[FOM] The Lucas-Penrose Thesis
rsmithjr at covad.net
Mon Oct 2 22:04:35 EDT 2006
>>Robbie Lindauer wrote...
>>An inconsistent machine will DEFINITELY prove 0=1. A human, even if
>>they're inconsistent on some matters, may not prove 0=1. They could do
>>this, for instance, by stubbornly refusing to recognize proofs that
>>they ought to believe that given the other things they believe.
>>I think the conclusion I draw is somewhat less strong than "Humans are
>>not machines", I draw the conclusion that humans are not LOGICAL
>>Machines (in the sense of modern-mathetmatico-logic).
I have thought that we need a notion of logic that deals with inconsistences
in a belief system in a way that is more psychologically persuasive yet
formally rigorous. Some ideas in Quine's "Web of Belief" may be relevant.
Consider the behaviour of "practical" people upon encountering some famous
paradox such as the Russell or liar's paradox. The reaction is something
like to regard the argument with suspicion, and not allow it to enter into
further arguments, but also not to be overly concerned about what is
considered a "logician's trick".
It is rather like policemen "cordoning off" a section of a city where the
streets are under repair so as not to allow accidents.
Mathematicians commonly do this as well when they use "naive set theory" in
normal mathematical exposition. They neither allow the Russell paradox to
be invoked in further proofs (allowing the contradiction to spread
arbitrarily) nor do they consider that they have to provide a thorough
analysis of the issues at hand.
The dictim "from a contradition one may derive anything whatsoever" is not a
principle people actually practice in a general way (although in limited
ways in reductio arguments it is used). The correct principle is
"suspicious contraditions should be set aside and not used in further
It is inaccurate to say that it is just a matter of "stubbornly refusing to
recognize proofs that they ought to believe". It is a matter of wisely
rejecting inferences that would lead them manifestly astray.
>From this perspective, the point is neither that "humans are not machines"
nor that "humans are not logical machines". The point is that a more
sophisticated handling of contradition is required by both humans and
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