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Etica
& Politica / Ethics & Politics, 2003, 1 http://www.units.it/~dipfilo/etica_e_politica/2003_1/10_monographica.htm   This Gödel is killing me (*) (**) Anthony Hutton Department of Mathematical and Physical
Sciences Plymouth Polytechnic In this paper I am going to concern
myself with the concept of consistency, Gödel’s second theorem, and the
anti-mechanistic argument of J. Lucas. (1) My approach to the issue involved will be somewhat
different from the usual logical/philosophical approach which always seems to
provide the mentalist with a convenient loophole. I will not attempt to give a
detailed explication of the relevant logical theory, as this is covered
adequately elsewhere, but some brief statement of my terms of reference is
obviously essential. Let us begin with a formal logical
system S which is adequate for the proofs of all the basic results of
elementary number theory. Such a system is said to be simply consistent
if and only if no well-formed formulas A of S is such that A and its negation
are both theorems of S, and absolutely consistent if and only if at
least one formula of S is not a theorem of S. Obviously simple consistency
implies absolute consistency and vice versa. Now Gödel’s second theorem, a
corollary of his first, may be expressed approximately as follows: if Con Lucas’s paraphrase of Gödel’s second
theorem is “Gödel has shown that in a
consistent system a formula stating the consistency of the system cannot be
proved in that system”, (2) and
whilst such an interpretation is open to serious criticism I will not pursue
the matter in this context. Using this interpretation as his basis, Lucas claims
that no consistent machine is capable of producing a computable or computably
enumerable expression of its own consistency and whilst such a limitation
applies to the formal reasoning of a human computer, the latter, by means of
informal reasoning, is able to transcend this limitation and assert his own
consistency. Lucas unfortunately does not clearly state what he means by
informal reasoning or informal logic. One could take it as a reference to that
vast number of inferences drawn in everyday life which are not, and could not
be, codified in any formal logic system. Alternatively he could mean by informal
logic ‘a theory of valid inference’, but whether this sheds any more light
on the matter I do not know. Nevertheless, the message is clear and simple; by
means of informal logic, which Lucas knows is consistent, he can informally
assert his formal consistency, and since he is formally consistent, so is
mathematics. But how does Lucas know this? “In saying that a conscious being
knows something we are saying not only that he knows it, but that he knows that
he knows it, and so on, as long as we care to pose the question: there is, we
recognize, an infinity here, but it is not an infinite regress in the bad
sense, for it is the questions that peter out, as being pointless, rather than
the answer.” (3) There are however one or two
questions I wish to pose before I peter out. I will begin by taking Lucas at his
word and define my null hypothesis as “my informal logic (reasoning) is
consistent”; in case my subsequent analysis leads me to reject this I will define as alternative
hypothesis “I do not know whether my informal logic is consistent or not.”
Since it is Lucas’s contention that the consistency of our informal reasoning
is the ultimate guarantee of the consistency of mathematics, my null hypothesis
allows me to assume the consistency of that discipline also. Hence I can use it
with confidence. In order to test my hypothesis it is
then essential that I direct my attention to that disorderly field which Lucas
claims, establishes my mental superiority over all possible individual
artefacts. To begin let us assume that our informal mental life-space consists
of n propositions, A (a)
1
axiom giving 4 choices (b)
2
axioms giving 6 choices (c)
3
axioms giving 4 choices (d)
4
axioms giving 1 choice. Within
this model it is also easy to demonstrate that
if a mental system consists of 1
axiom it can be inconsistent 0 ways if a mental system consists of 2
axioms it can be inconsistent 2 ways if a mental system consists of 3
axioms it can be inconsistent 4 ways if a mental system consists of 4
axioms it can be inconsistent 1 way. Table 1 summarizes the relevant outcomes
for n = 1, 2,…, 7 and in general, the generating formulae are as follows: – Total number of sets of axioms
(consistent and inconsistent) = 2 Number of sets of consistent
axioms = 3 If we now make the assumption that
all sets of axioms have equal probability of acceptance by a mind then the
probability of a mind being inconsistent is given by the function Nevertheless, the exercise has not
been futile. Consider once again a mind operating in a space consisting of n
axioms and their negations. Its informal mental calculus (and it is Lucas who
says that exists, not I) must consists of 1 axiom or 2 axioms … or 2n axioms
together with the necessary inferential rules which I am not questioning at
this stage. Now the (2 And at this point we run serious
difficulties. Despite the fact that we know that for n
I will now consider an obvious
criticism that can be levied against this model. It could be argued that no one
who would be counted as rational, in the general sense of the word, would
accept any proposition and its negation of axioms; for example, no one would
believe that God exists and God does not exist, at one and the same time. But,
on reflection, one has to admit that in real life, propositions are not always
presented to us in this clear-cut from; indeed one holds such a bewilderingly
complex array of axioms that the possibility of detecting all inconsistencies
must be infinitesimally small. In order to be certain of consistency it would
be necessary to ascertain the implications of each and every belief and this,
surely, would be a task beyond human endeavour. Nevertheless, one may press the
point and ask whether a mind exists which is, conclusively, inconsistent. Some
may well regard this question as a joke (with perhaps the exception of those
who have been on jury service and witnessed informal logic applied by
respectable citizens of sound mind), but, since most people take exception to
being called "inconsistent", it must be considered. I will discuss
the question initially in a general sense before returning to a statistical
analysis, the latter of course being a sub-domain of consistent mathematics. Let us consider, in the first
instance, a subject who holds axioms which are patently false. Our model is a
person with a high level of creativity, who is also extremely competent in the
field of mathematical logic, but is a confirmed gambler to the extent that he
leaves himself penniless. That such a person could exist cannot be denied and
we must remember that, according to Lucas’s schema, this subject can assert the
consistency of his formal work by reference to his informal reasoning. Now our
subject’s ‘gambling axioms’ must belong to the latter classification and these
would probably consists of one more of the following: – G1
I am certain I am going to win, eventually. (Like Lucas and consistency he ‘just
knows’ this and ‘just knows that he knows it’ and so on for ever backwards
until the questions or the money peter out). G2
I am much more intelligent in the gambling field than most other (or
all) gamblers. G3 Life itself is but a gamble and no one accumulates great
fortunes who is not a gambler. Now try as we might with arguments
from the arsenals of logic and statistics, we cannot get him to recognize that
these axioms are false. But this does not mean that he is inconsistent; in a
society of gamblers he would be perfectly consistent. That is my view but I do
not think Lucas would share it. He has argued that anyone who uses a rule of
inference in one situation and refuses to use it in another one in order to
avoid an inconsistency is inconsistent, tout a fait. “We do not lay it to a man’s credit
that he avoids contradiction merely by refusing to accept those arguments which
would lead him to it, for no other reason that otherwise he would be led to it.
Special pleading rather than sound argument is the name for that type of
reasoning. No credit accrues to a man who, clever enough to see a few moves
ahead, avoids being brought to acknowledge his own inconsistency by
stone-walling as soon as he sees where the argument will end. Rather, we account
him inconsistent too, not, in his case, because he affirmed and denied the same
proposition, but because he used and refused to use the same rule of inference.
A stop-rule on actually enunciating an inconsistency is not enough to save an
inconsistent system from being called inconsistent.” (4) This leaves us with a rather pretty
problem. Acting on my null hypothesis that mathematics is consistent (via
informal reasoning) I have come across a person who could well assure me (again
via his informal reasoning) that mathematics is consistent but whom Lucas would
have to classify as informally inconsistent. However there is a loophole.
Perhaps our gambling logician needs informal axioms A Nevertheless all these arguments are
irrelevant to Lucas’s thesis as I will demonstrate. In fact I will go further
and prove that anyone who believes in his own consistency, and thereby in the
consistency of mathematics, and has also given serious consideration to any of
the arguments expressed in this paper is, by that very act, inconsistent. I began this investigation by
defining as null hypothesis that ‘my informal reasoning is consistent’ and by
implication mathematics, and considered an informal axiom space consisting of
2n independent elements. This space was divided into consistent and
inconsistent subsets of axioms and on ascribing probability p “…cannot tell anybody else exactly
what it is.” (5) He believes he is consistent because
“it is, in the language of
mathematical theology, an act of faith.” (6) Lucas, in his paper, seems to
interchange the words ‘believe’ and ‘know’ which is reasonable since if one knew
something to be true than, I presume, one could believe it to be true,
although one could believe something to be true without knowing
it to be true. I am now going to make the
assumption that I know I am informally consistent, for the same reasons
as Lucas does. In this paper I have considered some arguments which make me
doubt this belief and, being a person who thinks he is striving very hard to
apply logic in a consistent fashion, I must accept these arguments as counter
evidence. Representing the probability that I am informally consistent (C)
because of my Lucasian reasons (L) by Pr (C/L) it follows that Pr (C/L, F) Pr(C/L, F) = Pr(C, L) Again, relying on the consistency of
mathematics, it follows that the term The immediate implication is of
course that no evidence of any kind can refute the claim that I am (know myself
to be) consistent. But as a person who thinks of himself as being rational, one
of my informal axioms must be that I make a proper use of reason: and this
implies, among other things, that I endeavour to correctly estimate the
strength of evidence. My knowing that I am consistent however obviates
this latter estimation. If l is to be a conclusive reason for C then L must
confer upon C the probability of 1, and this means that no fact F can serve as
an argument against C, and it seems to me that Lucas or I could not know
ourselves to be informally consistent unless the evidence L could be expressed
in the form of a deductive proof. This, as we know, is impossible. Let us finally review the situation.
I began by accepting as null hypothesis that ‘I am informally consistent’,
considered a number of arguments against this belief which were not and could
not be conclusive, but which were strong enough to make me feel happier
believing the alternative hypothesis that ‘I do not know whether or not I am
informally consistent’. I then made the assumption that I knew I was
consistent (and by implication that I knew mathematics to be
consistent), expressed my argument in mathematical terms and arrived at a
conclusion which could only be considered as a travesty of rational thought.
The case may be expressed symbolically as follows: – A:
I know I am informally consistent. B:
I know formal logic to be consistent. C:
I know mathematics to be consistent. 1.
A 2.
A This is Lucas’s axiom from (1), (2) 3.
B by
Modus Ponens and tautology. 4.
C 5.
C argument. 6.
A 7.
–C from
(2) and (6). 8.
C The axiom to be rejected is A, that
is, for consistency I must conclude that I do not know whether or not I am
consistent. And that concludes my case. The
result, it seems to me, is much more intuitively plausible that either of the
extremes adopted by Lucas or Putnam. (8) It
means of course that Gödel’s theorems have no implications for the mechanistic
thesis; if we ever do produce a machine which, it is claimed, replicates a
rational being’s mental functioning, then there is one unaskable question. Notes (*) Philosophia, 1976, 1, pp. 135-144. © Philosophia.
Republished by permission. back (**) I am indebted to John Watling,
U.C.L., for criticism of an original version of this paper. back (1) Lucas, J.R., Minds,
machines and Gödel, The Modeling of Mind, R.H. Sayre and F.J.
Crossan, eds., Univ. of Notre Dame Press, 1963. pp. 255-271. back (2) Ibid., p. 267. back (3) Ibid., p. 268. back (4) Lucas, J.R., The freedom of
the Will, Clarendon Press, 1970, p. 160. back (5) Lucas, J.R., Satan
Stultified, The Monist, v. 52, 1968, p. 148. back (6) Ibid., p. 158. back (7) Dretske, Fred I., “Reason,
Knowledge, And Probability”, Philosophy of Science, Vol. 38, No. 2,
1971. back (8) Putnam, H., “Minds and
machines”, Dimensions of Mind, Sidney Hook, ed., New York, 1961. back |
, A
, …, A
and their negations, –A
, –A
, …,
, where n is a finite or denumerably infinite, natural
number, and that an individual mind, in the course of its life, can acquire any
number of propositions and hold them as axioms. Furthermore, consistency will
be taken as the analogue of simple consistency in a formal system i.e., a mind
will be considered inconsistent if it holds any proposition and its negation as
axioms. For the purpose of illustration, consider the case where n = 2, i.e.
the mental life-space consists of four axioms
, A
, –A
,
. Now a mind in such an environment could hold 
, p
, …, p
+
+ … + p
.
. The ascription of values depends on what I will loosely
describe as ‘another source’ and this we could not possibly hope to analyse
mathematically. It is important to stress here that we are not talking of the
probability of an axiom being true but only of the probability of an individual
mind accepting a particular group composed of independent axioms. It is also to
be noted that in the case of Lucas, Pr(Con) = 1 i.e. his ‘other source or
sources’ must confer a probability of unity on his acceptance of a particular
axiom set and this again cannot be the null set since Lucas does believe
certain things to be true.
, …, A
plus some set of inferential rules to convince himself of
the consistency of mathematics and this is a different schema from that applied
in his gambling world. The one could be consistent and the other inconsistent
.Or, again, perhaps it is an analogous situation to my being able to show that
a small portion of the formal system representing Principia is consistent
whilst being unable to show that the entire system is consistent. But if we do
assert that the informal logic necessary for knowing mathematics to be
consistent, is consistent, what grounds have we for stopping there? Could not
another philosopher make an equal claim regarding his philosophy of life /
religious axioms?
to the event that a particular mind accepts subset S
as its informal axioms or beliefs I found that the
probability of such a mind being consistent was, Pr(Con) 
Pr (C/L) where F represents my arguments for not –C. Now I
know ‘that C’, hence Pr (C/L) must be unity. I know also that mathematics is
consistent and since Bayeslan Statistics is a branch of mathematics I can (using
Bayes’ Inversion Theorem) 
–A
–C