Inferences with Ignorance: Logics of Questions (Ukázka, strana 99)

98

3 and 4) admits to having not only declaratives among direct answers. This was important for ‘questions about questions’ and for the analysis of I don’t know answers. We did not study this problem in the chapters on epistemic logic, but we mentioned it in the last paragraph of Section 1.4.

5.2 MAIN RESULTS The 󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 ni󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 approach make itpossible tomodelquestionsbytheir semantic range.1 The bene󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 interestandstudywasprimarilyfocusedoninferenceswith questions.Many theoremsprovedinChapter2provide similarorthe same resultsasthe theorems of the original inferential erotetic logic. However, we often obtained the results in another way.2 Letusnowlistsome interestingandnewresultsthat canbe understoodasourcontributiontothe developmentofIELvariants:

• The semantic range of questions plays a role in pure erotetic implication.

The relationofpure e-implicationbetweentwoquestionscausesboth questionsto have the same semanticrange (Proposition 6).Asaconsequence we obtainthatclassesofquestions(normalandregular)are closedunderpure e-implication(Propositions 8 and 9).

• We were exceptionallyinterestedinrelationshipsamongconsequence relationswith questions. Proposition 12showsthat evocation is preserved under regular e-implication.

• The concept of giving an answer establishessimilarrelations. For example, Proposition 15 connectsregular e-implicationand evocation.

• Inthe lastpartofChapter2, we bringintoplaythe reducibility ofaninitial question to a set of questions. We wanted to be general enough and combine erotetic consequences with the concept of reducibility. Regular e-implicationand evocation(with respectto ‘givingananswer’)are connectedwith reducibility, see Propositions 19 and20.Inthe lastsubsectionofChapter2(Reducibilityandsetsofyes-noquestions), we studied the generalizedconditionsofthe reducibilityofaninitialquestiontoaset ofyes-noquestions. The establishedresultscorrespondtothe resultsin the originalIEL, see the referencesinthe subsection.

The secondpart(Chapter 3 andChapter 4)isdifferentinitssubstance. The epistemicframeworkformsthe background, andaquestioncorrespondstoa

1 The inspirationcomesfromamixture ofIELandpartitioningconceptin Groenendijk-Stokhof’s approach, cf.Section 1.3.2.

2 Referencescanbe foundinthe Chapter2.

inferences.indd 98

04.03.2016 23:07:40 Ukázka elektronické knihy, UID: KOS228208

Conclusion

99

certain state of ignorance and presuppositions of an agent. This approach is a novelty inspired by epistemic terminology often used by erotetic logicians.3 Questions become a part of epistemic language and they can be considered as ‘satis󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 a stateof a Kripkemodel(weuse the term askability in the state) aremore complex than in the case of declaratives, and they are based on three items (non-triviality, admissibility, and context). Although we introduced a general epistemic-like approach in Chapter 3, wehavedecided then touse a 󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 ant of SAM. Thedecisioncorresponds toour idea of ‘well-arranged’communications. At the beginning, wehad also wanted toprovide a very general approach to questions’formalization inepistemicframework, but werecognized that wehad to becautious of it. In theend, as an appropriate backgroundformalizationforpublic announcement logic with questions themulti-modallogic S5 was chosen, which is common in this area (Chapter 4). Somelogicians might consider S5 undesirable. Irrespectiveofobjections against epistemicproperties (e.g., thelogicalomniscienceproblem), our questions’ analysis is ‘reductionism’ in its nature—questions are representable by a complex modal formulas in S5.

However, it has some advantages. Wecanfully exploit theproperties ofpublic announcement logic, in particular, successful formulas and updates that can be problematic and sometimes must berede󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 Chapter 3 tries to develop the basics of erotetic epistemic logic:

• The starting erotetic term in the single-agent epistemicframework is the askability of a question by an agent in a state. This concept establishes individual answerhoodconditions, i.e., conditions under which questions arecompletely orpartially answeredfor an agent in a given state. The relationshipof bothpartial andcomplete answerability depends on the formof a set ofdirect answers (Fact 30 and Proposition 26). As a result we have questions, whichmeet Hamblin’s thirdpostulate, that areprominent.

• In building theepistemiclogicof questions, we tried to 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 notions as wehave in the IEL framework. Wede󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 presuppositions and prospective presuppositions of a question(Section 3.2.1). The term safe question in global as well as localform is basedon it (De󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 and 22).

• Inferential structures with questions arede󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 dard implication(Section 3.3). In Section 3.5 we studied theproperties of ‘erotetic implications’ withrespect to a set of auxiliary formulas (relativized askability) and, in thelast section(3.6), we introduced a variant of ‘reducibility’of a question to a conjunctionof yes-no questions.

Multi-agent epistemiclogic with questions in Chapter 4 introduces group erotetic terms.

3 The semantic approach is inspired by Groenendijk’s and Stokhof’s papers.

inferences.indd 99

04.03.2016 23:07:40 Ukázka elektronické knihy, UID: KOS228208

100

• We de󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮 󰅮󰅮󰅮󰅮󰅮 (De󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 answerhood conditions. Implicit answerability, stated in De󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 wasbased onthe distributive knowledge operator. Commonknowledge isthenimportantinthe case ofthe answerabilityofaquestionforagroup ofagents.

• The dynamic steptakesplace inthe frameworkofpublic announcement logic (builtoverthe modallogic S5). We proved that

– publicly asked questions are successfulformulas, sotheyare commonlyknownafterbeingpubliclyannounced (Proposition35), – anaskable questioninastate isasuccessfulupdate there (Fact38).

• Inadditiontothatwe introduced a conceptofformulainformative foran agentwithrespecttoaquestion (De󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮 swerabilityafteranupdate.

5.3 WEAK POINTS AND PROBLEMS 5.3.1 SAM AND CONTEXT CONDITION OF ASKABILITY

One ofthe problematic pointsisthe introduced set-of-answers methodology.

Criticshave aimed atthe impossibilityofsucharepresentationto modelall kindsofquestions. Our effortwasprimarilytobuild transparent communicationamongagents. Thisled ustoa versionof SAM where questionshave 󰅮󰅮󰅮󰅮󰅮󰅮󰅮 setsof directanswers. However, there are manyquestionsthathave well de󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 Whichnaturalnumberisgreaterthan 13?

canrequire ananswerfrom anin󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 number(s)greaterthan 13). Ourintentiontoprovide complexinformationon the ignorance and expectationsofanagent, publiclyaskingaquestion, connect the problem of SAM withthe conceptof askability in erotetic epistemic logic. The expectationofanagentis mirrored inthe third conditionof De󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 text condition)whichrequiresthat, ina 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮󰅮 󰅮󰅮󰅮󰅮󰅮 the disjunctionofall directanswers. Thisplacesarestrictiononthe epistemic possibilitiesofanagent-questioner. Nevertheless, agentsoftenaskquestions withoutbeingaware ofallpossible directanswers.4 We metthisproblem in anapplicationof erotetic epistemic logic in [49] and resigned ourselvestothe context condition. However, we feelthe need to discussthe role of ‘context’ as 4 Personal communicationwith Andrzej Wiś niewskiand Igor Sedlá r. Igor mentioned aquestion Where is Bill? Askingit, anagentneed notknowallthe possibilitiesof Bill’sposition.

inferences.indd 100

04.03.2016 23:07:40 Ukázka elektronické knihy, UID: KOS228208