DEONTIC LOGIC AND CONTRARY-TO-DUTIES
1 INTRODUCTION
Deonticlogic is concerned with thelogicalanalysis ofsuch normative no-
tions as obligation, permission, right and prohibition. Althoughits origins
lie in systematic legal and moral philosophy, deontic logic has begun to
attracttheinterestofresearchersinotherareas,particularlycomputersci-
ence,managementscienceandorganisationtheory. Amongtheapplication
areaswhichhavealreadyreceived someattention in theliteratureare: is-
suesof knowledgerepresentationin thedesignof legalexpert systems;the
formalspecicationofaspectsofcomputersystems,forinstanceinregardto
securityand accesscontrol policies, fault tolerance,anddatabaseintegrity
constraints; the formalcharacterisationof aspects oforganisational struc-
ture,pertainingforexampletotheresponsibilitiesandpowerswhichagents
arerequiredorauthorisedtoexercise. The\EON"workshopproceedings
providesomeillustrationsofworkinthese areas(see[EON91;EON94;
EON96]).
Deonticlogic is one of theformal tools neededin the designand spec-
ication ofnormative systems, where thelatter are understood to besets
ofagents(humanorarticial)whoseinteractionscanfruitfullyberegarded
asnorm-governed; thenorms prescribehowthe agentsideally should and
should not behave, what they are permitted to do, and what they have
a right to do. Importantly, the norms allow for the possibility that ac-
tual behaviour mayat times deviatefrom theideal, i.e. that violationsof
obligations,orofagents'rights,mayoccur.
In [Jones
and Sergot, 1992; Jones and Sergot, 1993]
Jones and Sergot
arguethatitispreciselywhenthepossibilityofnormviolationiskeptopen
thatdeonticlogichasapotentiallyusefulroletoplay. Ifagentscanalways
be assumed to behave in conformity to norm, the normative dimension
ceasestobeofinterest: theactualdoesnotdepartfromtheideal,sonoth-
ingislostbymerelydescribingwhat theagentsinfactdo. Thus,although
itiscorrecttosaythatdeonticlogicdealswiththelogicofobligation,per-
mission and other normative notions, a more insightful characterisation,
JonesandSergotsuggest,viewsdeonticlogicasessentiallyconcernedwith
representing and reasoning about the distinction between the actual and
theideal. Systemsforwhichthat distinctionisrelevantaregenuinely nor-
mativesystems,andtheirspecicationwillordinarilyinclude \secondary"
norms which indicate whatis to bedone incircumstancesin whichactual
behaviourhasdeviatedfromtheideal. Themethodologicalguidelinespro-
[Jones 1993]
that \secondary" norms of this kind (rst dubbed \contrary-to-duty" in
[Chisholm, 1963])
will be a prominent feature of normative systems, and
thus that anyadequate deontic logic mustaccommodate them. However,
theanalysisofcontrary-to-dutyobligationsentenceshasprovedtobeatask
ofsomeconsiderablecomplexity. Anditisthisissue|at theverycoreof
deonticlogic|whichthis chapteraddresses.
The plan is as follows: we rst (Section 2) describe Standard Deontic
Logic, and a numberof its defects, includingproblems regarding therep-
resentation of conditional obligationsentences. Inthe course ofSection 3
weexamineanumberofdierenttheorieswhichhaveattemptedtoaccom-
modatecontrary-to-dutyobligationsentences(CTDs),andinthecourseof
thisexaminationweidentifyseveralcriteria|eightinall|which,wear-
gue,anadequatetreatmentofChisholm'spuzzleaboutCTDsshouldmeet.
Someofthese criteriaare nottied toChisholm'sproblem,but applyquite
generally to the analysis of CTDs. Section 4 presents a revised, and in
parts considerablymodiedversionofthe
[Carmo
and Jones, 1997]
theory
of CTDs; its application to a number of CTD \scenarios" is investigated
in somedetail in Section 5, and this provides afurther impression of the
broadrangeof representationaland reasoningissues which aCTD theory
must address. Section 6 examines somepossible counter-examples to the
proposedanalysis,therebyrelatingitstreatmentofCTDproblemstoother
well-known issuesin deontic logic, concerning | in particular | the clo-
sureofdeonticoperatorsunderlogicalconsequence,andtherepresentation
ofconicts ofobligations. Section7oersfurther observationsonalterna-
tiveapproachesbasedontemporallogic,thelogicofaction,andpreference
orderings,respectively. Theoverallaimofthechapteristosupplyarather
detailed overviewof agroupofproblems attheheartof deonticlogic,and
aguideto existingattemptstosolvethem.
2 DEONTICLOGIC:THESTANDARDAPPROACH
2.1 Standard Deontic Logic
The standardapproach to deontic logic takesit to be a branch of modal
logic,interpretingthenecessityoperator
asexpressingethical/legalne-
cessity, i.e. as meaning \it is obligatory that", and denoting it by O;
accordingly,the dualpossibilityoperator
=:
:is interpretedasex-
pressing\itispermittedthat"(andisdenotedbyP),andtheimpossibility
modal contruction
: is interpretedas expressing\it isforbidden that"
(andisoftendenotedbyF).
Axiomatically, the weakest deonticlogic (called standard deontic logic,
SDL for short) is then obtainedby replacing the modal necessity schema
(T) ( A ! A: unacceptable for a deontic interpretation, since what is
obligatorymayfailtobethecase)bythe(D)schema(which requiresthat
what isobligatory ispermitted). Thus, followingtheChellasclassication
[
Chellas,1980 ]
,SDListheweakestnormalmodalsystemoftypeKD;that
is, its theorems can be characterized as the smallest set of formulas that
includes all instances of the following axiom schemas, and that is closed
under theO-necessitationruleandModusPonens(MP).
Axiomschemas
(PC) Allinstancesoftautologies
(PCstandsforPropositionalCalculus)
(K) O(A!B)!(OA!OB)
(D) (OA!PA)
Rules
O-necessitation:
A
OA
ModusPonens(MP):
A; A!B
B
Wehereemploycapitalletters(A;B;C ;:::)tostandforarbitraryformulas
(well-formedsentencesoftheunderlyingpropositionalmodallogic),andwe
uselowercaseletters(p;q;:::)forarbitraryatomicsentences,and?and>
todenote,respectively,acontradictionandatautology;parentheseswillbe
omittedfollowingtheusualprecedencerulesfortheoperators;theBoolean
connectiveswillbedenoted by:,^,_,!and$;inthemeta-languagewe
denotesuchconnectivesby\not",\and",\or",\if... then..." (or\implies")
and \i"(if andonlyif), andin themeta-languagewealsoavailourselves
of theuniversaland theexistentialquantiers(these donotappear in the
object language: we are concerned only with propositional modal logics).
Moreover,asusual,wewilluse`A(respectively6`A)todenotethatAisa
theorem(respectively,Aisnotatheorem)oftheunderlyinglogicalsystem;
and, followingthetraditionalphilosophical/logical approach todeduction,
wesay(cf.
[
Chellas,1980;HughesandCresswell,1984 ]
)thatAisdeducible
from aset ofhypotheses ,written `A(or simplyA
1
;:::A
n
`Aif is
nite),iAbelongstothesmallestsetofformulasthatcontains andthe
theoremsandthat isclosedunder(MP).
1
1
In this way we get a Boolean, compact, deductive system (see e.g.
[
Bull and
Segerberg, 2001]).
Non-Boolean axiomatic approaches to deduction, where non-
tautologicalrulesmayalsobeappliedtothehypotheses,andnotonlytothe theorems,
maybefoundinsomeworksintheeldofmathematicallogic,suchas
[Hamilton,
1978;
1979].
Semantically,themodelsMofSDLarestandardmodels
[Chellas,1980]:
M = (W;R ;V), where W is a non-empty set (the set of worlds), R is
a binary relation on W and V is an assignment to each atomic sentence
of a set of worlds; informally, V(p) denotes the set of worlds where p is
true. Inorderto validatethe schema(D) werequirethatthe accessibility
relationRisserial,i.e. (8w)(9v)wR v (usingw;v;:::todenoteworldsand,
asusual, writing wR v instead of hw;vi 2 R ). The deontic interpretation
ofthe accessibilityrelation isasfollows: wR v iv isadeonticalternative
to, or an ideal version of, w. The truth of a formula A in a world w of
amodelM is denoted byM j=
w
A and isdened asusual: for instance,
Mj=
w
OAi(8v) (if wR vthen Mj=
v
A); thus, informally,OA istrue
inaworldwiAistrueinallidealversionsofw. AformulaAistruein a
model M,written Mj=A, iAistruein alltheworldsofthemodelM;
andaformulaA isvalid,writtenj=A,iAistrueinallmodels.
2.2 SDL and its problems
Itis widelyacceptedthat SDLis notadequateasabasicdeonticlogic. In
fact,fewsystemsoflogichavebeenasheavilycriticisedasSDL;SDLgives
riseto aset of\paradoxes"(theoremsof SDL that many havedeemed to
becounter-intuitive)andtherearesomedeonticconceptsandconstructions
whichapparentlycannotbeexpressedinSDLinaconsistentmanner. Some
of the main examples will be given below. We have essentiallytwo aims
here: rst, withoutany claimsto originality, wecomment on thereasons
underlying the so-called paradoxes; secondly, we indicate which of these
problems have a counterpart in other areas of applied modal logic (e.g.,
epistemic, doxasticand actionlogics), and which seemto beparticular to
deonticlogic.
ArstgroupofparadoxeshasitsoriginintheclosureoftheO-operator
under logicalconsequence (that is, in the fact that SDL, like any normal
modal logic, containsthe(RM)-rule: \if `A! B then ` OA !OB").
Somewellknownexamplesare:
Ross paradox: (`OA!O(A_ B))
\If itis obligatory tomail the letter, then itisobligatory to mailthe
letter ortoburnit"
Thequestionofthesignicanceofthisparadoxhasbeenthesubject
of considerabledispute. Whereassomeclaimthatthe secondobliga-
tion(the oneintheconsequent)isacounter-intuitiveconsequenceof
the rst, since it seems to leave open to the agent achoice to mail
or to burn the letter, others maintain that the consequent doesnot
leavea choice of this kind, because burning theletter is clearlynot
theperspectiveon deonticlogicadvocated inSection 1,however,we
should also look at the problem from the point of viewof violation:
supposingthat AisobligatoryandthatAisnotthecase,howmany
obligationshavebeenviolated? IfweaccepttheRosstheorem, then
notonlyhastheobligationthatAbeenviolated,but|inaddition|
foreachstateofaairsBwhichactuallyfailstoobtain,anobligation
thatA_Bhasalsobeenviolated. Thisisapeculiarresult;itcontrasts,
of course, with how things look from a fullment perspective; for if
theobligationthat Aisfullled,thensoarealltheother obligations
which can be derived by application of the Ross theorem. (We are
assuming, asis natural, that within SDL violation of an obligation
OC istobeexpressedastheconjunctionOC^:C.)
FreeChoicePermissionparadox:
This paradox has to do with the fact that (in SDL) 6` P (A_B) !
(P A^PB),whereas|ordinarily|ifitispermittedthatAorBthis
wouldbeunderstoodtoimplythatAispermittedandBispermitted.
Weincludethisparadoxinthisgroup,sincethereasonwhywecannot
addP (A_B)!(PA^P B), asanewaxiom,toSDL isthefact that,
bythe(RM)-rule,`PA!P(A_ B),whichtogetherwithP(A_ B)!
(P A^PB) would imply PA ! (P A^PB); so permission to go to
thecinema would imply permissionto kill the President! Moreover,
fromanypermissionwecouldthendeduceP ?,whichisinconsistent
with the fact that ` O>. However, in common with some other
researchers,wethinkthatthis\paradox"isapseudo-problem: ifwhat
wewanttoexpressisthatbothAandBarepermitted,thenweshould
simplyrepresentthatformallybyPA^P B(instead ofbyP (A_B)).
Good Samaritanparadox:
\If it is obligatory that Mary helps John who has had an accident,
thenitisobligatory thatJohnhas anaccident"
Ontheassumptionthat\MaryhelpsJohnwhohashadanaccident"
isrepresentedastheconjunction\MaryhelpsJohnandJohnhashad
anaccident", thentheantecedent oftheaboveconditional takesthe
form\O(A^ B)". Since,tautologically,aconjunction implieseachof
itsconjuncts, the(RM)-ruleyields theSDLtheorem: `O(A^B)!
OB. Inourview,theformalconceptsneededtodealwithproblems
about contrary-to-duty obligations can also provide an appropriate
analysis ofthe Good Samaritan problem. So wereturn to thisissue
below,in Section6.
Deontic/epistemicparadox:
\Ifit isobligatory that Mr. X knowsthat his wife commitsadultery,
We hereassume,asisusual, that the(T)-schemaholds fortheepis-
temic operator. So this problem is again a result of the fact that,
in SDL, any logicalconsequence of that which is obligatory is itself
obligatory.
Wenotethattheclosureofthenecessityoperatorunderlogicalconsequence
isalsoasourceofproblemsforotherapplicationsofmodallogic,forinstance
epistemicanddoxasticlogics,wheretheassumptionthateveryagentknows
(believes) everylogical consequenceof what he knows (believes) is anex-
treme idealisation. In the logic of action, too, it is surely not acceptable
to supposethat anagentbrings aboutallthelogicalconsequences of that
whichhebringsabout(cf.
[
Elgesem,1993 ]
).
2
AsecondproblemofSDLhastothewiththeO-necessitationruleitself,
accordingtowhich anytautology (moregenerally,anytheorem) isobliga-
tory,whichisincompatiblewiththeideathatobligationsshouldbepossible
tofulllandpossibletoviolate. Similarproblemsoccurwiththisruleinthe
epistemicanddoxasticlogics,whereitrequiresthatanagentknows(orbe-
lieves)alltheorems(calledin
[Hintikka,1975]
the\logicalomniscienceprob-
lem"),andin thelogicofaction, whereitis ingeneralsupposed that that
which can be broughtabout mustbe avoidable(see, e.g.,
[Elgesem,
1993;
SantosandCarmo, 1996]).
A third problem of SDL is that, because of the (D)-schema, it is not
possibletoexpress consistentlyaconictof obligations,even though,as a
matter of fact, normative systems may indeed contain conicting obliga-
tions. We shall return to this issue later in this chapter. But, again, we
note at thispointthat this isnot aproblemonly ofdeonticlogic: similar
problemsmayappear,forinstance,in thelogicofbelief.
However,itisfairtosaythat, formostdeonticlogicians,theproblemof
howto representconditional obligation sentences has been their principal
reason for seeking an alternative to SDL. Let us denote by O(B=A) the
\conditional obligation of B, given A"; so O(B=A) is intended to mean
that \it is obligatory that B, if A is the case". In SDL there are two
possiblewaysto representsuchsentences:
(option1) O(B=A)=
df
A!OB
and(option2) O(B=A)=
df
O(A!B)
Noterstwhat thesetwooptionshaveincommon. With bothofthemwe
get(withinSDL):
(UN) `OB$O(B=>)
2
In
[Konolige
and Pollack, 1993]
it isargued that this problem, called the \side-
eect problem"in
[Bratman, 1987],
is even worsefor the logic of intentions -a logic
which,ithasbeensuggested,hasveryclosesimilaritiestodeonticlogic(see [P
orn,1977;
1991]).
(SA) `O(B=A)!O(B=A^C)
The rst theorem is generally seen as a good property, and has been
acceptedbymanyauthorsonthegroundsthat anunconditionalobligation
isaparticular(limiting)caseofaconditionalobligation,wherethecondition
is a logical truth. Here, however, we shall adopt the opposite view, in
line with the opinion expressed by Carlos Alchourron in [1993,
pp.
62],
who argued that (UN) wasone of thewrong steps followed by almost all
researchersindeonticlogic.
(SA) is knownas the\principle of strengthening of the antecedent"; it
isproblematic,sinceitappearstomaketheexpressionofdefeasible(condi-
tional or unconditional 3
) obligationsimpossible;but of courseitis acom-
monplacefeature of obligationsthat theyaresubjectto exceptions. Con-
sider, forinstance, aconditional obligationto theeect that, ifyouraged
mother is sick, then you should help her. Such a conditional obligation
mightwellleaveroomforexceptions,justas penguinsmightbetheexcep-
tion to the generalisation that birds y; supposing for example that your
youngchildhasbeeninjuredinacaraccident,andurgentlyneedsyouatthe
hospital,theobligationtohelpyoursick,agedmothermaywellbedeemed
to havebeen defeated, oroverturned. But againthis problem (which has
someconnectionswith theproblem ofhowto dealwith conictingobliga-
tions)isnotaspecicissueofdeonticlogic;theproblemofhowtodealwith
defeasibleconditionals appearsin manyotherareasandhasbeenasource
ofintensiveresearch.
So far we have not yet found a problem that sets deontic logic apart
from otherbranchesof modallogic. But we herereturn to thepointem-
phasisedintheintroductionandsuggestthattheissueofhowtorepresent
contrary-to-duty obligation sentences (CTDs) | obligations which come
into force whensomeotherobligationis violated|seemsto be aspecic
problem of deontic logic. It has sometimes been proposed, however, that
CTD obligationsmaybeseen ashandlingexceptionsto (primary)obliga-
tions. Although we accept that there may be someconnections between
theproblemofhowtodealwithCTDsandtheproblemsconcerningallow-
ableexceptions anddefault reasoning,it shouldbestressed that thereare
alsocrucially importantdierences(cf.
[Prakken
andSergot, 1994]):
when
a CTD obligation comes into force because of some violation, we do not
wantthentosaythat theviolatedobligationhasbeendefeated; ithasnot
beenoverturned,it has been violated! We needto beableto integratein
a singlelogical framework the ability to makedeductions at twodierent
levels: on thelevelof what ideally should bethecase, and onthelevelof
whatactuallyshouldbethecase,giventhecircumstances(where,ofcourse,
thecircumstancesmightincludethefact thatwhat hashappened deviates
3
Notethatcombiningthetwoprevioustheoremsweget`OB!O(B=A),andso
fromtheideal). Thesimultaneous specicationofbothidealbehaviorand
ofwhattodowhenactualbehaviordeviatesfromtheidealisacentraltask
ofdeonticlogic.
3 CONTRARY-TO-DUTIES
3.1 Chisholm's CTD-paradox and SDL
Considerthefollowingsetoffoursentences,formulatedbyChisholmin1963
[
Chisholm,1963 ]
:
EXAMPLE1.
(a) Itoughtto bethat acertainmangotohelphis neighbours.
(b) Itoughtto bethat ifhegoeshetellthemheiscoming.
(c) Ifhedoesnotgo,heoughtnottotellthemheiscoming.
(d) Hedoesnotgo.
Thereiswidespreadagreementintheliteraturethat,fromtheintuitivepoint
of view, this set is consistent, and its members are logically independent
ofeachother; andthere isagood dealof disagreement in theliteratureas
regardswhichfurtherrequirementsanadequateformalrepresentationofthe
Chisholm set should meet. We start by discussing whether the Chisholm
setcanberepresentedinSDLinawaythatmeetsthissetoftwominimum
requirements,leavingthediscussionofotherfurtherrequirementstolater.
Itisstraightforwardtorepresentsentences(a)and(d)inSDL;theques-
tionishowtorepresent(b) and(c), sincetheyexpressconditional obliga-
tions. Letus leavethat open forthe moment, and representthem bythe
use of our binary conditional obligation operator above; we then get (us-
ing \tell"and \help" in anobviouswayas abbreviationsof the sentences
concerned):
(a) Ohelp (orO(help/>),sinceinSDL` Ohelp $O(help/>))
(b) O(tell /help)
(c) O(:tell /:help)
(d) :help
InregardtotherepresentationofconditionalobligationsinSDL,recallthe
twoalternatives:
(option1) O(B=A)= A!OB
and(option2) O(B=A)=
df
O(A!B)
With (option1)wegetthefollowingresults:
`:A!O(B=A)
(FD) `A^O(B=A)!OB
GiventhatanexpressionoftheformO(B=A)isintendedtomeanthat,
in circumstancesA, B isobligatory,therstofthese tworesultsisclearly
problematic. From the fact that it is not raining we should not be able
to deduce that, in circumstances where it is raining, it is obligatory that
thePresidentbeassassinated. Theother theoremhasto dowith thefun-
damental issueofhowwecandetachnew(unconditional)obligationsfrom
conditional obligations, andit statesakindof \factualdetachment" prin-
ciple,allowingthedeductionoftheactual obligationsoftheagent,thatis,
theobligationswhicharisegiventheactualfactsof thesituation.
With (option2)wegetthefollowingresults:
`O:A!O(B=A)
(DD) `OA^O(B=A)!OB
Accordingto thersttheoremeverythingisobligatoryonthecondition
that someforbiddenfactisthecase: thus(option2) clearlydoesnotallow
us to express CTDs. The second theorem represents a kind of \deontic
detachment"principle,allowingthedeductionoftheidealobligationsofthe
agent,i.e. thefurtherobligationswhichariseifhebehavesinawaywhich
conformswithsomeexisting setofobligations.
The surfacestructures of lines (b) and (c) in the original Chisholmset
might be takento indicate that,within SDL,(option 2) should bechosen
for(b),and(option 1)for(c),giving:
(a) Ohelp
(b) O(help !tell)
(c) :help!O:tell
(d) :help
This wasChisholm'schoice,and he rightlywenton to pointoutthat this
formalisation yieldsaninconsistency,sinceOtellisderivablefrom(a)and
(b),whilstO:tellisderivablefrom(c)and(d)(andaninconsistencyfollows
bythe(D)-schema).
If, alternatively, weuse (option 1) for both lines (c) and (b), then the
resulting set is consistent, but logical independence is lost, since (b) will
(b)and (c),then (c)wouldbeaconsequenceof (a)bythe(RM)-rule. So,
inSDL,theconclusionisthat theChisholmsetcannotberepresentedin a
waywhich satises both of thetwominimum requirementsof consistency
andlogicalindependence.
3.2 Some further requirements on the representation of CTDs
A number of deontic logicians have argued that the problems raised by
CTDs involvein an essentialwayeither atemporal dimension oractions.
Weshallhaveagooddealmoretosayabouttheselines ofapproachlater
on (especially in Section 7), but for the moment we just wantto register
agreement with Prakken and Sergot
[Prakken
and Sergot, 1994; Prakken
and Sergot, 1996],
who have indicated that there are examples of CTD
scenarioswhereitisfarfromobvioushowconsiderationsofthetemporalor
actiondimensionsmightbeapplicable. Consider:
EXAMPLE2.
(a) There oughtto benodog.
(b) Ifthere isnodog,thereoughttobenowarningsign.
(c) Ifthere isadog,there oughttobeawarningsign.
(d) There isadog.
EXAMPLE3.
(a) There mustbenofence.
(b) -
(c) Ifthere isafence,thenitmustbeawhitefence.
(d) There isafence.
ExamplesofthesekindssuggestthatatreatmentofCTD'swhichistied
totemporaloractionaspectswillnotbesuÆcientlygeneralinitsscope.
A furtherquestionwhich existingtreatmentsofCTD'sraiseis this: are
lines(b)and(c)oftheChisholmsettobeassignedfundamentallydierent
logicalforms? Thetheorywedevelopbelowgivesanegativeanswertothis
question, and supplies a uniform treatment of deontic conditionals. Our
viewisthat,intheabsenceofstrongargumentstothecontrary,thesurface
forms of (b) and (c) should be deemed to be merely stylistic variants of
essentially the same type of underlying logical structure. In particular,
we reject the position taken in
[Prakken
and Sergot, 1994; Prakken and
1996],
logical representations just because (c), unlike (b), is a contrary-to-duty
conditional, expressing as it does the obligation which comes into force
whentheobligationexpressedbyline(a)isviolated. PrakkenandSergot's
approach makes the assignment of logical form to deontic conditionals a
highly context-dependentmatter, withtheconsequencethat any insertion
ordeletionofanormmayrequirethatsomerevisionthenhastobemadeto
theformalisationofsomeothernormintheset;(e.g.,deletingline(a)ofthe
Chisholmsetwouldrequire,ontheirapproach,achangeintheformalisation
ofline (c)). Likewise, theforminitiallyassigned toagivensentencemight
have to berevised in virtue of what turns out to bederivable from other
sentences;suppose,for instancethat \if A thenitis obligatorythat B"is
intheinitialset,andisassumednottobeaCTD;ifitthentranspiresthat
\it isobligatorythatnotA"isderivablefrom othermembersoftheinitial
set, then the conditional becomes a CTD and its logical form has to be
changedaccordingly. Thischange may,in turn, havefurther repercussions
regardingwhat can bederived...andso on. Nowwith asmallinitial set,
suchasChisholm's,itwillofcourseberelativelyeasytoseewherechanges
need to be made; but with a large corpus of norms it is not diÆcult to
imagine that the problem could become intractable. The disadvantages
which accrue from this kind of context-dependence of logical form are so
great, in our opinion, that any approach to the analysis of CTDs which
managestoavoiditis-otherthingsbeingequal-tobepreferred.
Thus, wehavesofar identied thefollowingrequirementsthat an ade-
quateformalisation oftheChisholmsetshould meet:
(i) consistency;
(ii) logicalindependence ofthemembers;
(iii) applicability to (at least apparently) timeless and actionless CTD-
examples;
(iv) analogouslogicalstructuresforthetwoconditionalsentences,(b)and
(c).
Oneimportantgroupof deonticlogicsthat satisfythese requirementsem-
ploys a primitive dyadic conditional obligation operator O(/), where
O(B=A) isread\it isobligatorythat B,giventhatA". These logicsusu-
ally take the unconditional obligation OB to be equivalent to O(B=>),
andtheyrepresenttheChisholmsetasfollows:
(a) O(help />)
(b) O(tell /help)
(d) :help
Following [Lower
and Belzer, 1983]
we can distinguish between two main
\families" of dyadic deontic logics, according to the kind of detachment
principles they support: one supports the \factual detachment" principle
(FD), and we call it the \FD-family";
4
the other supports the \deontic
detachment"principle(DD), andwecallitthe\DD-family".
5
Returningagainto theChisholmset,itisclearthat(asforitsproposed
representationwithinSDL)acceptanceofboth(FD)and(DD)wouldpermit
thederivationofO:tell(by(FD)onlines(c)and(d))andOtell(by(DD)
on lines (a) and (b)). If the (D)-schema is accepted, then the situation
arising from adoption of both (FD) and (DD) would of course be one of
logical inconsistency. But even ifthe (D)-schema is notaccepted, so that
theconjunctionOtell^O:tellisnotdeemedto belogically inconsistent,
thederivationfrom theChisholmsetofaconict ofobligationsofthetype
expressedbythisconjunctionissurelyunacceptablefromtheintuitivepoint
ofview. Thesituation describedbytheChisholmsetdoesnotpresentthe
agentconcernedwithamoraldilemma,onourview.Requirement(i),above,
should be understood as oneto the eect that aconjunction of the form
OA^O:Ashould notbederivablefromtheformalrepresentationofthe
set,regardlessofwhetherthatconjunctionisdeemedlogicallyinconsistent.
Of course, neither the FD-family northe DD-family accepts both (FD)
and (DD). Nevertheless, it mightbe suggestedthat afully adequaterep-
resentationof theChisholmset should beableto capture, in awaywhich
generatesneitherinconsistencynoramoraldilemma,boththefact that|
given the circumstances, and particularly the occurrence of the violation
of the obligation expressed by line (a) | the agent's actual obligation is
not to tell his neighbourshe is coming, and the fact that | under ideal
circumstances, in the absence of violation of the obligation expressed by
line (a) | the agent's obligation would be to help his neighbours and to
tellthemheis coming. Acceptingthesesuggestions,weoerthree further
requirementswhich webelieveanadequaterepresentationoftheChisholm
setshould meet:
(v) capacityto deriveactualobligations;
4
In general the logics in this family have a semantics based on minimalmodels
(proposed,independently, byDana Scottand Montague, and popularisedby
[Chellas,
1980]).
Asrepresentativesofthisfamily [Lower
andBelzer, 1983]
mention [Mott,
1973;
al-Hibri,1978;Chellas, 1974];
however,asregards
[Chellas,1974]
itisnotentirelyclear
whetherChellascommitshimselftoacceptanceof(FD).
5
[Lewis, 1974]
presents an overview of several members of the DD-family. These
logicsintroduce,inthesemantics,apreferencerelationbetweentheworlds,thatorders
theworldsaccordingtotheirideality;thenO(B=A) istrueataworldithereissome
worldwhere A^BistrueandthatismoreidealthananyworldwhereA^:B istrue.
(vi) capacitytoderiveideal obligations;
6
(vii) capacity to represent the fact that a violation of an obligation has
occurred.
NeithertheFD-familynortheDD-familymeet both(v)and(vi).
7
Wewill
returnlatertotheissuesraisedby(vii).
3.3 The\pragmatic oddity"
One of the logics that fullls all these requirements is the one proposed
in [
Jones and Porn, 1985 ]
. Jones and Porn adopt a completely dierent
approachfrom thatof thedyadicdeonticlogics, anddeneadeonticlogic
where non-normalobligation operators are obtained asBoolean combina-
tionsofnormalmodaloperators,followingastrategythathadalreadybeen
usedin theeldofactionlogicbyKangerandbyPorn.
Taking as its point of departure the observation that SDL fails in its
attempt to capture CTDsbecause| from the semantical point of view-
SDLconsidersonlytheidealversionsofeachworld,JonesandPornpropose,
in addition to SDL's accessibility relation, a second accessibility relation
which picks out the sub-ideal versions of agiven world (and they further
requirethateachworldiseitheranidealorasub-ideal versionofitself).
Thentheyintroduceintothelogicallanguagetwomodalnecessityoper-
ators, 8
heredenoted by
! i
and
! s
. Therstoftheseisjust theobligation
operatorof SDL,sothat anexpression oftheform
! i
A is trueatagiven
worldw iAis trueat allofthe idealversions of w. Bycontrast,
! s
Ais
trueatagivenworldwiAistrueatallofthesub-idealversionsofw. (A
sub-ideal versionw
1
, ofw,is informallyseenasaversionofwin which at
leastoneoftheobligationsin forceat wisviolated.) Theduals of
! i
and
! s
are,respectively,
! i
and
! s
.
Finally theyintroducebotha deonticnecessityoperator
!
, dened as
follows:
!
A=
df
! i
A^
! s
A
6
Somemightcallthem primafacie obligations. However,weavoidusing thisterm
heresinceitsmeaninginthe literatureseemstousto befarfromclear. Furthermore,
[Prakken
andSergot, 1997]
providegoodreasonsforsupposingthatthetermismostat
homeinthediscussionofdefeasibility,ratherthanCTDs.
7
In
[Jones,1993]
itisarguedthatafurtherproblemofthe (FD)-familyisthatthey
rejectthe \principleof strengtheningof theantecedent" (SA)whilst atthe sametime
accepting unrestricted factual detachment. The problem is that one of the reasons
for rejecting (SA) isthat onewants to be able to represent conjunctions of the form
O(B=A)^ O(:B=A^C),withoutgettinglogicalinconsistencyormoraldilemmaofthe
formOB^O:B,evenincircumstancesinwhichbothAandC aretrue.
8
Note thatthe notation employedhere forthe operators diersinmostcasesfrom
[Jones 1985].
andanactual-obligationoperatorOught,dened by:
Ought A=
df
! i
A^
! s
:A
(the secondconjunctguaranteesthat`:Ought >).
TheChisholmset isthenrepresentedin [Jones
andPorn, 1985]
asfollows:
(a) Ought help
(b)
!
(help!Ought tell)
(c)
!
(:help!Ought :tell)
(d) :help
Theset,onthisrepresentation,is consistentand itsmembersarelogically
independentofeachother. Lines (c)and(d)implyOught :tell(notethat
!
is a \success" operator, i.e. it satises the (T)-schema), and lines (a)
and (b)imply
! i
Ought tell. Furthermore,theconjunctionof (a) and(d)
may be taken asexpressing the fact that the unconditional obligation to
helptheneighbourshasbeenviolated;and,haditbeenthecasethat\(d')
help",ratherthan(d),weretrue,thenfrom(b)onecouldhavededucedthe
actualobligationtotell,Ought tell. Apparently,alliswell!
However, [
Prakken and Sergot, 1994; Prakken and Sergot, 1996 ]
point
out that the Jones and Porn treatment of Chisholm, in common with a
number of others, generates what they call the \pragmatic oddity": line
(a), togetherwiththederivedactualobligationOught :tell,requirethat,
inallidealversionsofthegivenworld,theagentconcernedgoestohelphis
neighboursbut does nottellthem he iscoming| aresultwhich appears
highlycounterintuitive.
Prakken and Sergot correctly point out that, for a number of cases, a
reasonabletemporalinterpretationisavailablewhichenablesthepragmatic
oddity to be avoided. For instance, perhaps the obligation expressed in
line (a)would ordinarilybeunderstoodasan obligationto goto help the
neighboursnolaterthanaparticulartime,t.Then,ifline(d)weretobetrue
after timet,theaccessibledeonticallyidealworldswould becharacterised
in such awaythat,after timet,these worldswouldrequirethat theagent
does nottell his neighbourshe is coming(but they would not, of course,
alsorequirethat hegoesto help,sinceitwouldthenbetoolate).
However,as weindicatedabove,PrakkenandSergotalsopointoutthat
there are instances of theChisholm set which may beinterpreted in such
awaythat thetemporaldimensioniscompletely absent( [
Jones,1993,pp.
153-4 ]
makesasimilar point). Example 2,above,isonesuch case: avery
ordinaryway of understanding that set takeseach sentence to be true at
qualicationsconcerningwhenthereoughttobenodog,orwhenthereought
tobeawarningsign|allowstheconclusiontobedrawnthatthereought,
in thecircumstances,to beawarningsign,withouttherebygenerating the
pragmatic oddity, i.e., without forcing the further conclusion that, in all
ideal versions of the given situation, there is no dog but there is a sign
warningofone.
Unfortunately, Prakkenand Sergot oerlittleby way ofexplanation of
the pragmatic oddity: they say little about what it is that creates the
sense of oddity. In [
Carmo and Jones, 1997 ]
we suggest an explanation
which exploits a parallel betweenexamples of type Example 2, which on
the [
JonesandPorn,1985 ]
analysisexhibitthepragmaticodditysimpliciter,
andexampleslikeExample3above(alsoduetoPrakkenandSergot)which,
byvirtueofsomeassumedlogicaltruth,areinconsistentwhenformalisedin
thestyleof [Jones
andPorn, 1985]
(accordingtowhich,inallidealversions
ofthegivenworld,thereisawhitefenceandnofenceatall!). Thesuggested
parallel is as follows: as represented in the languageof [Jones
and Porn,
1985],
Example 2exhibits thepragmatic oddity becausean inconsistency
would begeneratedwereoneto addto theexamplethefurther constraint
thatitoughtnottobethecasethatthereisbothnodogandasignwarning
of one. The sense of oddity arises because there is an interpretation of
Example 2 accordingto which itremains consistentevenif supplemented
with that further constraint; and the problem with the [Jones
and Porn,
1985 ]
approachisthatitfailstocapturethat interpretation.
Thus weadd another requirement which anadequate representationof
CTDsshould satisfy:
(viii) capacitytoavoidthepragmaticoddity(interpretedaccordingto the
previousdiagnosis).
3.4 Two attempts to resolve the \pragmaticoddity"
In
[Prakken
andSergot,1994;PrakkenandSergot, 1996],
PrakkenandSer-
gotarguethattheproperresponsetotheproblemsraisedbyExample2|
and in particular theproblem of pragmaticoddity | isto assign distinct
logicalformstoprimaryobligations,ontheonehand,andCTDobligations,
ontheother. ForCTDobligations,theyrelativiseanobligationoperatorto
aspecic \context ofviolation"; moreprecisely,an expressionof theform
O
A
Bisintendedtobereadas\thereisasecondaryobligationthatBgiven
that, or presupposing, the sub-ideal context A", or \given that A, which
isaviolationofsomeprimaryobligation,thereisasecondary,compromise
obligationthat B"
[
Prakkenand Sergot,1996,section5 ]
. Theyemphasise
that expressions of the form O
A
B are not to be read as conditional pri-
maryobligations. \TheexpressionO
A
B:::representsaparticularkindof
canbedetachedfromtheexpressionO
A
B"[loc.cit.]. Theirrepresentation
ofExample2takestheform:
(a) O:dog
(b) :dog !O:sign
(c) dog !O
dog sign
(d) dog
WeshallnotpursuethePrakkenandSergot94treatmentofCTDshere
(althoughwereturntotheirworkbrieyinSection7). SuÆceittosaythat
theirapproach(in
[Prakken
and Sergot,1994; Prakkenand Sergot, 1996])
rejectsthefourth ofourrequirementsforasatisfactorytheoryinthisarea.
For them, thechoiceof logicalform foran apparentlyconditional deontic
sentencewillitselfbedependentonwhichothernormsarecontainedin,or
derivablefrom,thesetofnormsbeingformalised.
9
In
[Carmo
and Jones, 1995]
10
weattempted adierentkindofapproach
totheproblemofthepragmaticoddity,distinguishingbetween\idealobli-
gations"(line(a)inExamples1,2and3,forinstance)and\actualobliga-
tions",whichindicatewhatistobedonegiventhe(perhapsless-than-ideal)
circumstances. Theoperator O
a
, for representingactualobligations, was
dened inthesamewayastheOught-operatorof [Jones
andPorn, 1985],
described above. As regards ideal obligations, the basic model-theoretic
idea was to distinguish between ideal versions of a given world (the fun-
damental feature of SDL), and ideal worlds themselves. Accordingly, we
dividedtheset of possible worldsW into twomutually exclusivesub-sets,
theset ofidealworldsandtheset ofsub-ideal worlds;importantlyforour
purposes, weallowedthat aworld w
1
could be anidealversionof agiven
(sub-ideal) world wwithout also itself being anideal world. And wexed
truth conditionsfor expressions of theform O
i
B (\it oughtideally to be
thecasethat B")in termsofthetruthof B inallideal worldsandfalsity
insome sub-ideal world.
11
9
TherearealsosomediÆcultiesinunderstandinghowO
A
Bshouldbeinterpreted,
particularlysincePrakkenandSergotinsistthatA(in O
A
B)necessarilyrepresentsa
contextofviolation. Forinstance,the formula(P A^O
A
B)!OBisvalid,on their
account(wherePisthepermissionoperator),butnottriviallyso.Asweseeit,intuitively
theconjunctionintheantecedentofthisconditional(giventheirreadingofO
A
B)could
onlybefalse,soitshouldimplyanything.Furthermore,whatcantheypossiblymeanby
theclaimthatOBisanabbreviationofO
>
B? Arewetosupposethatitisobligatory
thatBonlyifthetautologyrepresentsacontextofviolation?
10
Wethereadaptthe logic proposedin [Carmo
and Jones,1994;Carmoand Jones,
1996]
fortheanalysisofdeonticintegrityconstraints.
11
Thesecondconjunct simplyguaranteesthe violabilityof idealobligations(i.e. j=
:O
i
>).
[Carmo
andJones, 1995]
containsdiscussionofpossibleconnectionsbetween
thenotionsofideal/sub-idealworldandideal/sub-idealversionsofaworld,butweomit
WerepresentedExample 2inthefollowingway:
(a) O
i :dog
(b)
!
(:dog!O
a :sign)
(c)
!
(dog!O
a sign)
(d) dog
All therequirements(i){(viii) are metby this analysis. Inparticular, the
pragmatic oddity disappears because the conjunction O
i
:dog ^O
a sign,
which is clearly derivable from (a){(d), does not imply that, in all ideal
versionsofthegivenworld,thereisnodogbutasignwarningofone. What
the conjunctiondoes say, essentially, isthat in allideal worlds there isno
dog,but in allidealversions ofthegiven(clearlysub-ideal)worldthereis
a warningsign. The proposal worked well forthis and anumberof other
examples,andwehavedened acompleteaxiomatizationforthelogic.
However, as we now see things, this approach suered from a defect
similar to the one we have criticised in relation to [
Prakken and Sergot,
1994; Prakken and Sergot,1996 ]
: theassignmentof logicalform for some
ofthenormsin thesetisdependentontheother normsinit. In [
Prakken
and Sergot, 1994; Prakken and Sergot,1996 ]
this was reected in theuse
of dierent obligation operators for representing the deontic conditionals
expressedbylines(b)and(c);in
[Carmo
andJones, 1995]
itisreectedin
theuseofdierentobligationoperatorsforrepresentingline(a)andlines(b)
and (c). So, in orderto capturethegeneral issuemotivating theadoption
ofadequacyrequirement(iv),itshouldbereformulatedasfollows:
(iv) theassignmentoflogicalform toeachofthenormsin theset should
beindependentoftheothernormsin it.
Arelatedobservationisthatproblemsappearwithinthe
[Carmo
andJones,
1995]
approachifweadd totheChisholmsetother normsthatinteract in
somesignicantwaywiththenormsintheoriginalset. Inparticular,serious
diÆcultiesariseassoonasa\second-level"ofCTDsisconsidered.Suppose,
forinstance, thatlines(e)and(f),below,areaddedto Example2:
(e) Ifthereisadogand nowarningsign,thereoughttobeahighfence.
(f) Thereisnowarningsign.
The
[Carmo
andJones, 1995]
representationofthisextended setis:
(a) O
i :dog
(b)
!
(:dog!O
a
:sign)
(c)
!
(dog!O
a sign)
(d) dog
(e)
!
(dog^:sign!O
a fence)
(f) :sign
Andthepragmaticodditynowre-appears,sincetheconjunctionO
a sign
^ O
a
fenceis derivable. So,in all idealversions of thegivenworld, there
is asignand afence. (If thisdoesnotseem\odd", imaginethat thesign
says\Bewareoftheunfenceddog": itmaywellbeforbiddentohaveboth
asignofthat kindandafence. Thusthepragmaticoddity,in thesenseof
ourproposeddiagnosis,re-emerges.)
The problem of \further levels" of CTDs would force the
[Carmo
and
Jones, 1995]
approach to allow the possibility of an innity of obligation
operators: the needto associate(in somewayorother) acontextto each
obligationoperatorseemsto re-appear.
4 CONTRARY-TO-DUTIES:ANEWAPPROACH
On one very common interpretation of the set (a){(f) above, the actual
obligationwhichapplies inthecircumstancesis theobligationto put upa
fence,and itappliesbecausetheothertwoobligations(notto haveadog,
and to put up a sign if there is a dog) have been violated. As we have
emphasisedabove,it would beincorrectto saythat theobligationsnotto
have a dog, and to put up a sign if there is a dog, have been defeated,
or overturned; they have been violated, and any proper representation of
the situation must register the fact that, because of these violations, the
obligationwhichbecomesactualistheobligationtoerectafence. Buthow
arethese pointstobearticulatedinaformaltheory? Tothatquestionwe
nowturn.
4.1 Motivation
Consideragain Example 2,particularly lines (a), (c) and (d). The norms
governing, or in force in, thesituation are that there ought to beno dog,
andthatifthereisadogthereoughttobeawarningsign;andtherelevant
fact is that there is adog. So what is the actual obligation, of the agent
concerned,in these circumstances? Toerectawarningsign? Butwhynot
insist ongettingrid of thedog,rather thanon erectingawarningsign 12
?
Wewish to suggest that theanswerto such questions turns on thestatus
12
Rememberthat,inkeeping withour analysisofthe pragmaticoddity,weseek an
answerto thesequestions whichiscompatiblewithafurtherassumptionto the eect
assignedtothefactthatthereisadog|inthefollowingsense: solongas
there is a dog,but this, forone reasonoranother, is not deemed to be a
xed,unalterable featureof thesituation,thentheactualobligationwhich
appliesisthatthereoughttobenodog. However,assoonas,foronereason
oranother, thefactthat there isadogis deemedxed,i.e., itisseenasa
necessary, unavoidablefeature of the situation,so that | in consequence
| thepracticalpossibility ofsatisfyingthe obligationthat thereoughtto
benodoghastoallintentsandpurposesbeeneliminated,then theactual
obligationwhichappliesisthatthereoughtto beawarningsign.
13
Whatdowemeanwhenwesaythatforsomereasonoranotherafactof
thesituation |inthis casethat thereisadog|maybedeemedaxed,
necessary,unalterablefeatureofthatsituation? Well,therearevariousways
in which this \xity" mightarise; those whoproposed temporal solutions
to the problems associated with CTDs focussed on one of these ways. If
books shallbereturnedbydatedue, then ifyoustill havethebooks after
thedateduethereisnowaythatobligationcanbemet. Itistoolate! Itis
unalterablythecasethatthebooksarenotreturnedbythedate due,and
consequentlythepossibilityofsatisfyingtheobligationtoreturnthebooks
bydateduehasbeeneliminated.
But temporal reasons,although verycommon, arenottheonly reasons
whythingsbecomexed,inthesenseofnecessityorunalterabilitywehere
seek toexplicate;forinstance, itisnotfortemporalreasonsthatthedeed
of killing, once done, cannotbeundone. What explains xity in this case
is not temporal necessity, but rather causal necessity. Norneed temporal
considerationshaveanyroletoplayinexplainingwhythepresenceofadog
maybe,toallintentsandpurposes,anunalterablefeatureofthesituation;
itmay,forsomereason,bepracticallyimpossibleinthesituationtoremove
the dog; perhaps, for instance, its ownerstubbornly refuses to removeit,
andnobodyelsedares attemptthefeat. Thepresenceof thedog isaxed
fact: the dog remains unless the intervention of some agent leads to its
removal,andnoagentispreparedtoperformtherequiredaction. Fromthe
practicalpointofview|fromthepointofviewofdecidingwhichobligation
actuallyappliestothesituation|thekeyfeatureisthatthepossibilityof
satisfactionoftherequirementthattherebenodogiseectivelyeliminated.
As afurther illustration, consider nexttheexample ofthe \considerate
assassin".
14
EXAMPLE4.
(a) Youshould notkill Mr. X.
13
Someremarksinasimilarspiritaretobefoundin
[Hansson,
1971,sectionXIII:\on
theinterpretationofcircumstances"
]
.
14
Thisexamplecanalsobefoundin
[Prakken
andSergot, 1996]
(using\thewitness"
insteadof\Mr. X").Infact
[Prakken
andSergot, 1996]
providesanexcellentsurveyof
the principalexamplesofCTDs,andweusetheminSection5totestthe adequacyof
(b) -
(c) ButifyoukillMr. X,youshould oerhimacigarette.
WhendoestheassassinhaveanactualobligationtooerMr. Xacigarette?
Afterkillinghim? Butthenitistoolate! Oneintuitivelyacceptableinter-
pretationisthat theassassin'sactualobligationto oerMr. X acigarette
ariseswhenhermlydecidesthatheisgoingtokill Mr. X.It isthenthat
itbecomesasettled orxedfact that Mr. Xwill bekilled, andthen that
theassassin'sactualobligationisto oeracigarette.
Noticethattheexamplesindicate thattwodierentnotionsofnecessity
| and their associated notions of possibility | need to be considered.
Mr. X, once killed, cannot beoered acigarettebecausenobody hasthe
abilityor theopportunity to makeoersto thedead, just asnobody has
theabilityoropportunitytoreturnabookbydatedueifthedateduehas
passed. On theother hand,thedog-ownermay haveboth theability and
the opportunity to remove the dog, and the assassin may have both the
ability and the opportunity to refrain from killing Mr. X; but once each
hasmade armanddenite decision(to keepthe dogand to kill Mr. X,
respectively),thentoallintentsandpurposesthepersistentpresenceofthe
dogand thefuture performance oftheassassinationbecomexed features
of the respective situations; so questions about which actual obligations
arise in these situations have to be answeredin thelight of the fact that
alternativesin whichthere isno dog,orno assassination,arenotactually
available.
Nowitmaywellbethatthejudge,attheassassin'strial,insiststhatthe
assassin should neverhave decided to commit the murder, just as it may
bethat the managerof the housing estate refuses to acceptthat the dog
ownerwas entitled to decidethat he would keephis dog. Furthermore,it
is a well known feature of, for instance, disputes in legal cases, that the
partiestothedisputemaydisagreeaboutwhatanagentwasabletodo,or
what hehadthe opportunity todo. Buttheexistenceof disagreementsof
thesekindsisperfectlycompatiblewiththeapproachtoCTDscenarioswe
developbelow. Foritwillnotbethetaskofourlogicalsystemtodetermine
thereasonswhich justifytheclassicationofsomefactassettled. Rather,
whatthesystemwilldoisthis: rst,itwillspecifytheroleofassumptions
about twotypes of xityin reasoning aboutactual and ideal obligations;
and,second,itwillshowwhichactual/idealobligationscanbederivedfrom
agivenset ofnormswhensomefacts aretakentobexedin theone sense
orthe other.
15
15
OurthankstoLaymanAllenforraisingaquestionattheSesimbraEON96Work-
4.2 Thenew theory and its fundamental semantic features
Wenowpresentthebasicfeaturesof amodal-logicallanguagedesignedto
capture theapproach toCTDs describedabovein awaythat conformsto
theconstraints,orrequirements,(i){(viii). Weshallthenshowhowthenew
languagemaybeapplied to theChisholmset, andto theanalysisof some
otherproblematic CTD\scenarios".
We adopt the following approach to the formal representation of these
scenarios: theirdeonticcomponent(theobligationnormswhichtheyexplic-
itlycontain)willberepresentedthroughoutintermsofadyadic,conditional
obligationoperator O(/);theirfactual componentwill berepresentedby
meansof eitherunmodalised sentences,ormodalisedsentencesin twocat-
egories. These two categories correspond to the two notions of necessity
(andtheirassociateddualnotionsofpossibility)which weshallemployto
articulate theideasregardingxity,orunalterability,offactsalludedto in
theprevioussubsection.
Fromthe deonticand factual componentstakentogether, somefurther
obligation sentences may be derivable. The derived obligation sentences
are oftwotypes,pertaining to actualobligationsandideal obligations,re-
spectively. There is an intimate conceptualconnectionbetweenthese two
notionsofderivedobligation,ontheonehand,andthetwonotionsofneces-
sity/possibilityusedin characterisingthefactualcomponent,ontheother.
Consider rst the dyadic conditional obligation operator. How do we
wishtointerpretasentenceofthekind\ifthereisadogthenthereshallbe
awarningsign"? Onourview,this sentenceis tobeunderstoodas saying
that in anycontext in which the presence of adog is a xed, unalterable
fact,itisobligatorytohaveawarningsign,ifthisispossible. Wethinkof
a contextasa set of worlds| the set ofrelevant worlds forthe situation
at hand. Sotheabovesentenceisto beunderstoodassayingthat,forany
contextinwhichthereisadog(i.e.,foranycontextinwhichthereisadog
in each worldofthat context),ifitispossibletohaveawarningsignthen
it isobligatory to haveawarningsign.
16
Inorder to capturethis ideawe
introducein our modelsa function ob:}(W)!}(}(W))which picks out,
foreachcontext, thepropositionswhichrepresentthatwhichisobligatory
in thatcontext. Thatis,kBk2ob(X)(wherekBkdenotesthetruthsetof
B in themodelin question) ifand onlyif thepropositionexpressed byB
representssomethingobligatory in context X. Accordingly, wesay that a
sentenceO(B=A)istrueinamodelifandonlyif,inanycontextX where
AistrueandB ispossible(i.e. in anycontexthavingAtrueineachofits
16
Weare here usingthe term \obligatory"ina weak sense; inastrict sense, for a
sentenceBtobeobligatoryinacontextX wewouldalsoclaimthattheremustexist
atleastoneworldinX where Bisfalse(i.e.,wewouldinsist,forthestrictsense,that
obligationsmustbeviolable). However,ouractualandidealobligations,tobedened
worldsandB trueinatleastoneofitsworlds),itisobligatorythatB.
17
On the basisof this operator we could now derivethe obligations that
were applicable in each context, assuming that our languagecontained a
means of representing contexts. The question is: what are the types of
contextsthatweneedtobeabletotalkaboutinourformallanguage,given
thatwewantto beabletoderivesentencesoftwokinds,describingactual
obligationsandideal obligations, respectively? Weanswerthis questionin
termsofthetwonotionsofnecessity.
Therst ofthese will be denoted by
!
, and itsdual possibilitynotion
denoted by
!. Intuitively,
!
is intended to capture that which | in
a particular situation | is actually xed, or unalterable, given (among
otherfactors)whattheagentsconcernedhavedecidedtodoandnottodo.
In, for instance, the \dog scenario" (Example 2), if the agents concerned
have rmly decided that the dog is not going to be removed, then the
sentence
!
dogistrueofthatsituation(thepresenceofthedogisactually
anunalterablefact). Ontheotherhand,ifsomeactualpossibilityexistedfor
gettingridofthedog,thenthesituationwouldbeappropriatelydescribed
by
!
:dog(i.e. :
!
dog). Whichactualobligationsariseinthedogscenario
willdepend,in particular,onwhetherornot
!
:dog istrue.
Wemustemphasise oneimportantdierencebetweenthis notionofne-
cessity/possibilityandthesecondoneweemploy. Forthereasonsdiscussed
in theprevioussubsection,wedonotexclude, apriori, that asentence of
the form
!
A might be true even though the agents concerned have the
abilityandtheopportunitytosee toitthat :A. Thatis, weshallwantto
considerscenarioswhere,despitetheirabilitiesandopportunitiesforaction,
theagentshavermlyresolvednottoseetoitthat:A,andwhere|given
that this is what the agents have decided | there is (for all intents and
purposes)nowaythatAcouldbefalse.
In order to capture the semantics of thenecessity operator
!
, ourse-
manticalmodelswillcontainafunction,av,whichpicksout(foranygiven
world w) a set of worlds av(w) | the set of worlds which are the actual
versionsof w (the open alternatives forthe currentworldw),those which
constitute the context that it is actually relevant to takeinto accountin
determiningwhichobligationsareactuallyinforce,oractuallyapply,atw.
Accordingly,asentenceoftheform
!
A willbesaidto betrueatagiven
worldwifandonlyifAistrueat alloftheworldscontainedinav(w).
Given the way the function ob is understood, the set of propositions
ob(av(w))willbethesetofpropositionswhichrepresentthatwhichisoblig-
atoryinthecontextav(w)(thatistosay,inthecontextofthealternatives
thatareactuallyopenatw). Inlinewiththis,weshallsaythatasentence
17
WealsoaddthefurtherrequirementthattheconjunctionofAandBisnotcontra-
dictory,inordertoavoidsome\absurd"vacuousconditionalobligations,andasoneof
oftheformO
a
A(readas\itisactuallyobligatorythatA",or\itactually
oughtto bethe casethat A") istrue ata worldw onlyifthe proposition
expressed byA isoneof thosepropositionspickedoutbyob fortheargu-
mentav(w). Inaddition, thetruthof O
a
A at wwillrequirethat thereis
atleastoneworldinav(w)wherethesentenceAisfalse;thereasonforthis
secondrequirementisthatthatwhichisactuallyobligatorymightactually
fail toobtain.
The second of thetwonotions ofnecessity will be denoted by
, and
its dual possibility notion denoted by
. Intuitively,
is intended to
capturethatwhich|inaparticularsituation|isnotonlyactuallyxed,
but would still be xed even if dierent decisionshad beenmade, by the
agentsconcerned,regardinghowtheywere goingto behave. Forinstance,
certainfeaturesofthesituationwill besuch thatitisbeyondthepowerof
theagentstochangethem|theymaylacktheability,ortheopportunity,
or both. Of such features it is appropriate to say that they are xed in
the sense that theycould not have been avoided by theagentsconcerned,
no matter what they had done. It is noteven potentially possible for the
agentsto alterthem. IntheoriginalChisholmscenario,forexample,ifthe
bridge that leadsto theman's neighbours'house hasbeendestroyedby a
storm, and the man is unable to repair it, then clearly it is a necessary
feature of thesituation,in thissecond sense ofnecessity,that hedoesnot
helphisneighbours. Thisistobeunderstoodincontrasttothesituationin
whichitispotentiallypossiblefortheagenttogoto hisneighbours'house
to help them, but the agenthas made a denite decision, from which he
will not budge, notto go. His will is rm, and thus in allactual relevant
alternativesopentotheagenthedoesnotgotohelpthem(andsoactually
he oughtnottotellthem heiscoming). Butgiventhathe hastheability
andopportunitytogo,itispotentiallypossible|inthesenseexpressedby
the
operator|thathedoesso 18
,andsowewantthenbeabletoderive
that hisideal obligationwasto goand totellthem hewascoming.
Toarticulatethesemanticsofthesecondpairofnotionsofnecessityand
possibility, we introduce into the models a function, pv, which picks out
(for anygiven world w) aset of worlds pv(w) | the set of worldswhich
arethep otentialversionsof w. These worldswillconstitutethecontextit
18
So the best short readings for these two pairs of operators we can oer are the
following:
!
A: itisactuallypossiblethatA
A: itispotentiallypossiblethatA
!
A: itisnotactuallypossiblethat:A
A: itisnotpotentiallypossiblethat:A
(Inanumberofcases,thenaturalreadingofstatementsaboutpotentialpossibilitywill
isrelevantto takeintoaccountin determiningwhichidealobligationshold
at w (\what should have been done"). A sentence of the form
A will
besaidto betrueat agivenworld wif andonly ifA istrueat all ofthe
worlds containedin pv(w). Furthermore,giventhe waythefunction obis
understood,thesetofpropositionsob(pv(w))willbethesetofpropositions
whichrepresentthatwhichisobligatoryinthecontextpv(w). Thusweshall
saythat asentenceoftheform O
i
A (readas\itisideallyobligatory that
A", or \it ideallyoughtto bethe casethat A") is trueat aworldw only
ifthepropositionexpressedbyAisoneofthosepropositionspickedoutby
obfortheargumentpv(w). Inaddition,thetruthofO
i
Aatwwillrequire
that there is at least oneworldin pv(w) where thesentenceA isfalse |
sincethatwhich isideallyobligatory mightpotentiallyfail toobtain.
Finally,wedenethe notionof violationin termsof thenotionofideal
obligation,asfollows:
viol(A)=
df O
i A^:A
This choice is in accordance with the intuitiveidea that ideal obligations
expresswhatshould havebeendone,andtsinwellwithourtreatmentof
thepragmaticoddityand otherfeatures ofCTDscenarios,aswillbecome
clearerwhenweanalyseanumberofexamplesin somedetail. Briey,the
main pointsmayalreadybeexplainedasfollows: in, forinstance, thedog
scenario,ifitis axedfact that there isadog(i.e., if
!
dogistrue), but
it is actually possible that a sign may be erected and potentially possible
that there is no dog, then we shall be able to derive that it is actually
obligatory that a sign is erected and ideally obligatory that there is no
dog. The pragmatic oddity will be avoided because it will not, in these
circumstances, be possibleto derivean actual obligationthat there beno
dog. Nevertheless, westill of coursewant to say that an obligation (that
therebenodog)hasbeenviolated,andthisresultissecured ifviolationis
characterised as above. As theformal analysis of this example will show,
weshallalsobeabletoderiveasecondviolationinthissituation,ifnosign
hasbeenerected.
Thesemanticmodelsdescribednextwillbesubjecttovariousconstraints,
designedtoachieveaparticularpatternofrelationshipsbetweenthedyadic
obligation operator, the two types of necessity/possibility operators, and
theoperatorsforactualandidealobligations.
4.3 Syntax and semantics of the formal language
Syntax
aset of (natural language) terms (dog, fence, sign, ...) for atomic
sentences
:,^,_,!,$ (sententialconnectives)
(, ) (parentheses)
!
(dual:
!
=
df :
!
:)
(dual:
=
df :
:)
O(/) (dyadicdeonticoperator)
O
a
(monadicdeonticoperator-foractualobligation)
O
i
(monadicdeonticoperator-foridealobligation)
Rulesfor constructionof well-formedsentences: asusual
viol(A)=
df O
i A^:A
Semantics
Models:
M=hW;av;pv;ob;Vi,where:
1) W 6=;
2) V -afunction assigningatruthset toeach atomicsentence
3) av:W !}(W)
(alternatively: R aW W andav(w)=fw
1 :wR
a w
1 g)
suchthat:
3-a) av(w)6=;
4) pv:W !}(W)
(alternatively: R pWW andpv(w)=fw
1 :wR
p w
1 g)
suchthat:
4-a) av(w)pv(w)
4-b) w2pv(w)
5) ob:}(W)!}(}(W))
suchthat (whereX;Y;Z designatearbitrarysetsofmembersofW):
5-b) ifY \X=Z\X,then(Y 2ob(X)iZ 2ob(X))
5-c) ifY;Z 2ob(X);thenY \Z 2ob(X)
5-d) ifY XandY 2ob(X)andX Z,then((Z X)[Y)2ob(Z)
Truth inaworldwinamodelM=hW;av;pv;ob;Viischaracterisedas
follows(wherekAk=kAk M
=fw2W :Mj=
w Ag):
Mj=
w
p i w2V(p)
...(the usualtruthconditionsfortheconnectives:,^,_,!and$)
Mj=
w
!
A i av(w)kAk
Mj=
w
A i pv(w)kAk
Mj=
w
O(B=A) 19
i
kAk\kBk6=;and
(8X)(ifX kAkandX\kBk6=;,thenkBk2ob(X))
Mj=
w O
a
A i kAk2ob(av(w))andav(w)\k:Ak6=;
(i.e. i kAk2ob(av(w))andav(w)\(W kAk)6=;)
Mj=
w O
i
A i kAk2ob(pv(w))andpv(w)\k:Ak6=;
(NotethatthedenitionofMj=
w
O(B=A)entailsthatifMj=
w
O(B=A),
thenkBk2ob(kAk).)
AsentenceA issaidto betrue in amodel M=hW;av;pv;ob;Vi, written
Mj=A,ikAk M
=W;andAissaidtobevalid, writtenj=A,iMj=A
inallmodelsM.
Somecommentsabouttheconditions:
i) As would be expected, the set av(w) is required to be a subset of
pv(w), for any w (condition 4-a)), so that actual possibility entails
potentialpossibility. Conditions3-a)and 4-b)arealsoobvious.
In
[Carmo
and Jones, 1997]
20
we required that w 2 av(w) (which
implies 3-a)and, togetherwith 4-a),also implies4-b)). Althoughin
19
Analternativewouldbetodenethedyadic obligationoperatorinthestrictsense
referredtoinfootnote16,inwhichcaseMj=
w
O(B=A)ikAk\kBk6=;andkAk\
k:Bk6=;and(8X)(ifXkAkandX\kBk6=;andX\k:Bk6=;,thenkBk2ob(X)).
InthatcasewewouldrequirethatifY 2ob(X)thenX\(W Y)6=;;andthetruth
inaworldwofO
a
A(respectivelyO
i
A)wouldbedenedasfollows:Mj=wO
a Ai
kAk2ob(av(w))(resp. Mj=
w O
i
AikAk2ob(pv(w))). With bothapproacheswe
getexactlythesamesemanticsforO
a
AandO
i A.
20
Whereweuseva,vp andpi,instead ofthemoresuggestivenames,av,pvandob,
mostscenariositmakessensetosaythattheactualworldisalwaysan
actualalternativetoitself, wesometimes needto beableto describe
situationswhereAisnotyetthecase,but neverthelessinallrelevant
future alternatives open to the agent, A is the case. Consider, for
instance,thescenarioofthe\considerateassassin"(Example4): there
maybesituationswhere,althoughtheassassinhasnotyet killedMr.
X, in alltherelevantfuture alternativesopento the assassinMr. X
isgoingtobekilledbyhim(becausetheassassinhassodecided);the
naturalwayto representthissituation in ourlogicis: :kill ^
!
kill.
But of course this can only be expressed consistently if we do not
requirethat,forallw;w2av(w).
Thereexistotherconditionsthatitmayseemnaturaltoimposeonav
andpv,suchasthetransitivityofboththeactualandpotentialrela-
tions. But,forsimplicity,weconsiderhereonlythoseconditionswhich
appear to haveadirect bearing onthe analysis of thekeyexamples
ofCTDscenariosinSection 5.
ii) Condition5-a)meansthatwedonotacceptthatacontradictionmight
beobligatory.
iii) Condition5-b)meansthat if,fromthepointofviewof acontext X,
twopropositionsY and Z are indistinguishable,then oneofthem is
obligatoryitheotheris(thiscorrespondsto akindof\contextual"
RE-rule).
It is alsoappropriate at this point to statesome of themain conse-
quencesofcondition5-b),andto attachnumberedlabelsto them,in
ordertofacilitatelaterdiscussionofapossibleweakeningof5-b).
SinceY \X =(Y \X)\X,wegetasparticularcasesof5-b):
5-b1) ifY 2ob(X),thenY \X 2ob(X)
5-b2) ifY \X 2ob(X),thenY 2ob(X)
Ontheotherhand,using5-a)and5-b1)wegetthecondition(which
inturnimplies5-a)):
5-ab) ifY 2ob(X),thenY \X 6=;
iv) Condition5-c)requiresthattheconjunctionoftwoobligatorypropo-
sitionswithinacontextXisalsoobligatoryinthatcontext. Anatural
extensionofcondition5-c)wouldbetorequiretheclosureofobunder
arbitraryintersections(andnotonlyunderniteintersections):
5-c+) ifob(X)and6=;,then( T
)2ob(X)
(where isanysetofsubsetsofW,( T
)isdened as:
T
If we impose thisstrongerconditionthen wewould get( T
ob(X))2
ob(X), ifob(X) isnon-empty; note alsothat, by 5-b1), ifob(X)6=;
then ( T
ob(X))X; if ob(X)=; then, by denition,( T
ob(X))=
W. However,forreasonstobeexplained,whatweshallinfactpropose
laterisaweakeningofcondition5-c.
v) Condition5-d)statesthat ifasubsetY ofX isanobligatorypropo-
sitioninacontextX,theninabiggercontextZitisobligatorytobe
either inY orelseinthat partofZ whichisnotinX.
Takingintoaccountcondition5-b),itmaybeshownthateachofthe
followingconditionsisequivalentto5-d)-theycansometimesbeused
to simplifyproofs:
5-bd1) ifY X andY 2ob(X)andX Z,then((W X)[Y)2
ob(Z)
5-bd2) ifY 2ob(X)andX Z,then((Z X)[Y)2ob(Z)
5-bd3) ifY 2ob(X)andX Z,then((W X)[Y)2ob(Z)
5-bd4) ifY 2ob(X)andXZ,then((W X)[(X\Y))2ob(Z)
Usingconditions5-b),5-c)and 5-d),wecanalsoprovethat
ifZ2ob(X)andZ 2ob(Y),thenZ 2ob(X[Y).
4.4 Syntactic/axiomatic characterisation of the modal
operators
In what follows we introduce theaxioms and rules for the various modal
operators. Forsomeoftheaxiomsandtheoremsweintroducespeciallabels
| ifthere is no standardlabel| in order to facilitate reference to them
lateron.
Characterisation of
:
1.
isanormalmodaloperatoroftypeKT.
Characterisation of
!
:
2.
!
isanormalmodaloperatoroftypeKD.
Relationship between
and
!
:
3.
A!
!
A (axiomschema(
!
!
))
4. :O(?=A) (the schema:N forO :(O :N))
5. O(B=A)^O(C=A)!O(B^C=A)
(theschemaC forO :(O C))
6. Restrictedprincipleofstrengtheningoftheantecedent-1:
O(B=A)!O(B=A^B) (SA1)
7. theRE-rule withrespectto(w.r.t.) theantecedent:
if`(A$B)then`O(C=A)$O(C=B)
8. the\contextual RE-rule"w.r.t.theconsequent:
if`C!(A$B)then`O(A=C)$O(B=C)
Relationship betweenO and
:
9.
O(B=A)!
O(B=A) (
O !
O)
10. Restrictedprincipleofstrengtheningoftheantecedent-2:
(A^B^C)^O(C=B)!O(C=A^B) (SA2)
Characterisation ofO
a /O
i :
11. O
a A^O
a
B !O
a
(A^B) (O
a C)
O
i A^O
i
B!O
i
(A^B) (O
i C)
RelationshipsbetweenO
a
(respectively: O
i )and
!
(resp.:
):
12.
!
A!(:O
a A^:O
a
:A) (:O
a )
A!(:O
i A^:O
i
:A) (:O
i )
13.
!
(A$B)!(O
a
A$O
a
B) ($O
a )
(A$B)!(O
i
A$O
i
B) ($O
i )
RelationshipsbetweenO;O
a
(resp.: O
i )and
!
(resp.:
):
14. Restrictedfactual detachment:
O(B=A)^
!
A^
!B^
!:B!O
a
B (O
a
FD)
O(B=A)^
A^
B^
:B!O
i
B (O
i
FD)
15. O(B=A)^
!(A^ B)^
!(A^:B)!O
a
(A!B) (O !O
a
!)
O(B=A)^ (A^B)^ (A^:B)!O
i
(A!B) (O !O
i
!)
