negation ¬.

Then a yes-or-no question “Is it the case that S” is answerable only if S is

true or false. This is expressed by (S ∨ ∼ S). Hence the presupposition of the

yes-or-no question (in the sense of this chapter) is K(S ∨ ∼ S). Unlike K(S ∨

¬S), this is not logically true. Hence, in independence-friendly logic, yes-or-

no questions, too, have their non-trivial presuppositions, just like other kinds

of questions. These presuppositions must be established by previous inquiry

before the question may be asked. Thus yes-or-no questions are not always

permitted any longer. However, this does not break the strategic analogy with

deduction, in that disjunctions (S ∨ ∼ S) are not logically true. Rules like the

modus ponens and the cut rule hold for ¬ but not for ∼.

This change of logic does not change the theoretical situation, however.

What it does is to enrich the expressive potentialities of our logical language.

For instance, the range of answerable questions (see Section 3) in an inter-

rogative game now need no longer be speci¬ed from the outside as a part of

the de¬nition of the game. Available answers can be characterized as those

presuppositions for which the law of excluded middle holds.

In any case, the advantages of rules such as the tautology-introduction rule

and the cut rule in deductive logic are essentially strategic. In ordinary ¬rst-

order logic, they typically make it possible to shorten and to simplify proofs,

even though in such a logic, they do not allow proving any logical truths that

are not provable without them. They are dispensable, but only at the cost of

longer proofs.

At the same time, such additional rules as do not obey the sub-formula

principle complicate the strategic situation enormously. In deductive tableaux,

each formula introduced by the original rules was a sub-formula of one of the

earlier ones, and its introduction could therefore be thought of as a step in

analyzing the situation described in the initial premises or in the ultimate

conclusion. In contrast, the earlier history of a tableau construction does not

Presuppositions and Other Limitations 101

restrict the choice of the proposition S in the tautology (S ∨ ∼ S) being intro-

duced. This opens the ¬‚oodgates for a deluge of new strategies. Moreover,

since these new strategies are less strictly determined by the given initial data

of the problem, tautology introduction and unrestricted yes-or-no questions

allow much more scope for creative imagination and invention than the steps

satisfying the sub-formula property.

11. Presuppositions of Presuppositions

One thing we can now also see is that the presuppositions of yes-or-no ques-

tions K(S ∨ ∼ S) cannot normally be traced back to the initial premises that

express the epistemic starting points of inquiry. For facts as to what ques-

tions are answerable are normally determined by the context of the inquiry

independently of our thinking. And since they are sometimes indispensable

in inquiry, not all presuppositions have a precedent in the initial premises of

inquiry. There is no set of absolute presuppositions that would restrict inquiry.

Collingwood™s notion of absolute presupposition is not viable.

To return to the main theme of this chapter, we have reached an unequivocal

answer to the question as to whether rational epistemological inquiry is subject

to intransgressable restrictions such as are supposed to be dictated by Colling-

wood™s ultimate presuppositions or by Kuhn™s paradigms. The approach used

here has so far followed Collingwood in construing limitations of inquiry as

presuppositions of questions. What has been found by means of an exami-

nation of the presuppositions of questions is that these presuppositions do

play an important role in inquiry, but that they restrict strategies and do not

impose limits to what can be established by means of the inquiry. Moreover,

the effect of the restrictions can be compensated for, not only by liberalizing

the restrictions in the sense of assuming stronger initial premises but also by

increasing the range of answerable questions. This can be established perfectly

naturalistically”for instance, through improved experimental and observa-

tional techniques. Hence what has been found here tells against all theories of

unavoidable restrictions to inquiry, relativistic or not.

We have also seen in this section that there is no way of tracing all the

presuppositions relied on in an inquiry to its initial premises. The notion of

ultimate presuppositions is unworkable.

12. Presuppositions of Answers

But must all presuppositions of inquiry in the sense intended by the likes of

Collingwood be construed as presuppositions of questions in the logical sense

used earlier? This is a pertinent question. On the one hand, Collingwood

not only speaks in so many words of the presuppositions of questions but

often has obviously in mind the same sorts of presuppositions as have been

Socratic Epistemology

102

considered here. He even uses some of the traditional examples of violations

of such presuppositions”for example, examples of the type, “When did you

stop beating your wife?”

On the other hand, however, it seems to me that Collingwood is assimilat-

ing to each other presuppositions of questions in the sense used here and what

might be called “presuppositions of answers.” In order to avoid confusion, I

have nevertheless called them “uniqueness conditions.” They have been dis-

cussed elsewhere (see especially Hintikka 2007) and hence I can be relatively

brief here. An example can convey the main point.

As perceptive philosophers from Francis Bacon to Immanuel Kant to R. G.

Collingwood have pointed out, one can”and ought to”construe controlled

experiments in science as questions put to nature. Such a question has the form,

“How does the variable y for a certain quantity depend on another one, say

x, for a different variable?” (It is probably the very fact that the experimenter

can in fact control one of the variables that prompted Bacon™s metaphor of an

investigator forcing nature to reveal her secrets.) The experiment succeeds in

providing an answer to this question if the function expressing the dependence

is known. Technically this means that the desideratum of the question becomes

true. This desideratum can be expressed logically in any of the following forms:

(∃f)K(∀x) S[x, f(x)] (16)

K(∃f/K)(∀x) S[x, f(x)] (17)

K(∀x) (∃y/K) S[x, y] (18)

Here, K is the knowledge operator (“It is known that”) and / the independence

operator. What an experiment ideally achieves can be thought of as a function-

in-extension”that is to say, an in¬nite list of correlated arguments values and

function values. If this function-in-extension is g, then the purely observational

components of an experimental answer is

K(∀x) S[x, g(x)] (19)

One can think of (19) as being illustrated by an in¬nitely sharp curve on a

graph paper.

But (19) does not logically imply (16)“(18). This fact has a concrete inter-

pretation. Even if we abstract from all limitations of observational accuracy

and of the accuracy with which on can manipulate the controlled variable,

still “Nature™s response” (17) will not satisfy the experimentalist unless he or

she knows or ¬nds out what the function g is, mathematically speaking. In

terms of the hypothetical illustration, even if there is an arbitrarily accurate

curve on the experimentalist™s graph paper, he or she may still fail to know

what the function is that the curve represents. Only if the scientist knows or is

shown what that function is has he or she reached a conclusive answer to the

experimental question.

Presuppositions and Other Limitations 103

This additional information is expressed by what I have called the “conclu-

siveness condition.” In an example, this condition can be expressed in any of

the following four forms:

(∃f)K(∀x)(g(x) = f(x)) (20)

K(∃f/K) (∀x) (g(x) = f(x)) (21)

K(∃f/K) (g = f) (22)

K(∀x) (∃y/K) (g(x) = y) (23)

Even though (19) alone does not entail (16)“(18), it does so in conjunction

with (20)“(23). This shows that (20)“(23) constitute a kind of presupposition of

answers to an experimental question. Such “presuppositions” (conclusiveness

conditions) limit the possibility of answering questions somewhat in the same

way as the presuppositions of questions we encountered earlier. Hence they

are relevant to the over-arching theme of this chapter, which concerns the

limitations of inquiry. In what way does the need of conclusiveness conditions

like (20)“(23) limit our quest of information?

What kind of knowledge is it that the presuppositions of answers such

as (20)“(23) express? This knowledge concerns the identity of certain math-

ematical objects”namely, functions. It is hence conceptual and a priori in

character. Indeed, we have found one of the main gates through which math-

ematical knowledge enters into the very structure of empirical science. Such

knowledge is needed to answer scienti¬c questions, typi¬ed by experimental

questions. This knowledge does not come to a scientist automatically. It has

to be gained. But such knowledge is not uncovered in a laboratory or found

codi¬ed in a textbook of experimental physics. It is obtained in departments of