it might in fact be more appropriate to speak of the “logic of information”

than of the “logic of knowledge.” Unfortunately the term “information” has

other misleading overtones. But fortunately this issue does not affect what will

be done in this chapter.

Another problem area that I will not deal with fully in this chapter is the

behavior of identity. One reason why it would need a longer discussion is that

the received approaches are seriously off the mark. The source of the problem

is the fact that in epistemic and other intensional contexts, we have to consider

individuals as potential members of several scenarios. This is true in particular

Socratic Epistemology

64

of individuals considered as values of bound variables. Hence we must have”

or, rather, there must be implicit in the semantics of our language”criteria of

identity for denizens of different scenarios. They are sometimes called “crite-

ria of cross-identi¬cation.” How are we to deal with them? Many philosophers

and linguists have approached this problem by means of the notion or refer-

ence. This is especially true of Kripke and his acolytes. Kripke postulates a

special kind of direct or rigid reference to explain the identity in question.

This “new theory of reference” nevertheless offers us merely a good analogue

of Karl Krauss™s dictum about psychoanalysis. The so-called new theory of

reference embodies the very problem it is supposed to be a solution to. What

a system of reference does is provide criteria that tell us what the references

of our expressions are in each of the different possible scenarios that we might

want to approach by means of our language. Hence, almost by de¬nition,

such a system does not tell anything about the identities of individuals (or

of objects of a higher type) in different scenarios. For this purpose, we need

another system”or rather, there is another system embedded in our working

conceptual system”governing such identities. This system might be called the

“identi¬cation system.” It turns out to be largely independent of the reference

system. Hence, what is wrong about the new theory of reference in the ¬rst

place is that it is a theory of reference.

Epistemic logic plays an interesting role here in that it provides speci¬c

examples of the failure of the “new theory of reference.” For in whatever way

the reference of (say) a singular term is determined, it always makes sense to

ask, “Does b know what a is?” The answer cannot turn on the grammatical or

logical category of “a” used earlier. It is always a factual question.

Of course, a full treatment would here involve discussing the alleged mech-

anisms of creating direct reference that Kripke and others have proposed

(Kripke 1972). Suf¬ce it here merely to put Kripke™s idea of dubbing into a

historical perspective. Kripke™s idea is but a dramatized version of the old idea

that it is ostension that provides the basic semantical links between language

and reality. Wittgenstein held it for a while”rather a long while, if I am right.

However, he eventually came to reject it, for reasons that at least prima facie

apply quite as well against Kripke.

To put the same point in different terms, to say that “a” behaves like a rigid

designation in b™s knowledge worlds is to say that

(∃x)Kb(a = x) (1)

But what this expresses is the fact that b knows what a is (see later.) And this

cannot be guaranteed by the meaning of “a” alone. Whether or not (4) is true

depends crucially on the identi¬cation system one is relying on.

In this sense, epistemic logic provides strong evidence against any theory

of direct reference. No wonder, therefore, that the new theorists of reference

have studiously neglected it, in spite of its importance for applications of logic.

Second-Generation Epistemic Logic 65

2. The Promises

Now, what are the promises of such an epistemic logic? What questions does it

help us to answer? Well, what questions are we likely to ask in epistemology?

One of the ¬rst concerns surely the objects of knowledge. When one knows

something, what is the knowledge about? Interesting prima facie distinctions

are codi¬ed in the syntax of ordinary language. We speak of knowing truths,

propositions, and facts. Such knowledge is expressed by the knows that con-

struction that is incorporated in our epistemic logic. But how can we express

what might be called “knowledge of objects””that is, the kind of knowledge

expressed in English by what is known as simple wh-constructions such as

knowing who, what, where, when and so on. In the simplest cases, the answer

seems obvious. If K expresses it is known that and the variable x ranges over

persons, then the sentence

It is known who murdered Roger Ackroyd (2)

can be expressed by

It is known of some particular person x that x murdered

Roger Ackroyd (3)

Or, more explicitly, by

(∃x)K(x murdered Roger Ackroyd) (4)

which has the form

(∃x)KM(x, r ) (5)

For what else could be meant by knowing who did the dastardly deed than

knowing of some particular individual x that x did it?

Indeed, this is a viable analysis of simple wh-knowledge. The simplest case

of such simple knowledge is knowing the identity of an individual.

By the same token as before,

It is known who b is (6)

is equivalent to

It is known of some particular individual x that b is x (7)

which has the form

(∃xK(b = x) (8)

All that is presupposed by such analyses is some systematization of the logic

and semantics of the logic of knowledge along the lines indicated here. What

this amounts to is some version of what is known as “possible worlds” semantics

for epistemic logic, including a system of reference and a separate system

of cross-identi¬cation. (If one shares my distaste for the “possible worlds”

terminology, one may speak of possible scenarios instead.)

Socratic Epistemology

66

Knowing what an entity of a higher type is can likewise be expressed in terms

of K, but now we have to quantify over higher-order entities. For instance,

knowing which function g(x) is can be expressed by

(∃ f )K(∀x)(g(x) = f (x)) (9)

where f is a function variable. This might be abbreviated as

(∃ f )K(g = f ) (10)

which brings out the parallelism between (10) and (8).

Unfortunately, this analysis of what it means to know the identity of a

function is in terms of higher-order quanti¬cation (quanti¬cation over func-

tions). Such quanti¬cation promptly leads to an avalanche of dif¬cult prob-

lems. Which higher-order entities exist? If we know an answer to that question,

we could decide what axioms to posit in set theory. What does it mean for a

higher-order entity to exist, anyway? We would obviously be much wiser if we

could dispense with higher-order quanti¬cation.

The distinction between knowledge of propositions (or truths) and knowl-

edge of entities has many intriguing applications. Here I will mention only

one. It is obvious that intuitionistic mathematics is calculated to deal, not so

much with mathematical truths, as with our knowledge of mathematics. But if

this knowledge is assumed to be propositional, very little seems to be accom-

plished. Indeed, S is provable in the usual epistemic logics if and only if KS is

provable.

The real novelty is, I have argued, that intuitionists (the original ones, not the

soi-disant ones of our day and age) were not concerned with our knowledge of

mathematical propositions, but with our knowledge of mathematical objects.

Consider, for example, the axiom of choice. Does a choice set always exist?

We can consider this question till we are blue in the face without ¬nding an

easy answer. In contrast, it is easy for someone to admit that we do not always

know what a choice function would be. I will not pursue this matter here,

and use it only to illustrate the tremendous interest of the distinction between

knowledge of truths (propositions) and knowledge of objects (entities).

This distinction between the knowledge of propositions and knowledge of

entities is thus of considerable philosophical and other theoretical interest. For

another application, it shows that at least in the simplest cases, we can analyze

knowledge of objects in terms of knowledge of propositions. This is shown

by expressions like (8) (or perhaps also (5)“(7)), in that the only epistemic

element they contain is the knowing that operator K.

More generally speaking, one thing that epistemic logic seems to promise

is an analysis of different kinds of knowledge in terms of the single operator

K. This project can be carried out in some cases, in addition to the analysis

of simple wh-knowledge just outlined. For instance, why- and how-knowledge

is obviously more complex conceptually, but can be brought within the scope

of such analysis. (See, e.g., Hintikka and Halonen 1995.) Most importantly, an

Second-Generation Epistemic Logic 67

insight into the possibility (and indeed presence in our actual discourse) of the

different modes of identi¬cation opens up possibilities of analyzing different

types of knowledge by acquaintance”that is, of the kind of knowledge that is

in natural languages expressed by a direct (grammatical) object construction.

(see Hintikka 1975, chs. 3“4, and 1990; Hintikka and Symons 2003.)

Such analyses have a philosophical interest that goes way beyond whatever

logico-linguistic relevance they may have. For instance, the fact that the only