WCSWR@CARLETON.BITNET (Walter Roberson) (10/03/87)
In AILIST of October 2nd,
Christopher Lishka (uwslh!lishka@speedy.wisc.edu), and
J Storrs Hall (topaz.rutgers.edu!josh@rutgers.edu)
discuss whether human minds are inherently flawed. Chris proposes that
human minds just *are*, neither flawed nor unflawed; JoSh disagrees
strongly, and claims Chris's position to not be scientifically based.
Leaving aside for the moment the question of whether mathematics is a
science (at last notice, AILIST list hadn't resolved that one), I
believe that I can offer a mathematical basis for Chris's position.
Consider a set (possibly infinite) of objects, U, and at least two
single-place predicates over that set, P, and Q. Add n-ary predicates
and distinguished constants, if you like. Consider the following
first-order sentance over this language: "For all x in U, Px => Qx". Is
this sentance true? It depends on the relations P and Q. If Qx is
"false" for all x in U, and at least one Px is "true", then the sentance
is false -- for that P and Q. If Px is "false" for all x in U, then the
sentance is true -- again, for that P and Q. Thus, the truth of the
sentance depends upon the structure
(U, set(P, Q, etc), set(constants), set(n-ary functions))
in which it is evaluated. Now, as there is at least one such structure
in which this sentance is false, the sentance is NOT "logically implied"
by the language of its formulation. And, as there is also at least one
such structure in which the sentance is true, one can only talk about
the validity of the sentance in terms of its value in a particular
structure. Loosely speaking, the validity of the sentance varies with
the interpretation one gives to the relationships.
Consider now the above sentance, ("for all x in U, Px => Qx") with the
human intepretation that it denotes "all minds are flawed" -- that is,
Px being interpreted as the predicate "x is a human mind", and Qx being
intepreted as the predicate "x is flawed". Assigning the sentance a
human interpretation makes it no more true or false than before: the
difference is only in the emotional zing of the interpretation.
Assigning a validity to the sentance based on a religious set of values
corresponds to chosing a structure and evaluating the sentance within
that structure. The sentance may be valid or invalid within that
structure, but, in isolation, the sentance will still be neither true
nor false. Chris's position is that "human minds are flawed" is only
true within certain belief sets: that it is not a true statement because
it is not a logically implied statement. JoSH's position is that the
interpretations of the words "human minds", "are" and "flawed", are such
that the statement is implicitly true: that semantically, the statement
is automatically self-restricting to the class of structures in which it
is true.
Certainly the conventional wisdom is that "nobody's perfect". That has a
certain intuitive "rightness" to it which is very compelling. And if
nobody is perfect, then everyone is flawed, right? But what someone
saying, "Nobody's perfect" really means is, "There isn't anyone that
measures up to my standards of perfection". That, however, is more a
reflection of the utterer's standards of perfection than upon the
intrinsic qualities of any other given person. A lot of people have done
things which haven't pleased me, but that's a matter of my expectations,
rather than a question of whether they were "flawed" or not.
---
In part of his response, JoSH disapproves of Chris's position, based
upon operational grounds. Indeed, we do not -need- to study the
aerodynamics of angels in order to build an airplane. I don't believe,
though, that Chris implied that we needed to do so: rather, he favours a
position closer to the doctrine of necessity; that if X isn't necessary
in order to do Y, and Y is your goal, then don't do X. In this case, X
is "assign a definite truth value to 'human minds are flawed'", and Y is
"computationally model a human intelligence". Chris believes X to be
unnecessary (and impossible in finite time anyways). JoSH believes it to
be possible; I haven't been watching closely enough to determine whether
he believes it to be necessary.
---
Is 2+2=4 ? In the ring Z4, No: 2+2=0 instead. And since '=' is merely
the symbol for a binary operation, traditionally a certain well-known
predicate, then sometimes 2+2=5 afterall. Try, for example, reading '='
as denoting the binary predicate traditionally represented as '<'.
Is the broken radio flawed? Well, if it was hit by lightning while
playing "satanic rock music", and melted down into a representation of
"Jesus", I rather doubt people would call it "flawed" when they couldn't
get music out of it. Not much use in trying to decide whether an object
is "flawed" or "bad" or "evil" or whatever -- if it doesn't do what you
want it to, perhaps it'll make a dandy paperweight instead. Or bonfire
fuel, if you've found it particularily frustrating.
Is a dead person "flawed" because they are no longer living? I'm told
that death is a very natural process -- happens to everyone, they say.
But its not going to happen to me -- at least not during my lifetime!
(Thanks, Raymond!)
Walter Roberson <WCSWR%CARLETON.BITNET@WISCVM.WISC.EDU>
---
Reference: "A Mathematical Introduction to Logic", Herbert B. Enderton,
1972, Academic Press