(∃ f )KI (∀x)S[x, f (x)] (16)

Socratic Epistemology

120

The analogy with simple wh-questions is obvious. However, the desideratum

cannot now be expressed on the ¬rst-order level without the slash (indepen-

dence) notation, which is therefore indispensable for the purpose of generaliz-

ing the treatment just given of simple wh-questions to all other wh-questions.

What kind of reply can nature provide to an experimental question such

as (13)? In practice, an experiment usually provides a few points on the x-y

plane that represent the results of observations or measurements. They are,

furthermore, only approximations within the limits of experimental errors. The

scientist then tries to ¬nd a curve that best ¬ts these approximate observations.

There is an extensive epistemological and methodological literature on the

problems of such curve-¬tting. These problems are often thought of as the

paradigmatic epistemological problems of what used to be called “inductive

sciences.”

It is nevertheless important to realize that they do not exhaust the impor-

tant philosophical questions that arise here. Let us suppose that we idealize

boldly and look away from all the restrictions on observational accuracy and

on the number of observations that can be made. Then we can think of the

absolute limit (ideal case) of nature™s response to an experimental answer to

be a function-in-extension”that is, to say, a set of pairs of all possible argu-

ment values and the corresponding function values. Such an in¬nite class of

pairs (satisfying certain obvious conditions) of argument values and correlated

function values is, of course, what a function is, set-theoretically speaking. A

function in this sense can be thought of as being represented by a curve in the

x-y plane. No curve-¬tting is any longer needed, for the totality of the mea-

surement points is the curve. The result of such an idealized reply is that the

scientist can now say of the relevant function-in-extension g

K(∀x)S[x, g(x)] (17)

where the set of the pairs <x,g(x)> is the function in question (“function-in-

extension”).

But even if the scientist manages to ¬nd this function-in-extension”that

is, manages to make (17) true for some particular class of such pairs, can he

or she claim to know how the second variable depends on the ¬rst? In other

words, has the experiment conclusively answered the initial question (11)?

Obviously not. The result (17) will leave a serious scientist puzzled unless he

or she knows which function g(x) is, mathematically speaking. The situation

with experimental (AE) questions is precisely analogous to”nay, precisely

the same as”the case of simple wh-questions just examined. In both cases,

a response that only speci¬es a correct entity of the appropriate logical type

is not enough. The questioner must also know, or must be brought to know,

who or what this entity is. This quandary might be called, with an apology

to Heine, the Lorelei problem. (“Ich weiss nicht was soll es bedeuten. . . . ”)

If I am looking at the graph of a function without knowing which function it

represents, I am reduced to saying, “I don™t know what it is supposed to mean.”

The a priori in Epistemology 121

This requisite collateral knowledge can be expressed by a sentence of the

form

K(∀x)(∃y/K)(g(x) = y) (18)

which is equivalent to

K(∃ f/K)(∀x)(g(x) = f (x)) (19)

or

(∃ f )K(∀x)(g(x) = f (x)) (20)

which may be written elliptically as

(∃ f )K(g = f ) (21)

Here, (17) is analogous with (7), and (18) = (19) with (8) = (10). In other words,

experimental questions behave in the same way as simple wh-questions except

for the logical type of the entities (individuals versus functions) involved in

the two cases.

8. Identi¬cation of Second-Order Entities and Essentialism

We have thus found a strict parallelism between the identi¬cation of individu-

als and the identi¬cation of higher-order entities, such as functions. Admittedly,

I have explained the parallelism only for functions, but an extension to other

kinds of second-order entities is obvious. This parallelism, thought deceptively

simple, has major philosophical consequences. It shows once and for all the

hopelessness of all essentialist accounts of identi¬cation. These accounts rely

on the predicates and relations that individuals have for the purpose of deter-

mining their identity in different scenarios. Such an approach works only if

the identity of those predicates and relations is unproblematic. What has been

seen here shows that the identi¬cation of functions and”by parity of cases”

of predicates and relations”is not always automatic, but involves in principle

the same kinds of questions as the recognition of individuals. An essential-

ist approach hence merely replaces one identi¬cation problem with another.

Even if this move marks progress towards solving the problem”as I believe

it does not”it will not alone amount to a solution.

In principle, some philosopher might try to make use of the identity of

individuals picked up by a predicate as a means of identifying the predicate,

instead of basing the identi¬cation of individuals on their essential properties.

There is, in fact, a sense in which Aristotle can be said to have tried to do so.

Contrary to the usual super¬cial interpretations, a predication such as

A is B

is for Aristotle essential only if the is in it has identi¬catory force and hence

picks out a de¬nite substance or class of substances. The basis of this procedure

Socratic Epistemology

122

of Aristotle™s is that he does not make the Frege-Russell assumption that

words for being are ambiguous between expressing identity, predication, and

existence. An Aristotelian is or rather, estin can have all these senses as its

components, but it can also lack some of them. (See Hintikka, 2006.)

9. The Nature of the Conclusiveness Conditions

It is important to realize what the need of the conclusiveness condition (18)

means and what it does not mean. One thing it does not mean is that we are

considering functions otherwise than merely extensionally, let alone that we

are somehow smuggling intensional entities into mathematics. The values of

the function variable f in (19)“(21) are functions-in-extension, not any kind

of intensional entities, quite as fully as the values of the variable x in (8) or

(10) are ordinary ¬‚esh-and-blood human beings, not ghosts called “possible

individuals.” The additional force conditions such as (8), (10), or (18)“(21) is

merely that the identity of these ordinary extensional entities must be known

to the questioner.

There is admittedly a temptation to think that the idealized experimenter

who has constructed a perfect graph of the dependence of y on x on his or her

graph paper must know the function that codi¬es the dependence, so that (18)“

(21) are for him or her true, after all. For surely such an idealized experimenter

can be said to know the full set of pairs of argument values and corresponding

function values that is the function, extensionally or set-theoretically speaking,

or so it seems.

This temptation is a nice illustration of the complexity of the conceptual situ-

ation, and also an illustration of the distinction between two different senses

of identi¬cation that I have made and applied elsewhere. (See, e.g., Hintikka

1989 and 1996(b).) They have been called public and perspectival identi¬ca-

tion. A factually omniscient deity sees the entire function-in-extension in one

fell swoop and consequently can identify it perspectivally. But even such an

omniscience about matters of observable fact would not automatically enable

the deity to identify publicly the entity whose world line it is. For the deity in

question need not be mathematically omniscient. Even a ¬ctional compleat

experimenter is still in the same position as we humans are in that he or she sees

the function but does not see (visually know) which function it is, just as one

can see (in the sense of laying one™s eyes on) the murderer of Roger Ackroyd

without seeing who he or she is. Hence the correctly diagnosed temptation

under scrutiny does not in reality show that any non-extensional entities are

involved in the identi¬cation of functions.

The distinction between the two modes of identi¬cation and its applicability

here can be illustrated in different ways. One set of clues is yielded by the

re¬‚exions of the distinction in natural language. Yes, perhaps the experimenter

may be said to come to know the function, but that does not amount to knowing

The a priori in Epistemology 123

what function it is. This seemingly minor grammatical contrast signals in reality

a distinction of major importance between two modes of identi¬cation. (See,

e.g., Hintikka 1989 and 1996(b).) Suf¬ce it to point out merely that there is a

distinction between seeing a person and seeing who he or she is and likewise

between knowing a person (being acquainted with him or her) and knowing

who he or she is. The same distinction can be made”and must be between

knowing a function and knowing what function it is.

The point I am making here perhaps becomes clearer if one thinks of graphs

of functions as one notation among many in which one can deal with functions.

If this is legitimate, then an experimenter™s case is strictly parallel to that of

an inquirer who asks a simple wh-question and receives a reply. Like such an

inquirer, an experimentalist can refer correctly to nature™s answer to his or her

question; that is what the graph does. But that does not automatically (read: