When deduction is formulated in the abstract, in terms of assumptions and transformation, it is
immediately apparent that deductive reasoning is incapable of standing on its own. In isolation, it
is useless. For why would there be intrinsic value in determining which x lie in D(I,T)? Who
cares? The usefulness of deduction presupposes several things, none of them trivial:
1. the elements of I must be accepted to possess some type of validity.
2. it must be assumed that, if the elements of I are important
in this sense, then the elements of D(I,T) are also valid in this sense.
3. it must be the case that certain of the elements of D(I,T) are important in some sense.
The first requirement is the most straightforward. In mathematical logic, the criterion of
validity is truth. But this concept is troublesome, and it is not necessary for deduction.
Psychologically speaking, validity could just as well mean plausibility.
The second requirement is more substantial. After all, how is it to be known that the elements
of D(I,T) will possess the desired properties? This is a big problem in mathematical logic. Using
predicate calculus, one can demonstrate that if I is a set of true propositions, every statement
derivable from I according to the rules of Boolean algebra is also true. But Boolean algebra is a
very weak deductive system; it is certainly not adequate for mathematics. For nontrivial
mathematics, one requires the predicate calculus. And no one knows how to prove that, if I is a
set of true propositions, every statement derivable from I according to the rules of predicate
calculus is true.
Godel proved that one can never demonstrate the consistency of anysufficiently powerful,
consistent formal system within that formal system. This means, essentially, that if validity is
defined as truth then the second requirement given above can never be verified by deduction.
To be more precise: if validity is defined as truth, let us say T is consistent if it is the case that
whenever all the elements of I are true, all the elements of D(I,T) are true. Obviously, in this case
consistency corresponds to the second requirement given above. Godel showed that one can
never prove T is consistent using T. Then, given a deductive system (I,T), how can one
deductively demonstrate that T is consistent? — i.e. that the second requirement given above is
fulfilled? One cannot do so using T, so one must do so in some other deductive system, with a
system of transformations T1. But if one uses T1 to make such a demonstration, how can one
know if T1 is consistent? If T1 is inconsistent, then the demonstration means nothing, because an
in an inconsistent system one can prove anything whatsoever. In order to prove T1 is consistent,
one must invoke some T2. But in order to prove T2 is consistent, one must invoke some T3. Et
cetera. The result is that, if validity is defined as truth, one can never use deduction to prove that
the results of a given set of transformations are valid.
Yet we believe in mathematics — why? By induction, by analogy, by intuition. We believe in
it because, at bottom, it feels right. It’s never led us wrong before, says induction. It worked in all
these other, similar, cases, so it should work here — says analogy. Even if validity is defined as
truth, a recourse to induction and analogy is ultimately inevitable.
If validity is defined, say, as plausibility, then the situation is even worse. Clearly, any true
statement is plausible, so that it’s at least as hard to justify plausible reasoning as it is to justify
"certain" reasoning. And, furthermore, the very concept of "plausibility" refers to induction and
analogy. In sum, I contend that, in general and in specific cases, deduction is only justifiable by
recourse to induction and analogy.
ANALOGY GUIDES DEDUCTION
Finally, let us consider the third requirement for the usefulness of deduction: certain of the
elements of D(I,T) must be somehow important. Otherwise deduction would simply consist of
the haphazard generation of elements of D(I,T). This is not the case. In mathematics or in
everyday life, one wants to deduce things which are useful, beautiful, interesting, etc. This gives
rise to the question: how does one know how to find the important elements of D(I,T)?
It seems clear that this is a matter of analogical reasoning. For instance, suppose one has a
particular entity x in mind, and one wants to know whether x is an element of D(I,T). Or suppose
one has a particular property P in mind, and one wants to find an element x of D(I,T) which has
this property. How does one proceed? To an extent, by intuition — which is to say, to an extent,
one does not consciously know how one proceeds. But insofar as one makesconscious decisions,
one proceeds by considering what has worked in the past, when dealing with an entity x or a
property P which is similar to the one under consideration.
For example, when studying mathematics, one is given as exercises proofs which go very
much like the proofs one has seen in class or in the textbook. This way one knows how to go
about doing the proofs; one can proceed by seeing what was done in similar cases. After one has
mastered this sort of exercise, one goes on to proofs which are less strictly analogous to the
proofs in the book — because one has grasped the subtler patterns among the various proofs; one
has seen, in general, what needs to be done to prove a certain type of theorem.
Above I argued that deduction is only justifiable by analogy. Here the point is that deduction
is impotent without analogy: that in order to use deduction to work toward any practical goal,
one must be guided by analogy. Otherwise one would have no idea how to go about constructing
a given proof.
This is, I suggest, exactly the problem with automatic theorem provers. There are computer
programs that can prove simple theorems by searching through D(I,T) according to a variety of
strategies. But until these programs implement some form of sophisticated analogy —
systematically using similar strategies to solve similar problems — they will never proceed
beyond the most elementary level.
USEFUL DEDUCTIVE SYSTEMS
Another consequence of this point of view is that only certain deductive systems are of any
use: only those systems about which it is possible to reason by analogy. To be precise, let x and
y be two elements of D(I,T), and let GI,T(x) and GI,T(y) denote the set of all proofs in (I,T) of x
and y respectively.
Definition 8.1: Let (I,T) be any deductive system, and take a>0.
Let U equal the minimum over all v of the sum a%v%+B, where B is the average, over all
pairs (x,y) so that x and y are both in D(I,T), of the correlation coefficient between d#[St(x%v)-
St(x),St(y%v)-St(v)] and dI[GI,T(x),GI,T(y)]%. Then (I,T) is useful to degree U.
The relative distance dI[GI,T(x),GI,T(y)] is a measure of how hard it is to get a proof of x out of
a proof of y, or a proof of y out of a proof of x. If v were assumed to be the empty set, then
%d#[St(x%v)-St(x),St(y%v)-St(v)] – d[GI,T(x),GI,T(y)]% would reduce to %dI(x,y) –
d[GI,T(x),GI,T(y)]%. The usefulness U would be a measure of how true it is that structurally
similar theorems have similar proofs.
But in order for a system to be useful, it need not be the case that structurally similar theorems
have similar proofs. It need only be the case that there is some system for determining, given any
theorem x, which theorems y are reasonablylikely to have similar proofs. This system for
determining is v. In the metaphor introduced above in the section on contextual analogy, v is a
codebook. A deductive system is useful if there is some codebook v so that, if one decodes x and
y using v, the similarity of the resulting messages is reasonably likely to be close to the similarity
of the proofs of x and y.
The constant a measures how much the complexity of the codebook v figures into the
usefulness of the system. Clearly, it should count to some degree: if v is excessively complex
then it will not be much use as a codebook. Also, if v is excessively complex then it is extremely
unlikely that a user of the system will ever determine v.
Mathematically speaking, the usefulness of traditional deductive systems such as Boolean
algebra and predicate calculus is unknown. This is not the sort of question that mathematical
logic has traditionally asked. Judging by the practical success of both systems, it might seem that
their usefulness is fairly high. But it should be remembered that certain parts of D(I,T) might
have a much higher usefulness than others. Perhaps predicate calculus on a whole is not highly
useful, but only those parts which correspond to mathematics as we know it.
It should also be remembered that, in reality, one must work with dS rather than d#, and also
with a subjective estimate of % %. Hence, in this sense, the subjective usefulness of a deductive
system may vary according to who is doing the deducing. For instance, if a certain codebook v is
very complicated to me, then a deductive system which uses it will seem relatively useless to me;
whereas to someone who experiences the same codebook as simple, the system may be
DEDUCTION, MEMORY, INDUCTION
If the task of intelligence is essentially inductive, where does deduction fit in? One way to
approach this question is to consider a deductive system as a form of memory. Deduction may
then be understood as an extremely effective form of data compaction. Instead of storing tens of
thousands of different constructions, one stores a simple deductive system that generates tens of
thousands of possible constructions. To see if a given entity X is in this "memory" or not, one
determines whether or not X may be derived from the axioms of the system. And, with a
combination of deduction and analogy, one can determine whether the "memory" contains
anything possessing certain specified properties.
Of course, a deductive system is not formed to serve strictly as a memory. One does not
construct a deductive system whose theorems are precisely those pieces of information that one
wants to store. Deductive systems are generative. They give rise to new constructions, by
combining things in unforeseeable ways. Therefore, in order to use a deductive system, one must
have faith in the axioms and the rules of transformation — faith that they will not generate
nonsense, at least not too often.
How is this faith to be obtained? Either it must be "programmed in", or it must be arrived at
inductively. AI theorists tend to implicitly assume that predicate calculus is inherent to
intelligence, that it is hard-wired into every brain. This is certainly a tempting proposition. After
all, it is difficult to see how an organism could induce a powerful deductive system in the short
period of time allotted to it. It is not hard to show that, given a sufficiently large set of statements
X, one may always construct a deductive system which yields these statements as theorems and
which is a pattern in X. But it seems unlikely that such a complex, abstract pattern could be
recognized very often. What the AI theorists implicitly suggest is that, over a long period of
time, those organisms which did recognize the pattern of deduction had a greater survival rate;
and thus we have evolved to deduce.
This point of view is not contradicted by the fact that, in our everyday reasoning, we do not
adhere very closely to any known deductive system. For instance, in certain situations many
people will judge "X and Y" to be more likely than "X". If told that "Joe smokes marijuana", a
significant proportion of people would rate "Joe has long hair and works in a bank" as more
likely than "Joe works in a bank". It is true that these people are not effectively applying Boolean
logic in their thought about the everyday world. But this does not imply that their minds are not,
on some deeper level, using logical deduction. I suspect that Boolean logic plays a role in
"common sense" reasoning as in deeper intuition, but that this role is not dominant: deduction is
mixed up with analogy, induction and other processes.
To summarize: recognizing that deductive systems are useful for data compaction and form
generation is one thing; exalting deduction over all other forms of thought is quite another. There
is no reason to assume that deduction is a "better", "more accurate" or "truer" mode of reasoning
than induction or analogy; and there is no reason to believe, as many AI theorists do, that
deduction is the core process of thought. Furthermore, it seems very unlikely that deduction can
operate in a general context without recourse to analogy. However, because deduction is so
effective in the context of the other mental processes, it may well be that deduction is essential
Kaynak: A New Mathematical Model of Mind
When deduction is formulated in the abstract, in terms of assumptions and transformation, it is