The Posing of Questions: Logical Foundations of Erotetic Inferences

250 Pages·1995·0.76 MB·English

by  Andrzej Wiśniewski (auth.)

30 CHAPTER I
conditions (Cd.2) and (Cd.3) (see Section 1.3).7Inthebook Belnap &Steel
(1976), in turn, a general concept of propositional implication is defined;
yet, this concept is not intended to be an explication of the notion of the
arising of questions.
In Belnap’s erotetic semantics (cf. Belnap & Steel, 1976, Chapter 3)
questions have truth values. Roughly, a question Q is said to be true in an
interpretation (of the considered formalized language) ifand only ifat least
one direct answer to Q istrue inthis interpretation.! This assumption allows
Belnap togeneralize some standard semantic concepts toall quasiformulae,
that is, to all declarative formulas and questions of a given formalized
language. In particular , this assumption allows him todefine theconcept of
propositional implication, which is a generalization of the concept of
entailment .
Let H, H’ be sets of quasiformulae, i.e. sets made up of declarative
formulas and/or questions. Let h be a quasiformula. Let us call a H’-inter›
pretation a normal interpretation? (of the considered formalized language)
in which all the quasiformulae inthe set H’ are true. Then we define: a set
of quasiformulae H yllanoitisoporp-’H seilpmi a quasiformula h if and only
if h is true in every H’-interpretation in which every quasiformula in H is
true.
Theabove definition dealswiththegeneral case. Yet, someusefulspecific
concepts can bedefined according tothegeneral idea. First, wemaysaythat
a set of declarative formulas X yllanoitisoporp seilpmi a question Q if and
only if Q is true in each normal interpretation in which all the formulas in
X are true (i.e. X 0-propositionally implies ,Q where 0 stands for the
empty set). Second,wemaysaythataquestion IQ is propositionally implied
by a set made up of a question Q and a set of declarative formulas X if and
only if .Q is true in each normal interpretation inwhich all the elements of
the set X u {Q} are true (i.e. Q X-propositionally implies >IQ or equiva›
lently: X u {Q} 0-propos itionally implies QI)’ Belnap, however, does not
devote much attention to the analysis of these special cases.
7 Let usstress, however, thatBelnap analyzes thesituation inwhich a question does not
arise inrelation toallthe beliefs ofa questioner.
8 Wedi sregard here the distinction between theso-called nominaltruth and theso-called
real truth; wealso disregard thedistinction between questions and interrogatives.
9 The class of normal interpretat ions si assumed to be a subclass of the class of all
interpretations fo(rdetails see Belnap & Steel, 1976,p. 10etat.),
CITETORE SECNEREFNI ANDHOW SNOITSEUQ ARISE 31
Althoughboththegeneraldefinition proposedbyBelnapanditsexemplifi›
cations formulated above express clear logical intuitions, they nevertheless
cannot be used as adequate explications of the concepts in which we are
interested. It iseasily seen that every question is propositionally implied by
anyofitsdirect answers: sotheconditions (Cd.I) and(Cd.2) ofSection 1.3
arenotfulfilled. Furthermore ,there isnoguarantee thatthecondition (Cd.7)
of Section .51 is fulfilled for propositional implication of questions by sets
consisting of declarative sentence(s) and a question. Similar objections
pertain to Harrah’s proposal;moreover, inthe case of Harrah there iseven
no guarantee that the condition (Cd.3) of Section 1.3 is fulfilled.
Theconcept ofentailment andrelated conceptscanbeappliedtoquestions
if questions are reduced to expressions of syntactical categories whose
semantics iswell-defined. Reductionofthiskindisadoptedbymanylogical
theories of questions (cf. Chapter 2, Section 2.2).For example, inthelight
ofAqvist’stheoryquestionsareanalyzedasimperative-epistemic expressions
of some formalized languages supplemented with model-set semantics. To
be more precise, questions are paraphrased and then formalized as expres›
sion falling under the scheme "Let it be the case that where is a
I{) ", I{)
formula whichcontains, amongothers, epistemic operators anddescribes the
epistemic state of affairs whose achievement is demanded. Since the
semanticsoftherelevant imperative-epistemic languagesiswell-defined, one
mayspeakaboutentailment ofonequestion byanother; therelevant relation
of entailment holds between imperative-epistemic formulas. Kleiner (cf.
Kleiner, 1988, and Kleiner, 1993)applies this concept inthe philosophy of
science. Carlson (cf. Carlson, 1983,pp. 100-102)claims thatthe relation of
logical consequence between questions amounts to entailment of their
desiderata; bythedesideratum ofaquestion hemeans (following Hintikka’s
analysis; cf. Chapter 2, Section 2 .2.5) thedescription of theepistemic state
of affairs the questioner wants the respondent tobring about. The intuitions
which underlie the concepts of Aqvist and Carlson are, however, different
from that which enabled us to propose the conditions (Cd.6) and (Cd.7) of
Section 1.5. Let us add that some theorists speak about entailment between
questions without reducing questions toexpressions ofother categories; yet,
entailment is most often reduced here to containment. For example, in the
case of questions the basic intuition which underlies the (general) definition
ofentailment proposed byGroenendijk andStokhofisasfollows: aquestion
IQ isentailed by a question Q just incase every complete answer to Q also
32 CHAPTERI
gives a complete answer to lO
Ql ’
As we mentioned at the beginning of this Section, it is Jaakko Hintikka
who elaborated onthemost prominent alternative tothereceived view inthe
logic of questions.His considerations pertaining toquestionscanbesplitinto
two parts: atheory of questions andanswers, and the interrogative model of
inquiry. Hintikka accepts the imperative-epistemic view on questions (cf.
Chapter 2, Section 2 .2.5 for a short exposition). The problems dealt with
within the interrogative modelof inquiry, however, transcend thoseanalyzed
within the received view. Tospeak generally, the interrogative model aims
at showing how questions are answered in the sense of analyzing the
processes through which answers are found. The questioning processes are
viewed asquestion-answer dialogues; thesedialogues, inturn, are considered
as interrogative games between two parties, called "the Inquirer" and "the
Answerer" (the second party is also sometimes referred to as "Nature" or
"Oracle"). The aim ofagame istofind thecorrect answer tosome principal
or initial question; in the simplest cases an interrogative game isplayed by
means of a first-order language onagiven model ofthislanguage. There are
two kinds of moves: interrogative moves and deductive moves. In an
interrogative move the Inquirer addresses some auxiliary question to the
Answerer, who issupposed toanswer itasbest (she, it)can. Ina deductive
move, the Inquirer draws a deductive conclusion from some previously
accepted premise(s) and/or the answers so far obtained. The precondition
imposed on addressing questions is that a question cannot be addressed
unless its presupposition has been established. The interrogative games are
formalized by means of Beth-like tableaux; yet, certain modifications are
introduced tothe original Bethformulation on behalf oftheerotetic analysis
(for details see Hintikka, 1992a). We will not present here all the details of
the interrogative model, however: the work on this model is still not
completed and the consecutive publications of Hintikka and his associates
provide us with new ideas, enrichments and revisions.
II
Finally let us add that erotetic inferences were analyzed in the book
Wisniewski (1990a) and in the papers Wisniewski (1985), (1989), (1991a),
and Wisniewski & Kuipers (1994). The concept of generation of questions
10 Cf. Groenendijk & Stokhof(1984), pp.21, 466,478.
11 Fortheinterrogative’elmodand its applicationsseeHintikka,(1981),(1984a),(1984b),
(1985a), (1985b), (1987a), (1987b), (1988), (1989a), (1989b), (l992a), (1992b). See also
Hintikka & Hintikka (1982), Garrison (1988), Hintikka & Harris (1988), Sintonen (1993),
Halonen (1994).
EROTETICINFERENCESANDHOWQUESTIONSARISE 33
was analyzed in the dissertation Wisniewski (1986) and in the papers
Wisniewski (1987),(1989), (1991a);seealsothebookWisniewski (1990a).
Theconcept ofevocation ofquestionswasanalyzed inthebookWisniewski
(1990a) and in the paper Wisniewski (1991a)Y The concept of erotetic
implication wasintroduced inthepaperWisniewski (1990b)andanalyzedin
Wisniewski (1990a), (1991b), and (1994a).
21 InWisniewski (l991a) evocation iscalled kaew noitareneg .
CHAPTER 2
LOGICAL THEORIES OF QUESTIONS
The aim of this Chapter is two-fold . First, we shall introduce here the
basic terminology and notation used throughout this book. Second, we shall
present here an outline of the logical analysis ofquestions and answers. We
shall concentrate on the already existing proposals; the syntactic and
semantic assumptions of our further analysis willbe described indetail inthe
next chapters.
2.1. BASIC TERMINOLOGY AND NOTATION
2.1.1. smreT and evitaralced salumrof
Let us start from the characteristics of some formalized language J. To
speak generally , J is the language of some version of the first-order
predicate calculus with identity .
Let us designate by N the set of positive integers.
The vocabulary of the language J contains the logical constants: ,...
(negation), -. (implication), v (disjunction) , 1\ (conjunction), == (equiva›
lence), V’ (universal quantifier) , 3 (existential quantifier), and the identity
=.
symbol The vocabulary of J contains also an infinite list of individual
variables XI’ X 2, .. . , an infinite list of individual constants aI ’ a2’ and,
for each n E N, an infinite list of n-place predicate symbols PI" P/ and
an infinite list of n-argument function symbols F/, F In addition, the
2’... (cid:149)
vocabulary of J contains the auxiliary symbols: ( , ) (parentheses) and ,
(comma).
Byan noisserpxe of J wemean any finite sequence of the above-indicated
symbols .
The set of smret of the language J isthesmallest set which contains allthe
individual variables of J together with all the individual constants of J and
fulfills the following condition :if >lt ..., t, are terms of J, then an expression
of the form >lt(";F ..., )ot ’ where ";F isa function symbol of J, isalso aterm
of J.
A desolc mret is a term with no individual variables. The closed terms
will be also referred to as .seman
LACIGOL SEIROEHT OF SNOITSEUQ 35
=
Atomic formulas of J are expressions of J of the form t, t and of the
2
form t(";P t where t" t t are terms.
l , . .. , n ), 2 , (cid:149)(cid:149)(cid:149), n
The set I’, of declarative well-formed formulas (d-wffs for short) of the
language J is the smallest set containing all the atomic formulas of J and
r
having the following properties: (a) if A is in J , then expressions of the
form "’A , VX A, xI: A are also in r (b) if A, B are in r then expressions
i i J ; J ,
of the form A( B), (A v B), (A B), (A == B) are also in r
--+ /I J (cid:149)
Thed-wffsnotcontaining free variableswillbecalled sentences, whereas
the d-wffscontaining free variables will be called sentential functions . The
freedom and bondage of variables are defined as usual.
Each subset of the vocabulary of the language J which contains the
connectives .., and the universal quantifier V, both parentheses, all
--+,
individual variables, at least one predicate symbol, the identity symbol =
and possibly (and thus not necessarily) some other signs, such as other
predicate symbol(s), the connectives v, ,== the quantifier 3, individual
/I,
constant(s), functionsymbol(s)orthecommawillbecalledhere efirst-order
language with identity ; in what follows by a first-order language we will
meanafirst-order languagewithidentity. Theconceptsofterm, closed term,
atomic formula, declarative well-formed formula, sentence and sentential
function, aswell asthe remaining syntactic concepts aredefined for J inthe
same way as for any first-order language with identity.
One general remark is in order here: since we are going to consider
formalized languages whose meaningful expressions are both d-wffs and
questions, it is convenient to identify a language with the set of its signs
rather than with the set of its formulas.
2.1.2. Metalanguage . Notation
We shall use the symbols t, t., ... as syntactical variables which range
over terms. Since we will be frequently speaking about closed terms, we
shall introduce special metalinguistic variables for them. The letters u and
v, possibly with indices, willbesyntactical variables which represent closed
terms.
The letters A, B, C, D, with or without indices, will be metalinguistic
variables for d-wffs. An expression of the metalanguage having the form
Ax i, ... Xi. refers to the sentential functions whose free variables are exactly
the (explicitly listed) variables Xii’ .. ., .Xi’ Whenever an expression of the
form Ax il . .. .Xi is used, it is assumed that the variables Xi" ... , .Xi are
distinct. The letters P, R, possibly with indices, will be metalinguistic
variables representing predicate symbols. The symbolsX,XI’ ..., Y, Y
h . .. ,
36 RETPAHC 2
Z. ZI(cid:149) ... will be metalinguistic variables for sets of d-wffs.
The context of occurrence of the appropriate metalinguistic symbols will
always determine which formalized language we have in mind.
We adopt here the usual conventions for omitting parentheses .
An expression of the form f (cid:0)~ f isthe abbreviation of an expression of
the form ""(f( = (2 ). An expression l 2 of the form t, (cid:0)~ ... (cid:0)~ .nf in tum. is
the abbreviation of an expression of the form f) (cid:0)~ f2 II f) (cid:0)~ f3 II . .. II f)
(cid:0)~ fn II f2 (cid:0)~ f3 II ... II fn _1 (cid:0)~ fn’
We will also need the concept of lasrevinu erusolc of a d-wff. If A is a
sentence. then the universal closure of A is equal to A itself. If A is a
sentential function and X;I’ (cid:149)(cid:149)(cid:149)(cid:149) X;. (where i, < ... < iJ are the all free
variables of A. then the universal closure of A isof the form ’ixl" ... "Ix ;.A.
The universal closure of a d-wff A will be designated by .A
We assume that in the case of the analyzed languages the concepts of
(proper) substitution of aterm for a variable ina’ d-wff and of the substitut›
ivity of a term for a variable are defined in the standard manner. The result
of the substitution ofaterm f for a variable x, inasentential function Ax; will
be designated by A(X/f). By A(x,1f; >l ... , X;/fJ we shall designate the d-wff
which results from a sentential function Ax;1 (cid:149)(cid:149)(cid:149) x .i by the simultaneous
(proper) substitution of the terms f), ..., fn for the variables X; I’ .. ., x ..i
respectively .
Inthe metalanguage ofanyofthe considered languages weassume thevon
Neumann-Bernays-Godel version of the set theory (we choose this version
because we want to have the possibility of speaking about both sets and
classes). We shall use the standard set-theoretical terminology and notation.
In particular, by >lY{ ...(cid:149) }nY we shall designate the set made up of the
elements Y >I ..., Y ’n The singleton (or unit) set having Y as its element will
be referred to as }y{ . The symbol" willdesgineat" the empty set. The
symbol (cid:0)~ is the sign of inclusion, whereas the symbol C is the sign of
proper inclusion. The symbol E is, depending on the context. the predicate
=
of set membership or of class membership. The sign will be also used as
the metalingustic identity symbol. The symbols (cid:0)~(cid:0). g;. (cid:0)~ mean: "is not an
element", "isnot included in" and "isnot identical with". respectively . The
symbols u, n, - are the signs of the union of sets. intersection of sets, and
difference of sets, respectively. Sometimes we shall write X, Y instead of
X u Y and X, A instead of X u {A}. A metalinguistic expression of the
form :y{ })y(JI« denotes the set of all objects satisfying the (metalinguistic)
sentential function )y(JI« . The expression "iff" is an abbreviation of "if and
only if." The symbol 0 indicates the end of a proof.
LACIGOL SEIROEHT OF SNOITSEUQ 37
The remaining symbolsusedinthisbook willbeexplained directly inthe
contexts of their first appearance.
Let usfinally addthat thenames ofexpressions (ofanobject language or
ofametalanguage) willbe formedhere bypreceding theseexpressions with
words "term", "sentence", "question", etc. Phrases like "expression of the
form", "d-wff of the form", "question of the form", etc., are used here in
the role of Quine’an -quotes.quasi Thus, for example, the phrase "an
expression of the form (A -. B)" means "an expression made up of the left
parenthesis, a d-wff, the implication sign, a d-wff, and the right parenthe›
sis." Of course, these conventions will not always be kept, but they will
always beadopted incases inwhichthere maybeariskofamisunderstand›
ing.
In what follows we will be considering also some formalized languages
which are not first-order languages. Yet, the terminology and notation
introduced above will be also applied to these languages.
2.2. LOGICAL SEIROEHT OF SNOITSEUQ AND SREWSNA
Anyfirst-order languagecanbesupplementedwithaquestion-and-answer
system. This, however, can be done in different ways. Before we shall
introduce the concepts of question anddirect answer accepted inthe course
of analysis pursued in this book, let us pay some attention to the existing
logical theories ofquestions and answers. The followingpresentation isnot
anexhaustive one;weshallconcentrate onlyonsomemainideas andonthe
workofsomelogicians. Yet, itseemsthatsuchapresentation willmakeour
proposals more comprehensible and less arbitrary.
2.2.1. Questions: reductionism vs. non-reductionism
Asweread inoneof the very few monographs onthe logic ofquestions,
"Different authors developing logical theories of questions accept different
answers to thequestion ’What isa question?" ’(Kubinski, 1971,p.97).This
statement, written down more than twenty five years ago, still gives us a
realistic description of the situation within erotetic logic.
To speak generally, the approaches to questions proposed by different
logicians and formal linguists can be divided into reductionist and non›
reductionist ones. Inside the reductionist approach, inturn, the radical and
moderate standpoints can be distinguished.
2.2 .1.1. Radical reductionism. According tothe radical view, questions are
not linguistic entities. The reduction of questions to sets of sentences or
propositions is most often adopted here. Sometimes any setof sentences is
38 RETPAHC 2
allowed to be a question, but usually questions are identified with sets of
answers of some distinguished category. Stahl (cf., e .g., Stahl, 1962)
identifies questions with sets of their sufficient answers; these answers are
declarative formulas ofastrictlydefinedkind. Hamblin (cf.Hamblin, 1973)
identifies aquestion withthe setofitspossible answers, whereas Karttunen
(cf. Karttunen, 1978)identifies questions withsets of their true answers; in
both cases the relevant answers are propositions in the sense of some
intensional logic. Questions are also identified with functions defined on
possible worlds; thesetofvaluesofafunction ofthiskindconsists oftruth›
values, or of sets of individuals, or of sets of sets of individuals (cf., e .g.,
Tichy, 1978, and Materna, 1981). Also in this case some intensional logic
serves as the basis of analysis. An analysis of questions and interrogatives
interms of(someversions of)Montague intensional logic istobefounde.g.
in Groenendijk & Stokhof (1984). Some linguists developed the so-called
categorial approach toquestions: according tothisview, questions aretobe
considered as functions from categorial answers to propositions (cf., e .g.,
Hausser, 1983);acategorial answer maybeafullsentence, butmayalsobe
a part of it, e.g. a noun phrase, an adverb, etc. There are philosophers of
language (cf.,e.g., Vanderveken, 1991)whotendtoidentifyquestions with
speech acts rather than with expressions. Let usstress, however, that inall
of the above cases a distinction is made between an interrogative (or an
interrogative sentence) and aquestion:whereas interrogatives are linguistic
entities, questions are claimed not to be.
Let usfinallyaddthat somelinguists (cf. Keenan & Hull, 1973, andHiz,
1978)proposed theories inwhich the semantically meaningful units are not
questions,butquestion-answer pairs.Sometimesquestionsarealsoanalyzed
asordered pairs consisting ofinterrogative terms andstatements expressing
the relevant presuppositions (cf. Finn, 1974).
2.2.1.2. etaredoM msinoitcuder . The moderate reductionist view considers
questionsaslinguisticentitieswhich,however, canbereducedtoexpressions
of some other categories. Tobe more precise, it isclaimed here that every
question can be adequately characterized as an expression which is
synonymous (or synonymous to some reasonable degree) to a certain
expression ofa different syntactical category. Or, toput itdifferently, each
question can beadequately paraphrased asanexpression belonging tosome
other syntactic category and then formalized within some logic which,
although not primarily designed as the logic of questions, can thus be
regarded as providing us with the foundations of erotetic logic.
Sometheorists propose the reduction ofquestions todeclarative formulas
LOGICAL SEIROEHT OF SNOITSEUQ 39
of strictly defined kind(s). Sometimes questions are identified with declara›
tive formulas having free variables, that is, with sentential functions (cf.
e .g., Cohen, 1929, or Lewis & Langford, 1932). But questions are also
identified withsentences, thatis, declarative formulas withnofreevariables.
According to the early proposal of David Harrah (cf. Harrah, 1961, 1963)
whether-questions are to be understood and then formalized as declarative
sentences havingtheformofexclusive disjunctions, whereas which-questions
should be identified with existential generalizations . We shall present
Harrah’s proposal in a more detailed way in Section 2.2.3.
Questions are also identified with imperatives of a special kind. The
imperative -epistemic approach isthe most popular here with Lennart Aqvist
and Jaakko Hintikka as its most eminent representatives.
According to Aqvist (cf. Aqvist, 1965, 1969, 1971, 1972, 1975), a
question can be paraphrased as an imperative-epistemic expression of the
form "Let it (tum out to) be the case that where is a formula which
Ip", Ip
describes the epistemic state ofaffairs which should be achieved. Pragmati›
cally, a question isthus understood as an imperative which demands of the
respondent to widen the questioner’s knowledge. Questions are formalized
within the framework of some imperative-epistemic logic; on the level of
formal analysis wedealwith interrogatives. Each interrogative consists ofan
interrogative operator and its arguments. Interrogatives are defined as
abbreviations of certain formulas of the language of the considered
imperative -epistemic logic. We shall present some elements of Aqvist’s
theory in Section 2.2.4.
The imperative-epistemic approach toquestions isalsoadopted byJaakko
Hintikka inhis theory of questions and answers (cf. Hintikka, 1974, 1976,
1978, 1982a, 1982b, 1983). Hintikka interprets questions as requests for
information or knowledge: according to his view, each question can be
paraphrased as an expression which consists of the operator "Bring it about
that" followed bytheso-called mutaredised ofthequestion.The desideratum
describes the epistemic state of affairs the questioner wants the respondent
tobring about. Although the main ideas of tsivqA and Hintikka are similar,
they are elaborated onindifferent ways. Weshall present Hintikka’s theory
in a greater detail in Section 2 .2.5.
As far as the moderate reductionist view is concerned, the imperative›
epistemic approach is the most widely developed one. Yet, there are also
other proposals. In particular, there isanold idea (which goes back at least
to Bolzano) that the paraphrase of a question should contain an optative
operator. (It is worth noticing that Hintikka sometimes calls the operator