such abstract entities as functions in that part of mathematics.

It is thus instructive to study the role of the idea of function in our usual

logical notation involving quanti¬ers. This notation is also the basis of much of

the usual language of mathematics, as was ¬rst spelled out in the process of the

“arithmetization of mathematics” in the nineteenth century. In hindsight, this

process might almost as well have been called the logization of mathematics,

where the logic of that logization is essentially the logic of quanti¬ers. Hence

the key question here is the role of quanti¬ers in determining the ontology

needed in science.

Now, what both the defenders and the critics of the indispensability have

failed to see is the main reason why complex sentences cannot be tested directly

for truth. This reason is not always or even typically the presence of theoreti-

cal concepts in them or their being connected with testable sentences only

inferentially, but their quanti¬cational structure. One cannot verify or falsify

a sentence of the form

(∀x)(∃y)S[x, y] (2)

directly, in one fell swoop, not because it contains unobservable non-logical

concepts or because it is connected with observables only inferentially, but

because the semantical game it involves has at least one dependent move

made by the veri¬er. From this it follows that the strategies the veri¬er uses in

dealing with (2) can only be represented by functions. These functions cannot

be grasped at ¬rst one glance. The “theoretical concepts” that a sentence

such as (2) involves are its Skolem functions. These functions are imported by

quanti¬ers, not by non-logical theoretical concepts occurring in a proposition

or by its logical consequences.

Because of this crucial role of functions in logic and mathematics, the entire

project of eliminating numbers or other particular mathematical objects from

science is of a strictly limited interest. Such elimination uses our customary

logical apparatus, including quanti¬ers. Since the use of quanti¬ers was seen

to rely on Skolem functions, the apparent elimination merely replaces one

Socratic Epistemology

114

type of mathematical objects by another class of abstract entities”namely,

functions. One can dispense with Skolem functions only if one can dispense

with quanti¬ers, which in effect means dispensing with logic.

Thus, any elimination of mathematical objects in favor of theories involving

quanti¬ers (or equivalent) fails to get at the bottom of the problem of the role

of mathematics”and of other a priori knowledge”in epistemology. Such an

elimination leaves us with the most important logico-mathematical entities

playing a role in which they appear absolutely uneliminable. What will be done

next is to try to diagnose the role of mathematics in knowledge acquisition in

general.

4. Mathematical Knowledge in Knowledge Acquisition

Among the most important weaknesses of the indispensability dispute is the

way questions are posed in it. As was seen, the lion™s share of attention in it

is focused on ontology, ¬rst on the role (if any) of mathematical objects in

science. Because of this direction of interest, the role of mathematical knowl-

edge in science is easily obscured. And even when the discussion has conceived

(or can be interpreted as having conceived) the role of mathematical knowl-

edge in scienti¬c knowledge, the questions raised have concerned the role of

mathematical theories in scienti¬c theorizing. In spite of the lip service paid

to scienti¬c and mathematical practice, very little serious thought has been

devoted to the role of mathematics in the processes of knowledge acquisition

in science or elsewhere. This role is what is studied next in this chapter. It will

be shown that mathematical knowledge plays an essential role in the typical

procedures that introduce new information into scienti¬c reasoning”to wit,

in experiments and observations. The results of this inquiry will also show what

the nature of this indispensable mathematical knowledge is.

One reason why this study has not been attempted before is that there

has not existed an explicit logical and epistemological theory of knowledge

acquisition. It even used to be generally maintained that there cannot exist a

rational theory of such high-grade acquisition of new knowledge that can be

called discovery. The slogan was that only a theory of justi¬cation is possible

in rational epistemology, but not a theory of discovery.

But how, however, a logic of discovery is known to be possible, the reason

being that it is actual. This actual theory of discovery is an application of the

theory of information acquisition by questioning, also known as the “inter-

rogative model of inquiry.” (See, e.g., the papers collected in Hintikka 1999.)

This approach is a straightforward application of the logic of questions and

answers, otherwise known as “erotetic” logic. This logic at once reveals, when

correctly developed, the role of mathematical knowledge in empirical inquiry,

especially clearly in experimental inquiry.

The consequences of this analysis can be appreciated even without consid-

ering the details of erotetic logic used to uncover them. It will be shown that

The a priori in Epistemology 115

the results reached here put the paradigm case of ampliative empirical rea-

soning, experimental induction, and thereby the entire concept of induction,

into an importantly new light.

5. On the Logic of Questions and Answers

In order to carry out this project, it may be noted that there is a simple but

profound argument that shows that mathematical (logical, conceptual) knowl-

edge is indispensable in empirical science. As was indicated, this argument is

based on the logic of questions and answers. Even though the most general

formulation of this logic is not yet generally known, I can only, for reasons

of space, explain the main points by means of examples. (Brief expositions of

the underlying logical theory can be found in Hintikka 2003 and in Hintikka,

Halonen and Mutanen 1998.)

The crucial matter concerns the requirements for satisfactory (conclusive)

answers to questions, especially wh-questions. Ever since Aristotle, logicians

have recognized, more or less clearly, that in order to ask a factual question and

to be entitled to expect an answer, the presupposition of the question must

have been established. Nevertheless, not until the advent of contemporary

epistemic logic was it realized that in a different sense, satisfactory answers,

too, need presuppositions of sorts. These presuppositions of answers have to be

distinguished from the presuppositions of questions. In order to highlight the

difference, I have called the quasi-presuppositions of answers “conclusiveness

conditions.”

In order to see how these quasi-presuppositions of answers enter the prob-

lem situation, suppose that I ask:

Who will win the gubernatorial race?

In the eyes of an epistemic logician, the intended result of asking this question

is one that enables the questioner to say, truly,

I know who will win the gubernatorial race. (3)

Such a speci¬cation of the intended epistemic result of an adequate answer

to a question is called the desideratum of the question. It determines the logical

behavior of the question in question as well as the behavior of its answers. In

order to see how, suppose that you answer: “The democratic candidate.” Then,

assuming that I know that you are honest and knowledgeable, I can now say,

truly,

I know that the democratic candidate will win the (4)

gubernatorial race.

But (4) is not yet what I wanted to accomplish by my question. The outcome

of the response”that is, (4)”is not automatically the desired one. It is not the

desideratum (3), nor does it imply (3). The reason is that I may fail to know

Socratic Epistemology

116

who the democratic candidate is. In order for the answer to be conclusive, it

must be the case that

I know who the democratic candidate is. (5)

This requirement is called the “conclusiveness condition” of the answer.

It is the “presupposition” of an answer mentioned earlier. If your task is to

provide a conclusive answer to my question, you must bring about the truth

of (5) and not only of (4), unless you can assume it.

In the notation of epistemic logic (including the independence indicator/),

(3)“(5) can be said to instantiate the following forms:

KI (∃x/KI )W[x] (6)

KI W[d] (7)

KI (∃x/KI )(d = x) (8)

(See Hintikka 1992.) These forms have equivalents in the old-fashioned epis-

temic logic that tries to dispense with the independence indicator (slash). These

equivalents are:

(∃x)KI W[x] (9)

KI W[d] (7)

(∃x)KI (d = x) (10)

A clue to the meaning of the independence indicator is in order here. The

equivalence of (6) and (18) with (9) and (10) in fact illustrates this meaning.