However, the parallelism that has just been discussed apparently cannot

be generalized. It is not even clear in general what answers to more complex

questions will look like logically, nor is it clear what their presuppositions

might be. And even if answers to these questions were available, there appar-

ently are no rules of logical inference that could parallel the relevant complex

question“answer steps. This might seem to jeopardize the entire strategic anal-

ogy deduction and interrogative inquiry.

Other limitations are likewise conspicuous. Perhaps the most important

shortcoming of ¬rst-generation epistemic logic confronts us when we begin to

emulate Socrates and Aristotle and model all inquiry as a questioning process.

Such a model is straightforward to implement as long as the inquirer is given a

¬xed conclusion that it be established through an interrogative process starting

from given initial premises. This may be enough to answer why-questions

through a questioning process. However, there does not seem to be any way

of analyzing similarly the all-important method of answering questions”that

is, initial “big” or principal questions, by means of a number of “small” or

operative questions. This would be a serious limitation to any application of

the logic of questions and answers to epistemology.

In view of such applicational shortcomings of ¬rst-generation epistemic

logic, it might in fact look as if the philosophical community could be excused

when it has so far turned a deaf ear to the interesting and important philo-

sophical vistas suggested by the observations so far described.

3. Promises Ful¬lled by Means of the Notion of Independence

I have so far been telling a story that is partly historical, partly systematic.

That story has led us to a tantalizing impasse. On the one hand, by considering

simple examples, we can discover highly interesting philosophical suggestions

apparently implicit in our epistemic logic. On the other hand, these suggestions

apparently cannot be generalized from the simple cases, in which they are

more or less obvious, to more complex cases. Sometimes such an extension

can apparently be accomplished only by appealing to higher-order entities that

lead us to problems that are at least as recalcitrant as the ones we were trying

to overcome. What are we to do?

The most popular response in this day and age seems to be to throw up our

hands and claim that conceptual realities in epistemology are just too complex

and too context-dependent to be captured by the clumsy tools of epistemic

logic. Apparently it would be politically correct in this situation to evoke such

phrases as “family resemblance” or “fuzzy logic.”

Alternatively, some philosophers might decide to evoke more examples and

to develop much more detailed taxonomy and other theory for these realities,

in the style of empirical linguists.

Second-Generation Epistemic Logic 75

The main message of this chapter is that both reactions would be dead

wrong. Not only can all the dif¬culties I have described be solved, they can

be solved in one fell swoop. This swoop is provided by the same approach

that has prompted a revolution in the foundations of ordinary non-epistemic

¬rst-order logic. It is usually referred to as game-theoretical semantics, but

what is important in it is not the use of game-theoretical ideas per se. Rather,

the crucial insight is that the dependence of real-life variables on each other

is expressed in a logical notation by the formal dependence of the quanti¬ers

on each other to which they are bound. This insight motivates a change even

in the notation of ¬rst-order logic. In the usual notation, the dependencies

between quanti¬ers are indicated by the nesting of their scopes. But such a

nesting relation is of a rather special kind. Among other things, it is transitive

and asymmetrical. Hence not all patterns of dependence and independence

can be expressed by its means. And hence the received logical notation does

not do its job adequately, and has to be made more ¬‚exible. Since the problem

is to enable us to structure our formulas more freely, it could in principle be

solved without introducing any new notation and merely relaxing the scope

rules”that is, the formation rules for the pairs of parentheses that indicate the

dependence relations between quanti¬ers. In practice, it is more perspicuous to

introduce instead a special independence indicator (Q2 y/Q1 x) that expresses

the independence of the quanti¬er (Q2 y) of the quanti¬er (Q1 x). Game the-

ory comes in in that such independence can be modeled in game-theoretical

semantics by the informational independence of the move mandated by (Q2 y)

of the move mandated by (Q1 x), in the general game-theoretical sense of infor-

mational independence. The systematic use of this notation results in the ¬rst

place in what is known as independence-friendly (IF) ¬rst-order logic.

The use of the slash (independence) notation is not restricted to the usual

extensional ¬rst-order logic, however. One remarkable thing here is that this

notion of independence applies to all semantically active ingredients of a sen-

tence whose semantics can be formulated in terms of a semantical game rule.

The epistemic operator Ka is a case in point in that it mandates a choice by

the falsi¬er of one of a™s knowledge worlds with respect to which a semantical

game is to be continued. Accordingly, the slash notation makes sense also in

epistemic logic.

How, then, does it help us? Consider, in order to answer this question,

sentence (7):

(∃x)KM(x, r )

It expresses its intended meaning “it is known who murdered Roger Ackroyd”

because the “witness individual” value of x must be chosen before the choice

of a possible world (scenario) mandated by K. Hence this individual must be

the same in all of them.

But for this purpose the value of x must not necessarily be chosen in a

semantical game before the choice of the scenario (possible world) connected

Socratic Epistemology

76

with K. It suf¬ces to make the choice independently of the scenario choice.

Hence (7) says the same as

K(∃x/K)M(x, r ) (34)

Likewise in (17), the choice of the witness individual (the trusted person)

depends on the mother in question but not on the choice of the possible world

prompted by KI . Hence the right logical form of (17) is

K(∀x)(∃y/K)T(x, y) (35)

As can be seen, this is a sentence of IF ¬rst-order epistemic logic, and hence

independent of all the problems connected with higher-order quanti¬cation.

It is easily seen that (35) and its analogues do not reduce to a slash-free

notation. Hence the criticisms reported earlier are valid, albeit only as long

as the IF notation is not used. But as soon as this notation is available, we

can solve one of the problems posed in Section 2”namely, the problem of

expressing the desiderata of complex wh-questions on that ¬rst-order level.

The same notation can be extended to propositional connectives. For

instance, we can write

K(S1 (∨/K)S 2 ) (36)

This is readily seen to be equivalent to

(KS1 ∨ K)S 2 ) (37)

In more complex examples, however, the slash notation is not dispensable. An

example is offered by

K(∀)(F1 (x)(∨/K)F2 (x)) (38)

In general, we can take any sentence of the form

KS (39)

where S is a ¬rst-order sentence in a negation normal form. If in (39) we

replace one or more existential quanti¬ers (∃ x) by (∃ x/K) and/or one or more

disjunctions (F1 ∨ F2 ) by (F1 (∨/K)F2 ), we obtain an epistemic sentence that

can serve as the desideratum of a (usually multiple) question.

The possibility that there are several slashes in the desideratum means

the possibility of dealing with multiple questions. Their behavior in natural

language turns out to be a most instructive chapter of epistemic logic, as docu-

mented in Hintikka 1976. In this chapter, I will not deal with sentences with

multiple K™s”that is, with iterated questions.

Thus the received terminology already embodies a mistake. What happens

in quanti¬ed epistemic logic is not “quantifying into” an opaque context, but

quanti¬ng independently of an epistemic operator and the moves it mandates.

The resulting logic of K-sentences can be considered a second-generation

epistemic logic (or a fragment of such a logic). In that logic, the most important

Second-Generation Epistemic Logic 77

concepts relating to questions and answers can be de¬ned for all different

kinds of questions. For one thing, if the desideratum of a question is (39), its

presupposition is obtained by dropping all the slashes /K. A reply to such a

question brings about the truth of a sentence in which each slashed existential

quanti¬er sub-formula

(∃x/K)F[x] (40)

is replaced by

F[g(y1 , y2 , . . .)] (41)

where (Qy1 ), (Qy2 ), . . . are all the quanti¬ers on which the quanti¬er (∃ x/K)

in (40) depends in (39). Likewise, each disjunction (F1 (∨/K)F2 ) occurring as a

sub-formula of (39) is replaced by

((g(y1 , y2 , . . .) = 0 & F1 ) ∨ (g(y1 , y2 , . . .) = 0 & F2 )) (42)

where (Qy1 ), (Qy2 ), . . . are all the quanti¬ers on which the disjunction in ques-

tion depends in (39).

This reply amounts to an answer if and only if the conclusiveness conditions

of the form

K(∃ f/K)(g = f ) (43)

are satis¬ed. Instead of (43), we could write

K(∀y1 )(∀y2 ) . . . (∃z/K)(g(y1 , y2 , . . .) = z) (44)