In short: If one is a computer then one
does not believe that one believes that there is no program that puts
one into the state of belief that corresponds to this same statement
(that there is no program that puts one into the state of belief that
corresponds to this same statement) and one does not believe that there
is no program that puts one into the state of belief that corresponds
to this same statement (that there is no program that puts one into the
state of belief that corresponds to this same statement) and indeed
there is no program that puts one into the state of belief that
corresponds to this same statement (that there is no program that puts
one into the state of belief that corresponds to this same statement).
And therefore, since human beings
obviously are quite capable of believing that their own beliefs have
not been produced by some program (whether or not they are really
right), it follows human beings cannot be computers, since computers
cannot have such beliefs (if any). Up
2.1: The notation: The notation is logical on the left in the table, with
the reading on the right. The logic I use is a classically bi-valent
predicate logic with some extras. I start with explaining the
notation I use.
"f(a)" = "a has
the property f"
This is standard predicate logic, where
it is assumed that statements can be analysed as subject-predicate
structures, and the predicates may be binary relations between two
subjects or three-place relations between three subjects. In the
argument considered, all subjects are either human beings or computers.
"aBf(a)" = "a
believes the statement that a has the property f"
This is an extra compared to standard
predicate logic, for it introduces notation for propositional
attitudes. Propositional attitudes are terms like "believes",
"desires", "fears", "hopes" and very many others that are used in
Natural Language to attribute to users of language that they have
certain attitudes - beliefs, desires, fears, hopes etc. - to what is
expressed by certain statements.
= "the statement that a has the property f is the same as the statement
that b with the property h has the property g"
This conforms to standard predicate
logic, though this kind of expression is not commonly treated in
textbooks. An example of statements as intended is "my father (a) has a
pension (f)" is "the husband (h) of my mother (b) has old-age benefit
(g)". Note that "the same" in the notation is understood by reference
to the meanings of the statements on the left and right hand sides:
These are declared to be the same (rightly or wrongly).
"F(x) iff F(y)"
= "any two statements all the same except for having "x" where the
other has "y" or "y" where the other has "x" are either both true or
both not true"
This is again standard logic. The term
"F" in both cases refers to all of the statement except the terms "x"
or "y". An example is "Mary (x) never married (F)" and "Yonder spinster
(y) never married". The term "iff" is the standard logical and
mathematical abbreviation for "if and only if", and defined as stated.
This is standard notation in logic.
Here "a" is a name of something and "C" is the predicate "is a
"["f(a)"]" = "a
is in the state of belief that corresponds to the statement that a has
the property f"
Here a special convention is introduced
by way of the brackets "[" and "]", which are used to indicate that
what is contained within it is a state of belief that corresponds to
the statement within these brackets, which therefore does need to
contain some proper name of some supposedly animate entity (man,
animal, computer, angel, god, extra-terrestrial intelligence - you name
it) and occurs within quotes.
It should be noted that states of
belief are very everyday events for human beings, but that the states
of belief of other entities, whether those of one's pets or one's
computers are more dubitable. (Of course, logically speaking we may
assume anything we please.)
= "there is a program that a runs or that runs a that has put a in the
state of belief that corresponds to the statement that a has the
This is the last and most complicated
bit of notation we need. Formally, it combines the previous bit of
notation with the prefix "there is a program that a runs or that runs a
that". Informally, it states the claim that some program has given some
entity some state of belief that the same entity has a certain
property. It does so by explicitly listing the statement used to
express that belief, for which reason that statement appears within
quotes within the notation.
My reason for writing "a program that a
runs or that runs a" is simply to avoid discussing whether a human
being or a computer runs the programs it (presumably) uses or is run by
the programs it (presumably) uses. (This may be an interesting topic,
but not in the present context.)
Also, the reader should be aware that
it is an uncontroversial fact that both human beings and computers do
use programs for some tasks. What is at issue is not this fact, but
whether there is anything more involved in being a human or being an
animal than is involved in being a computer (in the sense of: a finite
2.2. The assumptions: Next, we must consider the assumptions we set up using
the notation we just discussed.
aBaBf(a) --> aBf(a)
These are three assumptions about the
propositional attitude "believes". The first says that if one believes
one has a property, then one does not believe that one does not have
the property. The second says that if one believes that one believes
one has a property, then one believes one has the property. And the
third one says that if one believes that one does not believe one has a
property then one does not believe one has the property.
All three are theorems in the logic of
propositional attitudes I have set up elsewhere, and make perfect
intuitive sense. It is especially (2) that is used in the argument, and
(2) is interesting in that its premise states a conscious belief of a,
to the effect that a believes that a believes that a has property f,
and its conclusion infers that therefore a simply believes that a has
This is interesting because it embodies
a way in which one may manipulate and change one's own beliefs: By
coming to have conscious beliefs about them.
--> F(f(a)) iff F(g(h(b)))
This is basically a standard convention
about the substitutions of two expressions supposed to represent the
same thing, as given in the hypothesis. It is stated as given because
we need this fairly complicated expression in the argument. A more
simple statement corresponding it is "a"="b" --> F(a) iff F(b).
One detail that is noteworthy is that
the hypothesis is explicit about its talking of expressions: What it
says is that the quoted expressions on the left and right sides of "="
represent the same thing. The conclusion, by contrast, uses the
expressions that occur quoted in the hypothesis.
An example of (4) is "The father of
Cesare Borgia" = "Pope Alexander VI" --> The father of Cesare Borgia
was a consumate liar iff Pope Alexander VI was a consumate liar. And
since the intent of "a"= "b" is: The expressions "a" and "b" represent
the same thing, the conclusion without quotation-marks justified, and
both sides of the equivalence must be both true or both false, since
they have the same predicate and refer to the same thing.
aBf(a) iff p(a,["f(a)"])
Sofar, we have made assumptions about
propositional attitudes and about the usage of quoted expressions. In
(5) we find the first application to computers. What (5) says is "If a
is a computer, then a believes a has property f iff there is a program
that a runs or that runs a that has put a into the state of belief
corresponding to the statement that a has property f".
Accordingly, (5) expresses the
assumption that the things computers do, including the having of
beliefs, are done by programs.
p(a,["f(a)"]) iff ["f(a)"]
Next, (6) is more precise about the
supposed relation between computers and their beliefs, if any, for (6)
says that "a is a computer only if there is a program that a runs or
that runs a that has put a into the state of belief corresponding to
the statement that a has property f iff a is in the state of belief
corresponding to the statement that a has property f."
What this expresses, then, is basically
that if a is a - properly working - computer that has states of belief,
then these states of belief are produced by a program and conversely
that a computer has a state of belief if this is produced by a program
the computer runs or is run by.
p(a,["p(a,["f(a)"])"]) --> p(a,["f(a)"])
This further precisifies what would be
involved in a computer running a program that manufactures its states
of belief, if any. What (7) says is that "a is a computer only if there
is a program that put a in the state of belief that there is a program
that put a in the state of belief that a has property f only if there
is a program that put a in the state of belief that a has property f.
In effect, (7) claims for computers and
their programs what (2) claims for human beings: That if one has an
iterated or conscious attitude, such as a belief that one has a belief
that 2+2=4, then one simply has the belief that 2+2=4. (The converse
need not hold for computers nor for humans: Humans, at least, have
quite a few beliefs they are not always or not ever conscious of.)
p(a,["~p(a,["f(a)"])"]) --> ~p(a,["f(a)"])
Like (7) corresponds to (2), so (8)
corresponds to (3). Hence what (8) says is that "a is a computer only
if whenever there is a program that put a in the state of belief that
there is no program that put a in the state of belief that a has
property f then there is no a program that put a in the state of belief
that a has property f.
Thus, (8) claims for computers and
their programs what (3) claims for human beings: That if one has an
iterated or conscious attitude, such as a belief that one does not have
a belief that 2+2=22, then one simply does not have the belief that
The reason (7) and (8) differ from (2)
and (3) is simply to be explicit about how computers are supposed to
reach their beliefs, if any: By some program that manufactured such
states of computed belief of a computer.
What is also noteworthy is that sofar
the assumptions introduced make intuitive sense for human beings, and
also make intuitive sense for computers on the assumption that they
generate their own states of belief, if any, by their own programs.
Here in effect we introduce the
notation that corresponds to a Gődelian diagonalization in Gődelian
incompleteness arguments (that the reader may take for granted if he is
not familiar with them: A good reference are the books of Raymond
Smullyan, such as "Gődel's Incompleteness Theorems" and "Forever
Undecided" - the former requires some knowledge of logic, and the
latter is an exquisite book of puzzles around Gődelian themes).
What (9) says and defines is the
property g: That something a has the property g refers to the same fact
as that there is no program that put a into the state of belief
expressed by the statement that a has the property g.
It should be intuitively obvious that
many human beings would insist that they have such a property, and that
some of their beliefs are not manufactured by a program (but, say, by
their own free will, that is no program and cannot be adequately
represented by a program).
What may be questioned is whether (9)
is a completely correct notation. This is a question I won't enter into
apart from noting that the apparent circularity of (9) is apparent
only, and corresponds quite well to such properties "g" like "is
akwardly self-conscious" or "is not an automaton" or "has a free will"
or "a speaks English iff a understands what "speaks English" means".
This is our last assumption and simply
uses (9) in the context of a propositional attitude. What (10) says is
that something a (whether a man or a computer) believes that the
expressions "g(a)" and "~p(a,["g(a)"])" represent the same, as (9)
2.3: The argument: We arrived at the explanation of the argument.
The idea of the argument is to deduce a
statement that humans know to be true of themselves that cannot be true
of computers with properties as earlier assumed. Therefore we argue all
the time on the hypothesis that a is a computer, and we use the fact
that our assumptions were framed about any thing to which we might
attribute propositional attitudes including computers.
aBaBg(a) --> aBg(a)
Accordingly, (11) simply substitutes
"g(a)" for "f(a)" in (2)=(aBaBf(a) --> aBf(a)), and adds the
hypothesis "Ca". This is simply applying standard logical principles of
inference to the assumptions made (as is all of the argument).
aBg(a) iff p(a,["g(a)"])
Here (5)=(Ca --> aBf(a) iff
p(a,["f(a)"])) is used as in step (11).
aBg(a) --> p(a,["~p(a,["g(a)"])"])
This step uses (12) to get Ca -->
aBg(a) --> p(a,["g(a)"])} and substitutes (9)=
("g(a)"=~p(a,["g(a)"])) into that to obtain (13).
aBg(a) --> ~p(a,["g(a)"])
This results from (13) by combining it
with (8)=Ca --> p(a,["~p(a,["f(a)"])"]) --> ~p(a,["f(a)"]).
aBg(a) --> ~aBg(a)
That ~aBg(a) is a direct consequence of
(12)'s (aBg(a) iff p(a,["g(a)"])) and (14)'s ~p(a,["g(a)"]).
The conclusion aBg(a) --> ~aBg(a) in
(15) is logically equivalent with ~aBg(a) V ~aBg(a) which is logically
equivalent with ~aBg(a).
Ca --> g(a)
By (12) we have Ca --> ~aBg(a) iff
~p(a,["g(a)"]) whence (17) by (9).
Ca --> g(a)
by (16, 17)
From (17) follows Ca --> (~aBg(a)
--> g(a)), whence (18) via (16).
From (11) follows Ca --> (~aBg(a)
--> ~aBaBg(a)), whence (19) via (16).
~aBaBg(a) & ~aBg(a) & g(a)
This simply gathers previous
conclusions and is the result we wanted.
The reason this is the result we wanted
Suppose you are a computer. Then (20)
implies that you do neither consciously nor unconsciously believe that
there is no program that puts you into the state of belief that
corresponds to this same statement (that there is no program that puts
you into the state of belief that corresponds to this same statement)
and indeed there is no program that puts one into the state of belief
that corresponds to this same statement.
But clearly you, being human, are quite
capable of believing that there is no such program that put you into
the state of belief that there is not such program. But then it follows
by (20) that as soon as you believe this i.e. believe that your beliefs
are not produced by a program, then you are no computer, for a computer
just cannot have such a belief, as has been proved (for a computer, if
it believes anything at all about its states of belief, must believe
these have been produced by a program, and especially cannot falsely
come to believe its program-manufactured beliefs are not
problems: One may well ask what an
argument such as the one just given really achieves.
According to some, such as the
philosopher Lucas and the mathematician Penrose, such an argument
plainly proves human beings cannot possibly be computers.
According to others, such as the
mathematical logicians Feferman and Boolos (both top in their field)
such arguments prove no such thing, for a reason we shall consider
According to yet others, including most
so called "cognitive scientists" the matter is open: The purported
proofs of Penrose and Lucas are either mistaken or cannot be
understood, and the developments in computer technology and
programming, including a chess-program that has beaten Gari Kasparov,
who is a genius at chess, show that the day may be close that computers
outperform humans on all tasks humans sofar uniquely excelled in.
I have given my own version of the
Lucas-Penrose line of reasoning, with the remark that, whatever its
status, it is clearer and briefer than the versions of Lucas and
Penrose. I will return to the merits and demerits of my argument below,
after first turning to the other positions I mentioned in the previous
The two basic reasons Feferman and
Boolos disagree with the Lucas-Penrose argument are that (1) the
arguments of both Lucas and Penrose are either not mathematically
rigorous or can be faulted and (2) both Feferman and Boolos have rather
deep doubts about semantical interpretations of formalisms, particularly of the present kind. Especially Feferman, in his
criticism of Penrose, takes a formalist stance, i.e. one that does not
go beyond rigorous formal logical proofs, except to express doubts
about such a semantical beyond.
Feferman and Boolos are certainly right
about (1), though it should be added in fairness to Mr Lucas that he
insists that his argument is informal and also must be
informal, the last essentially because it involves semantical
The formalist stances of Feferman and
Boolos are - to my mind - rather odd, and quite similar to one who
refuses to pronounce on moral questions apart from the literal
statements in books of law. For there certainly is a problem in making
sense of semantics, interpretations, etc. but these problems have at
least been somewhat resolved by mathematical logic and model theory,
and it seems rather prudish to object to their use in interesting cases
like the present one.
What remains true is that a supposedly
valid logical or mathematical argument should be capable of getting a
valid formal statement, and that as long as there is no such valid
formal statement there also is no valid proof of it.
Most "cognitive scientists" do not
understand much about mathematical logic, and arguments based on what
computers can do at present to what computers may be able to do in 10,
a 100 or a 1000 years are as safe as predictions about the course of
the stock exchange.
Also, it should be noted in the present
context that the notion of "the Turing Test" "cognitive scientists" are
prone to appeal to when discussing what computers may and may not do is
The idea was first stated by Alan
Turing, and comes to this: As soon as a computer is capable of
outperforming a human being, e.g. in chess, one (at least: Alan Turing)
concludes that "therefore" the computer thinks and plays chess at least
as well as a human being does.
The fallacy here is this: Even if a
computer mirrors all of one's behaviour completely, this is no reason
to conclude it produces this behaviour in the same sort of way as
a human being does - and indeed, it is an elementary fact that the same
result may be usually brought about in very many different ways.
Thus, the computer that beat Gari
Kasparov did not really play chess as human grandmasters of chess do
(which at present is not at all understood more than very superficially
and partially): it merely is capable of
extra-ordinarily fast processing and searching. And while it seems to
play chess, what happens inside it as it produces that behaviour, is
quite different from what happens in a human being when it thinks about
chess, for so much is certain.
Hence the Turing Test is about as
conclusive as is looking at the image of a man in Madame Tussaud's, and
concluding it "must" be a man because it looks externally like a man
and "therefore" must be like a man internally as well.
To turn to the argument I outlined
The argument I outlined above is also
not mathematically rigorous in a strict sense, because many of the
details that make a system and a proof into a logically rigorous system
and proof have not been given, either because this would be tedious or
because this would be difficult.
However, it is formally more rigorous
than what Penrose and Lucas offer, and seems to clarify what they
intended to prove.
The matters I left out that are
difficult are those that relate to propositional attitudes (e.g.
assumptions (1)-(3)) and to self-referential statements (assumption
(9)). So far, there just is no adequate logic of propositional
attitudes, and so far there is no adequate theory of self-reference,
though in either case there are promising beginnings and applications
of such beginnings. Up
My own position on whether human
beings are computers: Personally, I
don't believe I am a computer, i.e. a deterministic finite
state machine, and I don't believe you are, either,
whoever you are, if you can read and understand this.
My fundamental reasons are not the
Lucas-Penrose arguments, but the following - and perhaps I should add
that, unlike Mr Lucas, I am no religious believer and do not believe I
have an immortal soul:
A. There is to
this day no computer (Turing Machine) that has, supplies or generates a
semantics (for natural language, mathematics, or anything else of
B. Turing Machines are fundamentally very simple things.
C. There is no clear theory of either human qualia or human selves or of meaning.
These arguments are given in order of
Argument (A) has been best stated by J.
Searle. I have given my own version of his argument in a treatment I
wrote of Leibniz. Here is a link: Searle's argument in a Leibnizian
Briefly, the argument is that human
beings contribute meanings to understand symbolism, and all that
computers do is to shift about - what are for human beings - symbols without any understanding of their own, merely
based on the understanding programmers had and incorporated into the
programs that run on the computer their programs run on.
As I pointed out in my treatment, it is
not impossible that computers will have some semantical understanding
in some sense, but it is true that sofar they have none, and what they
seem to have is supplied by human beings, either in making the programs
of computers or in interpreting the output of computers. (Note that
supplying a computer with a ready-made computational semantics hardly
counts: The amazing thing about babies is that within a few years they
learn to talk without any programmer filling their heads with
Argument (B) is considerably stronger.
The underlying reasons are that (1) the mathematical principles
embodied in Turing Machines are the most simple there are and
(2) Turing Machines are finite in memory and in speed.
The first point is the most important.
It seems as if Nature involves all manner of continuous transformations
and operations, and as if Nature as a matter of course solves - or acts
according - the most complicated systems of differential equations. A
Turing Machine can mimic such mathematics, but only in a discrete way,
based on what are effectively finitely many natural numbers (for "the
real numbers" that a computer processes are
simply finite lists of integers that approximate real numbers).
If Nature really embodies continuous
processes, as it seems it does, a Turing Machine can at best
approximate aspects of such processes, but not really represent them.
And if the most complex thing known in Nature, such as a human being's
brain, essentially involves continuous processes, therefore such a
human can only be approximated in some of his or her aspects by a
Turing Machine but cannot be fully represented by a Turing Machine.
Reason (2) has a theoretical form that
is of some interest: The natural reply of anyone who knows anything
about computers is that "the sky is the limit" as regards limitations
of speed and memory of computers. This seems to me to be somewhat
optimistic, but is not my real point, which is this: It may be that
human beings (or bacteria or some other simple form of life) are not finite
Turing machines but are infinite Turing
machines. Note that for this it is not necessary that they are
infinitely large, but merely that they have something like an interval
of the real numbers accessible to them as memory, which then can play
the role of an infinite tape.
Argument (C) is the strongest, and
comes to this, in three steps.
C.1.: One essential set of qualities
human beings have is that they have human feelings, desires, ends,
fears, hopes etc., while no physical thing as reconstrued by
physics (sofar) has any feelings, desires, ends, fears or hopes: All
physical things have are charges, speeds, sizes, number, hardness
etc. - in short,
palpable physical qualities.
At present there just is no physical
explanation of the characteristiscs of human experiences
that makes them into human experiences: their aspects of
being feelings, desires, ends etc. that are in philosophy often
referred to as "qualia". And conversely, at present there also
is no explanation of physics in terms of qualia, i.e.
feelings, desires, ends etc. (although Aristotle believed there was,
and some modern philosophers, like Whitehead, followed him in this,
without much success).
C.2.: Another essential set of
qualities human beings have is that they all (or nearly all) believe
they are or have a self, a personality, a character, that is
more than is ever given in their momentary experiences, and carries
them from the past towards their future ends, across the present, and
that is and remains "what they really are" through all manner of bodily changes.
At present there just is no adequate
theory of what a human self is, and what makes such a theory
quite difficult is that it involves self-reference, levels of meaning,
human ends that go far beyond what is given, such as fantasies that function in one's character, and so on.
C.3.: A last essential quality all sane
human beings is that they know how to speak and understand natural
language, attribute meanings to marks, interpret
gestures, and are capable of symbolizing all manner of
possible and impossible things by arbitrary sounds or pictures, and may
understand such symbolizations when given them.
At present there just is no adequate
theory of what meaning is, beyond the simple level of first order
predicate logic, i.e. including self-reference, including reference to
universals and abstract entities such as classes, functions and
categories, and including a full and clear explanation of human
thinking and understanding and its various aspects, such as language,
mathematics, music and visual art.
Hence my own position is on the
question whether human beings are computers is this:
While I am an atheist, and don't
believe I am or have an immortal soul, I also do not believe that
"therefore" I am such a fundamentally simple thing as is a
Turing Machine, essentially because (1) Turing Machines embody so
little mathematics, and Nature, including human beings and other living
things, seems to embody very complicated mathematics and (2) there are
not even on the level of human experience adequate explanations of qualia
and of meanings and of selves, and so
there is no basis at all to attribute these to computers, and no basis
at all to start programming them into computers (for you can't program
what you don't really understand, even if it could be programmed in
principle when you would understand it).
So I prefer to think that what I and
other human beings are is a natural organism, that is not created by
any god or higher intelligence, that has evolved naturally, and that
embodies continuous mathematics such as can be mimicked but not fully
rendered by computers, and that probably has not been fully discovered
and is not fully understood by mathematicians, physicists and
biologists, at the present levels of their sciences.
Finally, it seems to me that the
problems of what is consciousness and of what is life are
rather intimately related, and that we know as much about the former as
about the latter: A little but by far not all of what there is to know.