cal physics. Hence we have found here a very real constraint on an empirical

inquirer™s ability to answer experimental questions, and indeed a constraint

on inquiry in general. Moreover, what is especially interesting here is that this

constraint is conceptual rather than factual in nature. In the case of an experi-

mental scientist, the restriction is imposed on him or her by the limit of their

mathematical knowledge. Any particular scientist labors under restrictions on

his or her knowledge of the relevant mathematical knowledge. But these per-

sonal restrictions are not inevitable or incorrigible. They can be overcome by

increasing one™s knowledge of the relevant functions or by consulting suitable

sources of mathematical information. Hence, once again we are not dealing

with absolute limits of inquiry, only epistemic ones.

These limits are nevertheless a very real factor in the history of science

and mathematics. Repeatedly, the need of knowing what a function is that

has been encountered by physicists and other empirical scientists has not only

prompted mathematicians to come to know it better, it has prompted them to

ask which function it is in the ¬rst place.

Socratic Epistemology

104

Likewise, the role of conclusiveness conditions (“presuppositions of

answers”) is highly important in experimental science. Everybody seems to

agree that one of the most important kinds of progress in science is the intro-

duction of new concepts. It is also thought fairly generally that such an intro-

duction typically happens when a new theory is introduced. What is seen here

is that mathematical concepts that facilitate the investigation of new and often

more complicated structures often take place for the purpose of answering

new experimental questions.

All these remarks can be extended from experimental questions to all com-

plex scienti¬c wh-questions.

An interesting feature of our results resolved so far is that the mathematical

or other conceptual knowledge that is required to reach conclusive answers to

empirical questions is not knowledge that”that is, knowledge of facts, proposi-

tions, or truths”but identi¬catory knowledge of certain kinds of mathematical

objects”in the ¬rst place, functions. This observation can be taken as further

evidence of the independence of the identi¬catory system of the referential

system in our actual semantics.

Knowledge of the identity of objects (of different logical types) can be

thought of as de¬nitory knowledge, at least in the sense that it answers ques-

tions to the form “What is . . . . ?” It is scarcely accidental that such questions

play an important role in the questioning method of the Platonic Socrates.

The importance of these questions is not due merely to the fact that their

answers are de¬nitions, but ¬rst and foremost to their role in all questioning

as unavoidable “presuppositions” (conclusiveness conditions) of answers to

all questions. More generally speaking, this role of the knowledge of identities

helps us to understand the important role of de¬nitions in the thought of Plato

and Aristotle. The more important questions and questioning are for a thinker,

the more important is identi¬catory knowledge likely to be for him or her.

The need to know the mathematical identity of the function-in-extension

g, as in (19), has a counterpart in the case of simple wh-questions such as (8).

There the conclusiveness condition that a response”say “b””has to satisfy

can be expressed in any of the following two forms:

K(∃x/K) (b = x) (24)

(∃x) K(b = x) (25)

where the variable ranges over persons. If so, (24)“(25) obviously amount to

saying that

I know who b is. (26)

The need of these requirements is obvious. If I do not know who b is, the

response “b” to the question (8) will not fully satisfy me.

When “b” is a proper name, (24)“(25) will express semantical knowledge.

They do not give anyone any factual information about the bearer of the

Presuppositions and Other Limitations 105

name “b.” It only tells one who is referred to by it. This kind of knowledge

is a counterpart to the mathematical knowledge expressed by (20)“(23). This

analogy throws some light on the nature of both kinds of knowledge, and raises

intriguing questions that I deal with in the next chapter.

It can be seen that this kind of identi¬catory knowledge is needed both

when the identity of a particular object is at issue and when the object in ques-

tion is a universal”for instance, a function. Indeed, one of the most interesting

results we have reached is the close parallelism between simple wh-questions

and complex experimental wh-questions. Such results throw interesting light

on the old interpretational problem concerning the what-questions of the Pla-

tonic Socrates, recently rehearsed in Benson 1992(b), as to whether he was

identifying particulars or universals.

Once again we have found genuine presuppositions of empirical inquiry.

Once again, they are factors operative in actual scienti¬c inquiry. The restric-

tions in question result from the limitations of our knowledge of mathematical

functions. But such restrictions are not unavoidable. They can be escaped by

means of whatever mathematical research it takes to come to know previously

unexamined functions typically resulting from a controlled experiment.

If there are any absolute limitations to empirical inquiry that result from

the presuppositions of answers, they can be traced back to the limitations of

our mathematical knowledge, in particular our knowledge of functions. Are

there such restrictions? An answer depends on what we take it to mean to know

which mathematical function it is that expresses a given mode of dependence”

in brief, what it means to know a certain mathematical function.

But even without having any detailed answer to this question (sic), it is

clear that the scope of one™s knowledge of the identity of various mathematical

functions can be enlarged and that the knowledge required for the purpose”as

well as the knowledge acquired in the process”is mathematical in character,

not empirical. Hence the limitations to our knowledge acquisition imposed

by the conclusiveness conditions of answers are not eternal and immutable,

but can be removed step by step by acquiring more mathematical and other

conceptual knowledge.

In sum, limitations to inquiry resulting from the initial premises can be

overcome by making more questions answerable. Limitations resulting from

presuppositions of questions present only strategic dif¬culties, not barriers

to what can be accomplished by inquiry, and limitations resulting from the

“presuppositions of answers” (i.e., conclusiveness conditions) can be overcome

by gaining more mathematical and other conceptual knowledge. And one

does not seem to face any intrinsic limitations in such a quest of mathematical

knowledge, either. Even restrictions on available answers can in principle be

removed by improved techniques of inquiry.

Acknowledging these limitations does not aid and abet in the least skeptical

or relativistic views, including the views of the “new philosophers of science”

a la Kuhn. I am even prepared to say more here, and to ask: “In view of the

`

Socratic Epistemology

106

results reported here, is it any longer intellectually respectable to hold rela-

tivistic views or otherwise believe in unavoidable restriction on our knowledge

seeking and knowledge acquisition”? The answer is, of course: “Only if you

can provide a better analysis of the presuppositions of questions and answers

than the one given here.”

References

Plato is cited in the Loeb Library translations (Harvard University Press).

Benson, Hugh H., editor, 1992(a), Essays on the Philosophy of Socrates, Oxford

University Press, New York.

Benson, Hugh H., 1992(b), “Misunderstanding the ˜What-is-F-ness” question,” in

Benson, 1992(a), pp. 123“136.

Beth, E. W., 1955, “Semantic Entailment and Formal Derivability,” Mededlingen van

de Koninklijke Nederlandse Akademie van Wetenschappen, Afd.Leterkund. N. R.

vol. 18, no. 13, Amsterdam, pp. 309“342.

Chomsky, Noam, 1959, “Review of Skinner: Verbal Behavior,” Language, vol. 35,

pp. 26“58.

Collingwood, R. G., 1940, An Essay on Metaphysics, Clarendon Press, Oxford.

Collingwood, R. G., 1993, The Idea of History, revised edition, Clarendon Press,

Oxford.

Hintikka, Jaakko, 2003, “A. Second-Generation Epistemic Logic and its Theoretical

Signi¬cance,” in Vincent Hendricks et al., editors, Knowledge Contributors, Kluwer

Academic, Dordrecht, pp. 33“55.

Hintikka, Jaakko, 1999, Inquiry as Inquiry, Kluwer Academic, Dordrecht.

Hintikka, Jaakko, 1998, “What Is Abduction? The Fundamental Problem of Con-

temporary Epistemology,” Proceedings of the Charles S. Peirce Society, vol. 34,

pp. 503“533. A revised version entitled “Abduction”Inference, Conjecture, or an

Answer to a Question” appears as Chapter 3 in this volume.

Hintikka, Jaakko, 1996, The Principles of Mathematics Revisited, Cambridge Univer-

sity Press, New York.

Hintikka, Jaakko, 1989, “The Role of Logic in Argumentation,” The Monist, vol. 72,

pp. 3“24.

Hintikka, Jaakko, Ilpo Halonen, and Arto Mutanen, 1998, “Interrogative Logic as a

General Theory of Inquiry,” in Hintikka, 1999, pp. 47“90.