the different relevant possibilities are. For a logical theory of information and

for many of its applications, it suf¬ces to rely on the distinctions between

different possibilities that are made by the very language we are using. These

distinctions give us a useful starting point for further analysis. Even though

there may be a price to pay for this reliance on language, it enables us to get

over Quine™s objections once and for all.

These possibilities are easiest to distinguish from each other in the case

of propositional language, conceived of as truth-function theory or Boolean

algebra. There, the ¬nest partition of the possibilities concerning the world are

expressed by what have been dubbed “constituents.” For instance, if there are

only two atomic propositions in our language, A and B, then the constituents

are:

(A & B)

(A & ¬B)

(¬A & B)

(¬A & ¬B) (1)

Any proposition of our truth-functional mini-language can be represented

as a disjunction of some of these constituents. This representation is called

“distributive normal form.” This form displays the logical status of the propo-

sition in question, and the respective normal forms of two propositions display

their logical relationships. When the normal form of S1 is a part of that of S2 ,

S1 logically entails S2 . As extreme cases, we have propositions whose normal

form is a disjunction of all constituents. Such a proposition is logically true but

does not convey any information. It is literally empty, or nugatory. Wittgen-

stein proposed calling such propositions “tautologies,” and the name has stuck.

Such tautologies are entailed by all propositions. More generally, when S1 logi-

cally entails S2 , the latter does not add anything to the information of the for-

mer, because it does not rule out any possibilities that the former doesn™t

already exclude. In this sense, purely logical inferences are tautological.

At the other end of the spectrum there are propositions that exclude all

possibilities. They are contradictory propositions, and they entail all the other

propositions.

More generally speaking, other things being equal, the longer the normal

form of a sentence S is, the more probable it is, and the less information it car-

ries. The reason is that the longer the normal form of S is, the more possibilities

it admits. One might even try to use the relative number of constituents in the

normal form of S as the logical probability of P(S) of S, and the relative number

of constituents excluded by S as a measure of the informativeness of S. Clearly,

Who Has Kidnapped Information? 193

I(S) = 1 ’ P(S). A differently calibrated measure of information inf(S) can

be de¬ned as ’log P(S).

In an applied propositional language, these measures are appropriate only

on the assumption that all the constituents are equiprobable, which means that

there are no probabilistic connections between the atomic sentences. Other-

wise, different constituents will have to be weighted differently. This would

mean that measures of probability and information are not determined purely

logically. Likewise, the status of logical truths as tautologies is predicated on

the absences of any logical connections between the atomic sentences in the

sense that all the constituents must be logically possible.

Subject to such quali¬cations, propositional languages constitute an

information-theorist™s paradise. The notion of information admits of a sharp

de¬nition, and it is related to the logical relations between propositions in a

clear-cut and natural way.

Wittgenstein believed initially that all logic could be considered truth-

functional logic. Then the paradisical situation just described would hold in

general. Unfortunately, such a belief is unrealistic, and was in fact given up

by Wittgenstein himself. The question now becomes as to whether, and if

so how, the treatment of the concept of information can be extended from

propositional languages to others. As a representative case of such “other”

languages, we can consider languages whose logic is the usual ¬rst-order logic

(aka quanti¬cation theory).

Logical positivists tried in effect to implement a generalization by giving up

the speci¬c idea of de¬ning notions such as tautologicity and information in

terms of admitted and excluded possibilities, and trying instead to rely on the

idea of truths based only on the meanings of our language. This approach has

been justly criticized. (See, e.g., Dreben and Floyd 1991.) Perhaps the worst

thing about it is that how such a determination by meanings alone is supposed

to work has never been spelled out in realistic detail.

Somewhat surprisingly”and perhaps also embarrassingly”it turns out that

means of generalizing the treatment of information from propositional lan-

guages to all ¬rst-order languages have been available to philosophers for

a long time, indeed almost precisely for a half century. This generalization

is accomplished by extending the theory of distributive normal forms to the

entire ¬rst-order logic. (See Hintikka 1953 and 1965.) Given a supply of non-

logical predicates (properties and relations), we can again de¬ne the con-

stituents that at a given quanti¬cational depth do specify the basic possibilities

concerning the world. It is even possible to describe in intuitive terms what such

constituents are. They are rami¬ed lists of all the different kinds of sequences

of d individuals that exist in a world in which the constituent in question is

true. By “rami¬ed” I mean, of course, lists in which we keep track of which

sequences share an initial segment of a length shorter than d members.

If we restrict our attention to propositions of a given depth d and to con-

sistent constituents, the situation can be said to be precisely the same as in

Socratic Epistemology

194

truth-functional propositional logic. With this quali¬cation, it is thus the case

that we can use at least a comparative (qualitative) notion of information in

all ¬rst-order languages. I will call this notion of information “depth infor-

mation.” We can also identify the truths of ¬rst-order logic with tautologies

in the sense of propositions that do not convey any depth information. (We

might call them “depth tautologies.”) Valid logical inferences do not increase

the depth information of our propositions. If we could restrict our attention

to consistent constituents only, the situation would be precisely the same as in

truth-functional logic.

But we cannot do so without further ado. What has been said so far, unfor-

tunately or fortunately, is not the end of the story. The rest of it has so far been

covered by the innocent-looking phrase “consistent constituent,” which I used

in introducing the extension of the idea of information from truth-function the-

ory to ¬rst-order languages. The notion of constituent is a syntactical (formal)

one. The distributive normal form of a sentence as a disjunction of constituents

can be thought of as being reached by a purely formal rule-governed trans-

formation. Only the excessive length of most constituents prevents us from

writing out the distributive normal form of any one sentence. But some of

such purely formally de¬ned constituents are inconsistent. And we cannot

simply look away from such inconsistent constituents, for there is no mechani-

cal (recursive) method of recognizing inconsistent constituents. Indeed, such a

recursive method of recognizing inconsistent constituents would yield a deci-

sion method for ¬rst-order logic.

Now a partial method of weeding out inconsistent constituents is implicit

in the very meaning of a constituent. As was indicated, a constituent is a list of

all the kinds of individuals that exist in the world. These kinds of individuals

are de¬ned by lists of all individuals that exist in relation to them, and so on.

In other words, a constituent C is a rami¬ed list of all the different sequences

of d individuals (down to the depth d of the constituent in question) that one

can “draw” from a world in which C is true. After i draws, i < d, and there still

are further individuals to be “drawn” from the world. Those different lists of

individuals must all match. If they do not, the constituent in question is incon-

sistent. Such a constituent is called “trivially inconsistent.” In other words, a

trivially inconsistent constituent is one whose several parts do not match.

The striking fact here is that there are inconsistent constituents that are not

trivially inconsistent. They look on the surface like consistent constituents”all

their different parts cohere with one another. The inconsistence of one of them,

say C, will emerge only when one transforms C into disjunctions of increasingly

deeper constituents. If C is inconsistent, then at some greater depth d, all those

deeper constituents will turn out to be trivially inconsistent. (This illustrates

the importance of the parameter that is here being called “depth.”) However,

there is no recursive function that tells us how deep we have to go in order for

such a hidden inconsistency to surface. This is the ultimate logico-mathematical

reason why we cannot simply dismiss inconsistent constituents. We do not

always know in practice which ones we would have to dismiss.

Who Has Kidnapped Information? 195

In order to convey to the reader a sense in which a constituent can be

inconsistent, I can perhaps offer some informal explanation. A constituent

is like a jigsaw puzzle or a domino set. It does not give one a picture of the

world; it merely presents one with a supply of pieces out of which one has

to assemble a picture. One is given a ¬nite number of inexhaustible boxes

of pieces of different kinds. The constituent states that at least one piece

from each box must be used and that the puzzle can be completed. Thus,

a constituent may imply the existence of an individual that cannot exist, or

imply the existence of two different possible individuals that cannot exist in

the same world. For instance, a constituent might imply the existence of a

person who loves all and only those people whose love is never requited.

(We might call such a person the universal consoler.”) Or a constituent might

assert the existence of both someone who hates everyone and the existence