observations illustrate the sense in which the consequence relation (1.11) can

be said to be “trivial” or “analytic.”

But in what sense does the interpolation formula IL provide an explanation

of why G follows from F”that is, why G must be true if F is? Let us assume,

for simplicity, that no individual constants occur in F or in G, and that F and

G are in the negation normal form. The atomic sentences in each alternative

branch describe a part of a possible scenario in which F is true. This scenario

involves a number of individuals. Where do they come from? A look at the

situation (see the L-rules shown earlier) shows that there are two kinds of

such individuals. Some of them are introduced by existential instantiation.

They are individuals that exist in a scenario in which F is true. They typically

depend on other such individuals. These dependencies are what the Skolem

functions of F codify. The introduction of these individuals hence amounts to

an analysis of the dependence relations among the individuals in models of

F. In any case, the individuals said to exist by IL must exist in the particular

scenario corresponding to the branch in question.

The other individuals are introduced to the con¬guration of individuals in

the scenario in question by applications of universal instantiation with respect

to individuals that occur on the right side of the tableau. They are introduced

into the right side by right-side applications of universal instantiation. What is

the intuitive meaning of such individuals? They instantiate “arbitrary individ-

uals” in the models of G, individuals whose identity is not known to us. There

is a notion that is not usually employed in logical theory but that is familiar

from routine mathematical thinking and routine mathematical jargon. They

are the unknown individuals used in the proof. If G is false, they are the indi-

viduals calculated to serve as counterexamples vouchsa¬ng its falsity. Hence,

if G turns out to be true even for such test case values of universally quanti¬ed

variables, G must be true in any case (if F is true). Naturally one does know

Socratic Epistemology

170

which individuals they are. Applications of universal instantiation on the left in

a tableau constructor mean that we are applying to such unknowns the general

laws we know hold in the relevant model of F.

Thus a scenario that corresponds to a branch on the left side involves two

kinds of individuals. On the one hand, there are the individuals that must exist

in the scenario in question. On the other hand, there are the “unknowns” of

which we need know that they have to conform to the general truths prescribed

by F.

Thus the left interpolation formula I tells us everything about the structures

in which F is true that is needed to understand the consequence relation from F

to G. Likewise, the right interpolation formula IR tells us everything about the

models of G that is needed to see the consequence. Seeing that IL IR then

completes the explanation. We can speak of “seeing” here because the validity

of IL IR can be established without considering any individuals already con-

sidered in IL or in Ir . In other words, the validity of IL IR can be seen by using

merely propositional reasoning, rather like what happens in the apodeixis part

of a proposition in Euclid. We can, as it were, argue about and the same the

con¬guration in question without having to take into account anything else.

Such reasoning is “analytic” in the etymological sense: It suf¬ces to analyze the

“data” of the given con¬guration or ¬gure, without carrying out any “auxiliary

constructions.”

If such a “seeing” does not qualify as an explanation of why G must be true if

F is, it is hard to imagine what could so qualify. Now it is known (and can easily

be proved) that any logical consequence relation in the usual ¬rst-order logic

can be proved by means of the simpli¬ed tableau method used here. Thus we

have proved the ¬rst main thesis of the chapter. Each ¬rst-order logical proof

of G from F can be transformed into a form where it provides an explanation

as to why G must be true if F is. Logical proofs yield (or can be transformed

so as to yield) explanations as to why the consequence relation holds between

them. “Logical explanations” can be rightly so called.

2. The Structure of Explanation

Needless to say, not all actual explanations are “logical explanations” in this

simple sense. To think so would be to hold an extreme form of what might be

called the “covering-law model” of explanation. According to such a view, to

explain what happens is to subsume it under a general law. This is an over-

simpli¬ed view. From a general law alone preciously little can be deduced.

A more realistic account is provided by Hintikka and Halonen (2005). Even

C. G. Hempel, who is usually credited with (or blamed for) the covering-law

model, acknowledged that further premises are needed in explanation. These

premises are not general laws but are ad explanandum in the sense that they

specify certain features of the explanatory situation, perhaps initial conditions

or boundary conditions. Discovering them is typically the crucial part of the

Logical Explanations 171

process of explanation, and typically consists in ¬nding them. Their status

might repay further study. For instance, it may be asked whether they are in

the last analysis needed because general theories are normally applied, not to

the universe at large, but to some actual or possible “system” (to use a physi-

cist™s word) within it. The role of boundary conditions is then to identify the

system in question.

If these ad explanandum premises are A, the underlying general theory T,

and the explanandum E, then T and A must together entail E. If so, A will

entail (T ⊃ E). Then the interpolation formula between A and (T ⊃ E) will

provide the explanation in the sense diagnosed earlier in the case of purely

deductive explanation.

At ¬rst sight, this simple schema perhaps does not seem to throw much

light on the nature of explanation. A closer examination nevertheless reveals

several interesting things, only two of which will be mentioned here. First,

assume for the sake of example that an explanandum E is an atomic sentence

of the form P(b), and assume that the particular individual a is not mentioned

in T but is mentioned in A = A[a]. Then we have:

A[a] I[a] (2.1)

(T ⊃ P(a))

I[a] (2.2)

Hence also

(I[a] ⊃ P(a))

T (2.3)

and therefore (since a does not occur in T),

(∀x)(I[x] ⊃ P(x))

T (2.4)

Thus, a byproduct of a successful explanation is in this case a covering law

(∀x)(I[x] ⊃ P(x)) (2.5)

implied by the background theory T.

Second, let us assume that there are no individual constants in T, and that

its quanti¬ers are all universal. Then the explanatory interpolation formula IL

has two kinds of ingredients, besides the usual logical notions. A number of

individuals are mentioned then that have to exist in the models of (T ⊃ E),

hence of (∼T ∨ E). Now ∼T does not introduce any individual constants that

might affect IL . Hence all the other steps in forming IL are applications of

the general laws holding in the models of F to the “unknowns” introduced by

applications of right-hand universal instantiations.

These results throw an interesting light on the logical structure of expla-

nation. The general theory of explanation will not be pursued much farther

here, however. A couple of general comments on this theory are nevertheless

in order.

Socratic Epistemology

172

First, in what sense does a successful explanation yield new information?

It is quite striking that one looks in vain for an answer to this simple ques-

tion in the literature. Admittedly, on the account of interrogative explanation

given earlier in this essay an explanation involves ¬nding the ad explanandum

conditions A that means acquiring new information. But surely a deductive

explanation also yields new information in some sense. The question is: In what

sense? Here the distinction between depth information and surface informa-

tion made in, among other places, Hintikka 1970 and Chapter 8 in this volume,

serves us well. The depth of information of the interpolation formula that cod-

i¬es an explanans cannot be greater than that of the premise F. However, the

increase of depth (i.e., in effect the number of individuals considered in their

interrelations to each other) typically increases when we look for the interpo-

lation formula, which enables the elimination of inconsistent alternatives and

thus increases surface information. In other words, the surface information of

a non-trivial interpolation formula can be greater than that of the premise The

new information yielded by explanation is surface information. Indeed, expla-

nation may be the most prominent example of the role of surface information

in epistemology.

This observation has a corollary concerning the discussions among philoso-

phers dealing with explanation in the last several decades. It is doubtful