David Corfield Logic of Question and Answer

Idea

Collingwood discusses the ‘Logic of Question and Answer’ while criticising the idea of freestanding propositions in a number of places. For him, propositions are not just bald statements, but rather answers to questions, which in turn presuppose further propositions. The idea of a question arising or not arising reveals the implicit presuppositions of the situation.

If presuppositions are well-treated by dependent type theory (cf. p. 57-8, 92 of Modal Homotopy Type Theory), then we would expect this logic to account well for questions. This is what Aarne Ranta proposes in Type-Theoretic Grammar (OUP, 1994), see below.

Quotations

A = Autobiography, EM = Essay in Metaphysics, PH = Principles of History

There have always been people who saw that the true ‘unit of thought’ was not the proposition but something more complex in which the proposition served as answer to a question. Not only Bacon and Descartes, but Plato and Kant, come to mind as examples. When Plato described thinking as a ‘dialogue of the soul with itself’, he meant (as we know from his own dialogues) that it was a process of question and answer, and that of these two elements the primacy belongs to the questioning activity, the Socrates within us. (A, 34-35)

a logic in which the answers are attended and the questions neglected is a false logic. (A, 31)

Meaning, agreement and contradiction, truth and falsehood, none of these belonged to propositions in their own right, propositions by themselves; they belonged only to propositions as the answers to questions: each proposition answering a question strictly correlative to itself. (A 33)

This attempt to correlate the logical proposition with the grammatical indicative sentence has never been altogether satisfactory (A 34)

For a logic of propositions I wanted to substitute what I called a logic of question and answer. It seemed to me that truth, if that meant the kind of thing which I was accustomed to pursue in my ordinary work as a philosopher or historian—truth in the sense in which a philosophical theory or an historical narrative is called true, which seemed to me the proper sense of the word—was something that belonged not to any single proposition, nor even, as the coherence-theorists maintained, to a complex of propositions taken together; but to a complex consisting of questions and answers. (A, 36-37)

What is ordinarily meant when a proposition is called ‘true’, I thought, was this: (a) the proposition belongs to a question-and-answer complex which as a whole is ‘true’ in the proper sense of the word; (b) within this complex it is an answer to a certain question; (c) the question is what we ordinarily call a sensible or intelligent question, not a silly one, or in my terminology it ‘arises’; (d) the proposition is the ‘right’ answer to that question. (A, 38)

In logic I am a revolutionary; and like other revolutionaries I can thank God for the reactionaries. They clarify the issue.

[A]ccording to my own ‘logic of question and answer’, a philosopher’s doctrines are his answers to certain questions he has asked himself, and no one who does not understand what the questions are can hope to understand the doctrines. The same logic committed me to the view that any one can understand any philosopher’s doctrines if he can grasp the questions which they are intended to answer. Those questions need not be his own; they may belong to a thought-complex very different from any that is spontaneously going on in his own mind; but this ought not to prevent him from understanding them and judging whether the persons interested in them are answering them rightly or wrongly.

…whenever anybody states a thought in words, there are a great many more thoughts in his mind than there are expressed in his statement. Among these there are some which stand in a peculiar relation to the thought he has stated; they are not merely its context, they are its presuppositions. (EM 21-22)

…[questions] must be asked in the right order. Descartes, one of the three great masters of the Logic of Questioning (the other two being Socrates and Bacon), insisted upon this as a cardinal point in scientific method, but so far as modern works on logic are concerned, Descartes might never have lived. Modern logicians are in a conspiracy to pretend that a scientist’s business is to ‘make judgments’, or ‘assert propositions’, or ‘apprehend facts’, and also to ‘assert’ or ‘apprehend’ the relations between them; suggesting that they have no experience of scientific thinking, and wish to palm off, as an account of scientific thinking, an account of their own haphazard, unsystematic, unscientific consciousness. (PH, p. 29)

Type-theoretic account (after Ranta’s Type-theoretic Grammar)

We should say that a judgement presupposes (MHTT, p. 59) other judgements.

Question types

Single answers

• $P$?, for $P$ a proposition.
• $P$ or $Q$?, which one? This isn’t quite $(P$ or $Q)$?, even constructively, as it’s not expected that you say ‘Neither’. (Ranta sees $P$? as a case of this kind, i.e., $P$ or not $P$?.)
• Who, when, what $X$, which $X$, whither, whence, whose, how, how much/many/ long, whether (indirect): $(Wh\; x: A)B(x)$, including requesting an element of a type, say a canonical term given a non-canonical term, What is 2 + 2? as $(Wh\; x: N)(x = 2+2)$?.
• Iterated question: Who read which book? Who did what to whom? This is $(Wh\; x:A)(Wh\; y:B)R(x,y)$?, etc.
• Why $P$? Highly dependent on the nature of $P$, and the way the type that is $P$ has been generated by type formation rules and generators. If $P$ states an activity is taking place by an intentional agent, then it may be asking for the achievement that will be the culmination of the activity (II.6 of MHTT). Why state? or Why activity? may request a previous activity or achievement. This can work in terms of types of activity. Cf. literature on explanation. In a sense then this still asks $(Wh\; x: A)B(x)$?, ‘which achievement does this activity aim for?’, ‘which state prompted this activity?’, ‘which activity brought about this state?’
• Could even $P$ or $Q$? be taken as $(Wh\; x: A)B(x)$? in the sense that $P(0)$ or $P(1)$? is $(Wh\;x: \mathbf{2})P(x)$?

Multiple answers

• Who are the $A$?, listing $A$ members.
• Which $A$ are $B$?, etc. $(Wh\; x: A)B(x)$?
• Whose glass is whose? $(Wh\; x:A)(Wh\; y:B)R(x,y)$?

Presuppositions

• Can only ask $P$? if we already have $P: Prop$
• Can only ask ‘Who is the $X$?’ if we already have ‘the $X$’ as an element of existing type $X$.
• Can only ask $(Wh\; x: A)B(x)$ if we have $A: Type$, $B(x): Type$. For single answer, that $\sum_{x:A} B(x)$ is contractible, often from $B(x)$ being a property that holds of just one $A$.
• Does $P$ or $Q$? presuppose $P \vee Q$ is true? Yes, according to Ranta (6.12 Fairness in a dialogue). (Ranta says a third use of ‘presupposition’, aren’t they related?). But then we also assume that $P \wedge Q$ is false.

Assertions

• $A$!, make $A$ true.

Complex question

Consider Collingwood’s ‘Have you left off beating your wife?’ This is $P$?, so presupposes that $P$ is a proposition, ‘You have left off beating your wife’. This proposition is a type formed according to the generators of the type 1-theory. (See p. 57 of MHTT, and blog post.)

We would have something like

If X: Habitual Activity Type, then leave off(X): Accomplishment Type.

Before this

If Y: Activity Type, then habit(Y): Habitual Activity Type.

If z, w: Person, then z beats w: Activity Type.

So habit(z beats w): Habitual Activity Type

then leave off(habit(z beats w)): Accomplishment Type, for z, w: Person.

So leave off(habit(you beat your wife): Accomplishment Type.

Unpicking further gives us Collingwood’s presuppositions, right back to that there be a wife.

We need to consider elements of types. An element of habit(z beats w) will require there to have been several instances of z beats w. The question is arises at a certain time after these.

Y: Activity Type, $y_1, y_2, .., y_n: Y$, some condition on $t(y_i)$s $\vdash f(y_1, y_2, .., y_n): habit(Y)$

This condition is related to the iteration map which takes momentaneous (point) event types to activity (process) types (p. 67 of MHTT).

He sneezed. He sneezed right through the performance.

Collingwood on why?

When a scientist asks “why did that piece of litmus paper turn pink?” he means “on what kind of occasions do pieces of litmus turn pink?” When an historian asks: “why did Brutus stab Caesar?” he means “what did Brutus think, which made him decide to stab Caesar?” (Idea of History, p. 214)

References

Kazuki Watanabe, Koji Mineshima, Daisuke Bekki Questions in Dependent Type Semantics (pdf)

Other approaches to a logic of questions

See SEP: Questions.

For a simple type-theoretic account, see

• Ciardelli, I., F. Roelofsen and N. Theiler, 2017, “Composing alternatives”, Linguistics and Philosophy, 40 (1): 1–36.