g(y1 , y2 , . . .) is replaced by an individual constant.

This shows also how knowing the identity of a function can be expressed on

the ¬rst-order level, thus removing one of the limitations of the ¬rst-generation

epistemic logic.

This generalizes the notions of presupposition, desideratum, conclusiveness

condition, and the question“answer relation to all questions, with the partial

exception of why- and how-questions, which have to be dealt with separately.

With this quali¬cation, it can be said that we have reached the ¬rst fully explicit

and general logic of questions and answers. This generality is of considerable

interest for the purposes of both philosophers and linguists. For philosophers,

one of the many interesting things about this second-generation epistemic

logic is that it is a ¬rst-order logic. All quanti¬cation is over individuals, thus

avoiding all the dif¬cult problems concerning the existence of higher-order

entities.

Linguists might be interested in the fact that in the framework of semantic

representation provided by epistemic logic, the slash (/) is the embodiment of

the question ingredient. Applied to disjunctions (as in (∨/’)), it creates propo-

sitional questions, and applied to the existential quanti¬ers (as in (∃ x/’)),

Socratic Epistemology

78

it creates wh-questions. This also throws light indirectly on the semantics of

the question ingredient in natural languages.

The independence notation has further uses. For one thing, it enables us to

express what is commonly referred to as common knowledge. In the case of

knowledge common to two agents, a and b, it means that they not only share

the same information, but that each of them knows that the other one knows

it, and that each knows that the other one knows that the other one knows it,

and so on. This is achieved by making the order of the initial K™s irrelevant and

also using as the wh-ingredient (“/Ka Kb ). For instance, the following sentence

expresses the idea that it is common knowledge between a and b whether it is

the case that S:

Ka (Kb/Ka )(S(∨/Ka Kb) ¬S)

This shows that in order to express common knowledge in general, we have to

make sure that the relevant knowledge operators are on a par in the slashes,

and not only when they are pre¬xed to a sentence. And, if so, we can see that

common knowledge could not have been formulated in general terms in the

¬rst-generation epistemic logic.

The result of this liberalization of epistemic logic is a simple but powerful

logic of knowledge, including a logic of questions and answers. This second-

generation epistemic logic ful¬lls the promises that the ¬rst generation sug-

gested but did not deliver. By its means, we can carry out the promised analyses

of all different kinds of knowledge and all different kinds of questions, using

the knows that operator K as the only epistemic ingredient.

Thus, for instance, knowledge de dicto and knowledge de re are but different

variants of the same basic notion of knowledge. Moreover, the important

difference between two different modes of identi¬cation can be seen not to

imply any differences in the kind of information (knowledge) involved in

them. This fact turns out to have interesting consequences even outside logic

and philosophy in neuroscience, where it helps to understand the two visual

cognition systems sometimes known as the where-system and the what-system.

(See Hintikka and Symons 2003.)

The notion of independence also enables us to deepen the analysis of the

de dicto versus de re contrast sketched earlier. It is possible to assume that in a

semantical game, the non-logical constants are not initially interpreted by the

participants. Instead, the references that are assigned to them are chosen by

the veri¬er as a move in the game. In fact, Hintikka and Kulas have shown that

this assumption serves to throw light on certain regularities in the semantics of

ordinary language. It is perhaps not obvious that such mileage can be obtained

from the assumption, for there is clearly only one choice that can win”namely,

to assign to the constants their linguistically determined references. The extra

mileage is nevertheless real, for once there is a game rule for the interpre-

tation (i.e., assignment of references), applications of this rule may or may

not be independent of other moves in the game. (This is the opening utilized

by Hintikka and Kulas. In particular, the choice of the reference of b in a

Second-Generation Epistemic Logic 79

sentence such as (21) may or may not be independent of the choice of the sce-

nario prompted by KI . Earlier, it was assumed to be dependent on the choice

of the scenario. It can be made independent notationally by replacing (21)

with

KI M((b/K)r ) (45)

This is obviously what is meant by taking b to be de re. Almost equally obvi-

ously, (45) is logically equivalent to the conjunction of (21) and (23).

General terms can be treated in the same way. This possibility is perhaps

somewhat more conspicuous in the case of belief than of knowledge, but this

is merely a matter of degree. For instance, consider the statement

Ka (P/K)(b) (46)

It says that a knows that b is one of the individuals who or what in fact are P.

It does not require that a know that they and only they have the property P.

These extensions of the independence notation show several interesting

things. It reinforces my earlier result concerning the de¬nability of the de dicto

versus de re contrast. It also shows that this contrast is not restricted to singular

terms. Furthermore, we can now see that the familiar contrast between the

referential and predicative uses of singular terms such as de¬nite descriptors

is not an irreducible one but is rather a matter of different construction in

terms of the same basic notions.

An especially interesting generalization of earlier insights that is now

brought out into the open is the partly conceptual character of the conclu-

siveness conditions of all wh-questions, simple and complex. For instance, a

reply to an experimental question whose desideratum is of the form (12) is a

function-in-extension”that is to say, a mere correlation of argument values

and functions-values like a curve on graph paper. Such a reply does not qualify

as an answer because the questioner might not know which function is repre-

sented by the correlation or by the graph. Knowing its identity is conceptual”

in this case, mathematical”knowledge. This throws into a strikingly sharp

pro¬le the role of mathematics in experimental (and hence presumably

empirical) science and even the indispensability of mathematics in science

in general. Likewise, the partial strategic parallelism between deduction and

questioning can now be generalized. As the ¬rst step, we can epistemologize

the questioning processes in the following way:

(i) Every initial premise S is replaced by KI S.

(ii) Every answer A is replaced by KI (A & C(A)), where C(A) is the

conjunction of all the conclusiveness conditions for A.

This does not yet generalize the parallelism between question“answer steps,

and logical (deductive) steps, which suggested a strategic near-identity of inter-

rogation and deduction. We have generalized the rule for question“answer

steps in inquiry, but we do not have any rule of deductive inference that would

Socratic Epistemology

80

match it. Such a parallel rule can be obtained by generalizing the rules of purely

deductive reasoning. The original analogy was between existential instanti-

ation and simply wh-questions. The existential quanti¬er whose variable is

instantiated must occur sentence-initially in the received rule applications

of existential instantiation. Why can we not instantiate also inside a larger

formula? (Surely an existential quanti¬er expresses there, too, the availabil-

ity of truth-making individuals.) The answer is that the “witness individual” in

question will then depend on certain other individuals. Hence the instantiating

term must be a function of those individuals. Formally speaking, we can extend

the rule of existential generalization so as to allow to replace a sub-formula of

the form

(∃x)F[x] (47)

of a larger formula by a formula of the form (41), except that now g must be

a new function symbol or a “dummy function symbol,” if they are used as a

separate category of symbols. Some people might want to call them names of

“arbitrary functions,” in analogy with “arbitrary individuals.”

This clearly serves to extend the parallelism between deduction and ques-

tioning discussed earlier. In a context of pure discovery, the optimal strate-

gies of questioning parallel the optimal strategies of deductive inference, in

the sense explained. Even though deductive steps and interrogative steps in

inquiry are different from each other, they are, in the case of pure discovery,

governed by the same rules. As I sometimes have put it, Sherlock Holmes was

right: All good thinking is logical thinking. More cautiously speaking, we have

located one important role of deductive logic in inquiry”namely, the role of

providing strategic advice for reasoning in general.

Along the same lines, we can also at once remove the most important limita-

tion to the applicability of the logic of questions and answers and hence of the

applicability of epistemic logic. We can now reconstruct the process of answer-

ing principal questions by means of operative questions. How can we do it?

Simply by using the desideratum of the principal question as the target propo-

sition to be established by the interrogative process. Naturally this involves

making the epistemic element explicit throughout, as indicated earlier. For