AllowedArguments",whichappears in this IJCAI.
Nowthe picture of the derivation of conveyedmeaningswhichl will develophere is in outline as follows.
Takesituation (I) above: someoneutters "the cat is not on the mat" in the presenceof an interpreter. The
interpreter observesthis utterance and his observationleads himto adopt, as a premiseof his interpretative
reasoning, ntis"the cat is not on the mat". Also, syntactic analysis contributes the premisel.~"l~¢tis-n0t on
u~en~’, "-~on(THEcat,THEreat)".Nowto these premises and to the pragmatic and other generafizations in
BTwe add as manyof the above assumptionsas we can without introducing inconsistencies. Westart with
the assumptionsin G1and f’md that all five can be added. Thenwe go on to add as muchof ~2 as we can
without introducing inconsistencies, finding that all of ~2 can be added, too. (Of course consistency must
be demonstrated, say, with a model construction.) So the assumptions we are able to makeare just
qlUq2. Nowit can be verified that relative to BT, our premises and assumptions entail, amongother
things, both cons3xcat x and cons-on(THEcat,THEreat).That is,
{Utts"thecat is not on the mat", lf("tbe cat is not on the mat", "--,on(TH~at,THEma0"
BTu gluq2
In this waythe interpreter is able to derive both 3xcat x and --on(THEcat,THEreat)as conveyedmeaningsin
situation (I).
Takesituation (II) above:the speaker utters "the cat does not exist" in the presenceof the interpreter. This
leads the interpreter onceagain via syntactic processingmpremises{Utts"thecat does not exist",/f("the cat
does not exist", "-~3xcat x")}. The assumption-making
process begins once again with the higher-priority
assumptionsin ~l, finds that they can all be addedwithout loss of consistency, and moveson to see how
muchof ~ can be added. Nownote that
{ Utts’the cat doesnot exist",/f(’the cat doesnot exist", "-axcat x" }
cons(-~xcat x)
Themainplayer in the demonstrationthat this is so is the sentence
Utts"the cat doesnot exist"^If("the cat does not exist", "~3xcatx" )
which with great foresight was included in ~1. Other elements of ~1 enable us to derive as well the
followingthree things:
cons(--~xcat x) --~ bels(-~xcat x)
cons(3xcat x) -* bels(3xcat x)
Putting all of these premisestogether (and assuming-- with less than full plausibility -- that the logic of
bel is suchthat believing contradictoriesentails believing .J.) we have
{ Utts"thecat doesnot exist",/./("the cat doesnot exist", "-~xcatx"}
BTv ~1
-,cons(3xcat x)
The effect of this is that the following sentence, whichis in ~2, cannot consistently be addedto these
Utts("the cat does not exist") >I_.> cons(3xcatx)
The other element of .q2 can be though. So in utterance situation (II) the result of assumingas much
possible while respectingpriorities is just
{utts"the cat doesnot exist",/f("the cat doesnot exist", "-~xcatx"}
BTuglU{utts("the cat is not on the mat") >1._, cons(3xcatx)}
Wehave seen that this set entails cons(~3xcatx) but (by its consistency) does not entail cons(3xcat x).
Thusby this assumption-makingthe interpreter is able to derive, from the observed facts of utterance
situation (II) and his backgroundtheory of interpretation, that the speaker has conveyed--,3xcat x. But
other than in utterance situation (I), the speaker’s choice of definite-description syntax does not lead the
interpreter to supposethat he has also conveyed3xcat x.
The previous informal discussion of assumption-making
illustrates the following precise notions. First a
definition. Letting .~-- <~1, G2,-.- 6> be a focal vector and letting g(be a set of assumptionsdisjoint from
every .qi, .~lg(is a useful notation for the focal vector <~1,~2, ... 0,o9(>-Definitionsand proofs using this
notation can proceedby inductionon the length of focal vectors. For a trivial start, a focal vector _9"canbe
"flattened" into ~ as follows: oV = { }; (~I~)~’J = ~ u ~ The following notion captures the idea of a
set £I of assumptions,taken from.2"in order of their priority and consistent with premisesr’, than whichno
other such set is moreinclusive:
[1 is maximalF-satisfiable within 2[is defined by induction on the length of 9":
Basestep: { } is maximalF-satisfiable within <>.
Induction step: £2 is maximalF-satisfiable within ~lMif
i. ~ .~ (~L~J~;
ii. ~’ln~4"-j is maximalF-satisfiable within .~
iii. IXA2~.1., and if [1 ~ [1" c (~lY~and KA’~*~.L, then [1 = t2*.
prioritized allowed entailment F ~2-¢pmeansthat for every t2 whichis maximalF-satisfiable
within y_, F~ ~=¢p.
The following continuation of the earlier examplesshowshow]=~ is used to formalize the derivation of
conveyedmeaningsfrom premises including pragmatic generalizations and the fact that given sentences
Let ~= <~l, [email protected]>be as in the earlier informal discussion.
(I) A speaker says "the cat is not on the mat" and nothing else. Let Fbe BT~utts"thecat is not on the
mat"]. Theearlier discussion is a demonstrationthat there is a unique.~-respecting and maximalFconsistent subset of ~po~, and that is ~1~.~2itself. Also, it was shownthere that rLTglV
~ ~-cons~xcat
x) and also it wasshownthat r~lv~2 ~-cons(-,on(THEcat,THEreat)).Thus uttering "the cat is not on the
mat"conveysthat there is a cat, whichis not on the mat:
01)A speaker
earlier discussion is a demonstrationthatthere
is a uniquemaximalF-consistent
setwithin ~ and that is
~lV{Utts("b"l:c~isnxmlherr~’) >/_.~ cons~xcatx)
Call this set £1. It was shownearlier that FUll ~cons(~3xcat x), though F~,.£1 ~econs(3xcatx). Thus
uttering "the cat does not exist" conveysthat there is no cat, andfails to conveythat there is a cat.
Before going on to analyze someexamplesthough, it maybe helpful to compare~ with Reiter’s [1980]
Default Logic. I will showhow~ can be characterized using a generalization of Reiter’s [1980] Default
Logicwhichallowsfor default rules to havedifferent priorities.
(Reiter): Let ¢p. Vand Z be sentencesof f-, default ru le is a rule of inference:
It is convenientto write this rule ¢P:W/Z.A normal"default is a default rule of the form
I will write such a rule ~W.
(Reiter): (normal) default th eory is a pair if, A) whe
re f.i s a s etof s entences and A isa set
of (normal)default rules.
This notion can be generalizedas followsto allow for a priority order on sets of default rules. Let A = <AI,
A2,... Ak>be a finite vector of sets of default rules. Then<F,~>is a prioritized default theory.
(Reiter): Anextension of(F,A) is any set ~of :,sentences such that ~= ~-,i. where each ~,i
is defined in terms of ~E and (F,A) as follows:
Ei+l = Cn(Ei)
u (g:forsome~ and¥, q~e~, ¢:~/g
e A and-,¥~~:}.
HereCn(3)= {q): S ~=~0}.This notioncanberecursivelyextended
to priodtizeddefault theories.As for
baseclause, wesay that E is an extensionof (F,<>) just in caseE = Cn(F).For the inductive clause,
say that ~; is an extensionof <F,<AI,A2 .... Ak+l>>just in casethere is someextension~;’of <F,<AI,
A2.... Ak>>suchthat E is an extensionof <E*,Ak+I>
is the original senseof Reiter’s.
Nowfor any sentence q) define A~the default rule correspondingto q~, to be the normaldefault Tiq). Note
that these are particularly trivia[ normaldefault rules. Nowgeneralize to sets of sentencesand focal vectors
as follows: for any set For sentences, AF={Aq~q~e
F}; and for any focal vector .2"= <FI, if2 .... Fk>,A.2"=
<Affl, Ay,2,...A ~. Then the following thing can without muchdifficulty be proven by induction on the
an extension
of <F,A~>.
anyis extension
of <F,A.7"F-consistent
> can be written
of if: Let
a focal vector.
Then for
any ~1 which
Cn(I’~2), for some II which is maximal 5r-respecting
equivalencetheoremcan be stated:
within E Thus the following
F ~9~iff for every default extension ~ of <F,A~>,q~e 2;.
In "AllowedArguments"[IJCAI1995] I showthat allowed entailment can also be characterized as what,
generalizing John McCarthy’s[1980] notion of Minimal Entailment (the model-theoretic pendant of
Circumscription) has been called a "preferential" entailment notion. The preference relation is in this
characterization an alphabetic preference order on possible worlds determinedby ~. (Thus one world, u, is
preferred to another, v, if u satisfies moreof FI than does v, or if they satisfy the samesentencesof F! but
u satisfies moreof F2 than does ~, or if they satisfy the samesentences of FIVF2 but u satisfies moreof
$:3 thandoesv; or if....)
Theinterest of these characterizations here is that interpretation, if it admitsan analysis like the one I
suggest, is of a kind with forms of nonmonotonicreasoning like Default Logic and Circumscriptionwhich
are already quite well understood.
4 Back
It is nowpossible to analyze the examplesI started out with (and manymorebesides whichI don’t have
roomto treat here.) Let F be a backgroundtheory containing the pragmatic generalizations and other
"knowledge"of section 2 above.
The sentences which appear in someFi are said to be in focus. Exactly which instances of modus
ponens will be in focus wheninterpreting any given utterance or text? Andwhat priority do they have?
Are there assumptionsto be madein interpretation whichare not instances of modusponens? I expect that
the factors determiningthe choice of focus for any given interpretation task will turn out to be manyand
varied. But for now,and by wayof illustration, I can makeexplicit the strategy whichwas followedearlier
on, in the discussion of example1. Taking a generalization, say, Utts("...the H...")>cons3xrl, an
instantiation is a sentence whichlike
UttsCtheking of Franceis bald")>cons_~xkin&-of.france
substitutes into the generalization, in this case, the nameof a sentence of English, and an appropriate
predicate expression. A sentence A>/--~B, whereA>Binstantiates the generalization y, can be said to
activate y. In the examplesbelow I will assumethat all instances of modusponens are in focus which
activate one or other of the abovepragmaticgeneralizations.
Havingdecided which sentences will appear somewherein our focal vector 2~it remains to decide in
which Y] they will appear. In the exampleI discussed above I supposed that some assumptions have a
higher priority than others in pragmaticreasoning. In particular I supposedthat assumptionsof a general
conversational nature, and those concerned with the speaker’s choice of conveyedmeaning,have higher
priority than assumptionsconcernedwith the speaker’s choice of syntax. Assumea hierarchy of pragmatic
generalizations in whichDEFINITE
and other generalizations deriving from presuppositional
triggers are ranked lower than QUALITY,
and the rest (compareGazdar’s [1979] suggestion).
Here, the lower the index i of the Fi of which q~>/...>~ is a member,the higher the priority of the
assumption ¥. In this way, through these indices, focal vectors can reflect the pragmatic hierarchy:
suppose there are pragmatic generalizations Yl and 3"2 of which q)l>~/l and q~2>¥2,respectively, are
instantiations. Andsupposeon the pragmatichierarchy YI is ranked higher than Y2. Andlet there be i and
j such that q)l>/._+V/l e Fi and (p2>/_~¥2 G Fj (which is to say, suppose we are focusing on these
generalizations). Thenit is required that icj.
I makeaboutwhichconveyed
canbe derived
the cat isn’t on the mat A speaker utters "the cat isn’t on the mat" (and nothing else), conveying
amongother things that there is a cat, and a mat. Wehave
F, Utts("thecat isn’t on the mat")
C Ons(3xcat
The followingdemonstrationsketch illustrates the general approach.It is not difficult to showwith a model
construction (and it is not surprising) that V, utts("the cat is not on the mat") is satisfiable together with
everything in ~. So there is just one fiwhich is maximal~-respecting and D~Utts("thecat is not on the
mat")}-satisfiable within .~ and that is _9v. Now
I’, Utts("tbecat isn’t onthe mat")
Utts("...the Fl...")>/4ons.~xl’l
x )Acons(3xmat
if the eat isn’t on the mat then it’s in the kitchen This example illustrates the "projection" of
presuppositionsinto a conditional context. A speaker utters "if the cat is not on the matthen it’s in the
kitchen" (and nothing else), conveyingamongother things that there is a cat, a mat, and a kitchen. Now
we have:
F, Utts("the cat is not on the mat") ~- cons(3xcat x)/~cons(3xmatx) Acons(3xkitclwnx)
there isn’t a eat A speaker utters two sentences: "The cat is not on the mat" and, by way of
explanation, "Thereisn’t a cat." (and nothingelse). In so doing he does not conveythat there is a cat, but
does conveythat there is a mat. Let F* stand for F~.7{utts("the cat is not on the mat"), Utts(There is
cat")}. Nowcons(--~xcatx) and cons(3Xmatx) follow from F* with ~9". cons(3xcat x) does not.
the eat is on the mat if there is a eat A speaker utters "the cat is on the mat if there is a cat." Let
I’* stand for 17~Utts("the cat is not on the matif there is a cat")}. Now
x), cons(~bels~3xcat x), cons(--,bel s (on(THEcat, THEreat))) and
cons(--&els(--~n(THEcat,THEreat)))follow with ~=~._ from U*. The speaker’s choice of the indicative
conditional conveyshis agnosticism about the antecedent and the consequent. The generalization QUALITY
plays an important part in the demonstration. On the other hand cons(3xcat x) does not follow. This
existential presuppositionof the expression"the cat" is overriddenby the conversationalimplicature that
the speaker is agnostic about the antecedent.
Abnormality Predicates and Default Rules
It is important to say whyI chose to represent pragmatic generalizations using the modalconditional
operator >, and not as fast-order sentencesinvolving "abnormality"predicates as in familiar applications of
McCarthy’s[1980, 1986] Circumscription,or as default rules in the sense of Reiter’s [1980] Default Logic.
Thereasonis that the object languagesof these theories are too restrictive.
Circumscriptionis definedwithin the extensionalsetting of classical fast-order logic and to generalize to
more expressive languages is not trivial. (See for example Thomason[1990].) This commitment
extensional logic makesit difficult to see howCircumscriptioncould capture, say, nonmonotonic
involvingpropositionalattitudes, conditionalsand the like, all of whichfigure in pragmaticgeneralizations.
Default Logicin Reiter’s original [1980] formulationalso takes a classical perspective thoughin this case
that is inessential. Anotherlimitation is howevernot so easily overcome.Thedefaults of Reiter’s theory
are metalinguistic rules, not a part of the object language, so it is not possible to reason about themin
Default Logic. In particular, default rules cannot be nested into other sentences. But a/l of the pragmatic
generalizations I stated hold only under the assumptionthat the speaker is being cooperative, so that
for exampleshould really be stated cooperative(s)--->(cons~
p > bels~P). For another example,my
principle of signification requires that one > can appearwithin the scopeof another>, whereasdefault rules
in Reiter’s theory cannot be nested one within the other.
This point that > is not to be thought of as a default rule of inference should of course not be confused
with the point expressedin an earlier theorem,that the notion of nonmonotonic
inference I have defined on
the languageof > can be characterizedusing defaults over this language.
Future Work
Priorities include integrating with Lascarides and Asher’sworkon rhetorical relations, and a comparison
with Gazdar[1979].
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