The essay shows that there are 'true contradictions' (ie real unresolvable paradoxes known as dialetheia or dialethia) at the limits of logical thought, but the limits of logical thought are not the limits of mind.
At least that's what I think it shows. When I came to the tetralemmas it made my brain hurt and I had to stop thinking and go to the pub.
So is there anything that the mind can do that a machine
cannot? Appreciate art? Compose poetry? Feel love and compassion? Lie detection...
Liar Paradoxes In philosophy and logic, the liar paradox or liar's paradox (pseudomenon in Ancient Greek) is the statement "this sentence is false." Trying to assign to this statement a classical binary truth value leads to a contradiction.
If "this sentence is false" is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on without end.
Similarly, if "this sentence is false" is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on without end.
There are similar sentences such as
The Ultimate Truth
is that there is no ultimate truth.
Postmodernism proves that all philosophical systems are invalid.
contains three erors.There are only two spelling mistakes and the sentence is
grammatical so what's the third error? Well it's obviously the factual error in counting
three errors when there are only two. So that explains how the sentence comes to have
three errors. In which case the sentence is factually correct, hence there are only the
two spelling errors. So the sentence is incorrect.
The barber shaves
everybody who doesn't shave himself. Who shaves the barber?
But what's all this got to do with mind versus
Any machine will represent truth and falsity as discrete states or numerical
values. A computer represents the truth of a statement as a single bit. For exampe
TRUE = 1 and FALSE = 0.
You can also have fractional truth values if
you are dealing with probabilities. Thus 0.5 is the truth value of an unseen tossed coin
being heads. But this actually is the average of the two possible truth values. An act of
observation will always resolve the coin as heads or tails. (We'll exclude the coin
remaining standing on its edge by assuming we're aboard a trawler in a force nine gale).
Artificial intelligence requires numerically encoded truth states to be manipulated and
combined to give long chains of logical reasoning.
But when we examine the statements in italics,
it's apparent that their truth-states are not numerically encodable and do not lie on a scale of 0 to 1, or even on a scale
of 0 to 42. In fact they're off in another dimension. These paradoxical truth-states
(known as dialetheia or dialethia) have
more of a qualitative, intuitive feel to them than anything quantifiable. It seems
that the human mind can access states which cannot be represented or manipulated within
any machine (remember a computer is a universal machine which can simulate any physical
system including any other machine).
If dialetheist truth states were purely
inconsequential curiosities, none of this would matter too much. But as Russell and
later Gödel were to show, such dialetheia lie at the very heart of mathematics. It's also possible that the ability to deal with indeterminate truth states is an important
factor in 'open-ended' or non-algorithmic mental processes such as freewill and artistic creativity.
So perhaps Boolean logical processes
implemented on any sort of physical machine are inadequate to describe the capabilities of
human mental processes.
This limitation will not be solved by hardware improvements.
No matter how many terabytes,
gigaflops, neural nets or iterations of Moore's law we throw at the problem of producing a
machine-mind, the difficulties will remain insurmountable as long as the hardware is only
capable of dealing with truth values which can be treated as binary or