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

extremely useful.

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

to intelligence.

Kaynak: A New Mathematical Model of Mind

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