Re: JSH: Logic and paradox
- From: Rotwang <sg552@xxxxxxxxxxxxx>
- Date: Sat, 06 Oct 2007 06:05:40 -0700
James Harris wrote:
[stuff about logic]
James, I hope you don't mind if I provide an excerpt from a related
blog entry of yours: from
http://mymath.blogspot.com/2005/06/3-logic-more-basics.html
consider the sentence:
This sentence is false.
It has a logical value of -1, as it is negatable:
This sentence is not false.
You have made a rookie error here. The negation of the sentence "this
sentence is false" is not "this sentence is not false". That is
because the word "this" in the two sentences refer to different
things. That is, suppose we refer to the first sentence as U, so that
U is the sentence "Sentence U is false". Then the negation of U is the
sentence "Sentence U is not false". But this is not the same as U, so
it is not equivalent to "This sentence is not false".
As consider the suppose paradoxical statement:
Consider a set of all sets that exclude themselves.
That's not a statement.
That is a malformed statement as it violates 1. and 2. above. But to
know that you have to hold a certain amount of information in your
mind, sort of in the working space you might say of your brain.
If you lack the mental capacity to do that then your mental wiring
prevents you from comprehending that reality.
Let me explain in detail and see if you can hold in all the info:
A set of all sets that exclude themselves cannot exist as it needs to
include itself, but if it includes itself it excludes itself, so the
statement is malformed, as a set cannot include and exclude itself.
Yes, we all understand Russell's paradox.
The easy fix is, consider a set of all sets that exclude themselves,
except itself.
So let me get this straight: when a theory such as Set Theory with
unrestricted comprehension contains a set which leads to a
contradiction, the "easy fix" is to consider a different set. Er, no.
That doesn't fix the problem, that just ignores the problem. You might
as well argue that... no, sorry. I can't think of an analogy that is
as dumb as this.
The exception creates a well-formed statement.
If you cannot follow that you may not be mentally capable of holding
enough information in mind long enough to connect the dots.
Yet the simple principle that controls what must be true is simply to
accept that equals means equal.
But to many people if you have x = y, you have DIFFERENT things, so in
their MINDS equal does NOT necessarily mean equal as x is not y, as
in, x is a different letter from y.
Get it, yet? I think that the human brain simply is less evolved than
most people realize and such considerations push its circuitry to
their maximum and for some people require capacity beyond their
maximum, so there is this notion of "logical paradox", when such a
thing is impossible.
Rather than spout this stuff about what logicians do and don't
understand, why not first learn a bit about what they've done? It's
painfully obvious that you don't know the first thing about first
order logic with equality, for example, which is highly relevant to
what you are talking about.
.
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