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The Logic of Interrogatives
THE LOGIC OF INTERROGATIVES M. J. CRESSWELL Victoria University of Wellington, New Zealand
This paper attempts to give a formal analysis of a concept which bears many resemblances to "p is the answer to d" where p is a statement and d a question. Let Spd represent this concept. The sense of "answer" which it characterizes is such that the following laws hold; Al
(Spd. Sqd)
~
(p = q)
Every question has at most one answer. This will exclude answers which give too much information. E.g., if! ask "Is it raining?" and receive the reply, "No it isn't; in fact the sun is shining." Then this would usually count as an answer. But in the sense of "answer" characterized by S the only correct answer to "Is it raining?" would be, "It is not raining" though the statement, "It is not raining and the sun is shining" obviously entails the answer to "Is it raining?" A2 (tlp)Spd, Every question has at least one answer. A3 Spd ~ p, The answer to a question is true. S then designates the true or correct answer to a question. A4
[0 (p == q). Spd]
~ Sqd,
If a statement is the answer to a question then any statement logically equivalent to it answers that question. This does make S diverge from many ordinary senses of "answer". It is a sense which, e.g. is of no use to the mathematician, since all mathematical truths are logically equivalent. But it does represent "answer" in that given a p which answers d wecan deduce an answer which satisfies any stricter criterion for "answer".
9
THE LOGIC OF INTERROGATIVES
There are cases where the same statement answers different questions. If I ask "Are both Mary and John here?" and someone else asks "Which of Mary and John are here?" then if both Mary and John are here the answer to both questions is "Both Mary and John are here" (though in the first case we would express it by saying "yes" and in the second by saying "both".) But they are different questions; for suppose Mary is here but John isn't. Then the answer to the first is, "Mary and John are not both here" and the answer to the second is "Mary is here but John isn't." (This last is an "answer" to the first question which gives too much information.) Identity of questions is defined,
Def=
(d
=
e)
=
df
(P) 0 (Spd
=Spe)
The following question-forming operator on questions is such that; Q
(d v e) is a question whose answer is the conjunction of the answer to d and the answer to e. If I ask, "Which of Mary and John is here?" then the possible answers are,
"John is here and Mary is here" "John is here but Mary isn't" "Mary is here but John isn't" "Neither Mary nor John is here" Q
and these are the only possible answers. The axioms for v are; A5
Q
(Spd.Sqe) => S(p.q) (d v e) Q
Q
A6 (d v e) = (e v d) Q
Q
Q
Q
A7
((d v e) V f) = (d v (e v f)
A8
(d
= e)
=>
Q
Q
[(fv d) = (fv e)J
The simplest kind of question is the yes/no question, i.e, the question which has the form "Is it the case that p'I", If I ask "Is it raining?" there are only two possible answers, "It is raining" "It is not raining"
Letting Qp = "Is it the case that p?" we have A9
SpQp v
S~ pQp
10
M. J. CRESSWELL
from which we may prove (using A3) SpQp == p, S"'pQp == "'p, and Qp = Q'" p. But yes/no questions form only a subclass of questions. A much more extensive class may be formed by introducing a questionforming quantifier. We use (Qa)A(a) to mean, "for which of the a's does A hold?" where a represents any variable and A a wff in which a occurs. I might ask, (Qp)fp, for which p does the propositional function/hold?
or
(Qx)¢x, which x's ¢?
or
(Q¢ )¢x, what properties does x possess? etc.
The answer to (Qa)A(a) will be the true conjunction of A(an)'s or '" A(an)'s for every an" To express this formally we express what it is for a proposition p to entail and be entailed by such a conjunction. p entails such a conjunction iff, (a) [p ~ A(a) . v -P -'5 '" A(a)] p is entailed by such a conjunction iff it is entailed by every q which
entails such a conjunction (since among these q will be the conjunction itself) i.e. iff (q){(a)[p ~ A(a) => q -'5 A(a):. :p -'5 '" A(a) => . q -'5 '" A(a)] => (q -'5 p)} 0
0
0
p is the answer to (Qa)A(a) if both these are satisfied and p is true.
AlO p. (a)[p
~
A(a). v.p ~ '" A(a)]. (q){(a)[p ~ A(a). => . q -'5 A(a):. :
p ~ '" A(a). => . q -'5 '" A(a)] => (q ~ p)).:
== :. Sp(Qa)A(a)
If we consider the class of questions definable in terms of (Qa) and restrict our logic to these we can define the expression (Sa)pA(a) (i.e. "p is the answer to the question 'which a's A?''') as; p.(a)[p -'5 A(a).v.p ~ ",A(a)].(q){(a)[p ~ A(a). =>.q ~ A(a):.: p -'5 ",A(a). =>.q -'5 ",A(a)] => (q
~
p)}
A wider class of questions can be considered by defining the schema Sp(Qna) [A 1(a), ... , An(a)] where (Qna) [A 1(a), ... , A,,(a)] would be equivQ
Q
alent to (Qa)A 1(a) v ... v (Qa)An(a). It is then possible, substituting (Qa)A 1(a)/d, (Qa)Aia)/e, etc., to prove as theorems the equivalents of Al-AlO by means of this definition.
THE LOGIC OF INTERROGATIVES
11
This method is sufficient to define, in a modal system with quantification, the concept "p is the answer to d" (in the sense of "answer" outlined) whenever the question d can be interpreted as asking, "Which a's A 7" I.e. it is sufficient for questions where a set of possible answers is so definable that the true member of the set is the answer to the question. If we can interpret questions like "When - - - - 7" as asking, "At what time- - - - 7", "Why- - - - 7" as asking, "For what reason - - - - 7", "How- - - - 7" as asking "By what means- - - - 7", etc. then the account given will suffice as an account of question logic in respect of "p is the answer to d", References [1] C. L. Hamblin, Questions. Australasian Journal of Philosophy 36 (1958) 159. Discussion-Questions aren't Statements. Philosophy of Science 30 (1963) 62. [2] David Harrah, Communication: A Logical Model (M.LT. Press 1963). [3] Henry S. Leonard, Interrogatives, Imperatives, Truth, Falsity and Lies. Philosophy of Science 26 (1959) 172. A Reply to Professor Wheatly. ibid. 28 (1961) 55. [4] J. M. O. Wheatly, Note on Professor Leonard's Analysis of Interrogatives. ibid. 28 (1961) 52.
This paper attempts to give a formal analysis of a concept which bears many resemblances to "p is the answer to d" where p is a statement and d a question. Let Spd represent this concept. The sense of "answer" which it characterizes is such that the following laws hold; Al
(Spd. Sqd)
~
(p = q)
Every question has at most one answer. This will exclude answers which give too much information. E.g., if! ask "Is it raining?" and receive the reply, "No it isn't; in fact the sun is shining." Then this would usually count as an answer. But in the sense of "answer" characterized by S the only correct answer to "Is it raining?" would be, "It is not raining" though the statement, "It is not raining and the sun is shining" obviously entails the answer to "Is it raining?" A2 (tlp)Spd, Every question has at least one answer. A3 Spd ~ p, The answer to a question is true. S then designates the true or correct answer to a question. A4
[0 (p == q). Spd]
~ Sqd,
If a statement is the answer to a question then any statement logically equivalent to it answers that question. This does make S diverge from many ordinary senses of "answer". It is a sense which, e.g. is of no use to the mathematician, since all mathematical truths are logically equivalent. But it does represent "answer" in that given a p which answers d wecan deduce an answer which satisfies any stricter criterion for "answer".
9
THE LOGIC OF INTERROGATIVES
There are cases where the same statement answers different questions. If I ask "Are both Mary and John here?" and someone else asks "Which of Mary and John are here?" then if both Mary and John are here the answer to both questions is "Both Mary and John are here" (though in the first case we would express it by saying "yes" and in the second by saying "both".) But they are different questions; for suppose Mary is here but John isn't. Then the answer to the first is, "Mary and John are not both here" and the answer to the second is "Mary is here but John isn't." (This last is an "answer" to the first question which gives too much information.) Identity of questions is defined,
Def=
(d
=
e)
=
df
(P) 0 (Spd
=Spe)
The following question-forming operator on questions is such that; Q
(d v e) is a question whose answer is the conjunction of the answer to d and the answer to e. If I ask, "Which of Mary and John is here?" then the possible answers are,
"John is here and Mary is here" "John is here but Mary isn't" "Mary is here but John isn't" "Neither Mary nor John is here" Q
and these are the only possible answers. The axioms for v are; A5
Q
(Spd.Sqe) => S(p.q) (d v e) Q
Q
A6 (d v e) = (e v d) Q
Q
Q
Q
A7
((d v e) V f) = (d v (e v f)
A8
(d
= e)
=>
Q
Q
[(fv d) = (fv e)J
The simplest kind of question is the yes/no question, i.e, the question which has the form "Is it the case that p'I", If I ask "Is it raining?" there are only two possible answers, "It is raining" "It is not raining"
Letting Qp = "Is it the case that p?" we have A9
SpQp v
S~ pQp
10
M. J. CRESSWELL
from which we may prove (using A3) SpQp == p, S"'pQp == "'p, and Qp = Q'" p. But yes/no questions form only a subclass of questions. A much more extensive class may be formed by introducing a questionforming quantifier. We use (Qa)A(a) to mean, "for which of the a's does A hold?" where a represents any variable and A a wff in which a occurs. I might ask, (Qp)fp, for which p does the propositional function/hold?
or
(Qx)¢x, which x's ¢?
or
(Q¢ )¢x, what properties does x possess? etc.
The answer to (Qa)A(a) will be the true conjunction of A(an)'s or '" A(an)'s for every an" To express this formally we express what it is for a proposition p to entail and be entailed by such a conjunction. p entails such a conjunction iff, (a) [p ~ A(a) . v -P -'5 '" A(a)] p is entailed by such a conjunction iff it is entailed by every q which
entails such a conjunction (since among these q will be the conjunction itself) i.e. iff (q){(a)[p ~ A(a) => q -'5 A(a):. :p -'5 '" A(a) => . q -'5 '" A(a)] => (q -'5 p)} 0
0
0
p is the answer to (Qa)A(a) if both these are satisfied and p is true.
AlO p. (a)[p
~
A(a). v.p ~ '" A(a)]. (q){(a)[p ~ A(a). => . q -'5 A(a):. :
p ~ '" A(a). => . q -'5 '" A(a)] => (q ~ p)).:
== :. Sp(Qa)A(a)
If we consider the class of questions definable in terms of (Qa) and restrict our logic to these we can define the expression (Sa)pA(a) (i.e. "p is the answer to the question 'which a's A?''') as; p.(a)[p -'5 A(a).v.p ~ ",A(a)].(q){(a)[p ~ A(a). =>.q ~ A(a):.: p -'5 ",A(a). =>.q -'5 ",A(a)] => (q
~
p)}
A wider class of questions can be considered by defining the schema Sp(Qna) [A 1(a), ... , An(a)] where (Qna) [A 1(a), ... , A,,(a)] would be equivQ
Q
alent to (Qa)A 1(a) v ... v (Qa)An(a). It is then possible, substituting (Qa)A 1(a)/d, (Qa)Aia)/e, etc., to prove as theorems the equivalents of Al-AlO by means of this definition.
THE LOGIC OF INTERROGATIVES
11
This method is sufficient to define, in a modal system with quantification, the concept "p is the answer to d" (in the sense of "answer" outlined) whenever the question d can be interpreted as asking, "Which a's A 7" I.e. it is sufficient for questions where a set of possible answers is so definable that the true member of the set is the answer to the question. If we can interpret questions like "When - - - - 7" as asking, "At what time- - - - 7", "Why- - - - 7" as asking, "For what reason - - - - 7", "How- - - - 7" as asking "By what means- - - - 7", etc. then the account given will suffice as an account of question logic in respect of "p is the answer to d", References [1] C. L. Hamblin, Questions. Australasian Journal of Philosophy 36 (1958) 159. Discussion-Questions aren't Statements. Philosophy of Science 30 (1963) 62. [2] David Harrah, Communication: A Logical Model (M.LT. Press 1963). [3] Henry S. Leonard, Interrogatives, Imperatives, Truth, Falsity and Lies. Philosophy of Science 26 (1959) 172. A Reply to Professor Wheatly. ibid. 28 (1961) 55. [4] J. M. O. Wheatly, Note on Professor Leonard's Analysis of Interrogatives. ibid. 28 (1961) 52.