Saturday, October 06, 2007

 

JSH: Logic and paradox

I think talking about a simple failure in what is usually taught as logic can give you a sure-fire way to understand how simple thinking failures can underpin disagreements with my research.

Like consider 1 = 1, a simple tautological statement which is called an identity in mathematics, and notice, the equal sign means you have the same thing on the left of the equals as on the right.

Even if you have x=y, it must be true that, what? x=y

That is, it must be true that x and y are equal, as consider

1 = 0

as in mathematics that is invalid, but modern logicians do not have an invalid type in standard logic.

So they might just say that 1=0 is false, not invalid.

But the expression is not so much false—though it is false—as it fails because it contradicts the use of the equals sign.

If you figure that out, you can work your way through supposed paradoxes in logic and figuring that out is what I did years ago, and I even posted about it years ago as consider:

Logical Formedness Axioms
  1. Identical sets are identical.

  2. Different sets are different.

  3. Statements contradicting axioms 1 or 2 are false or malformed.

  4. A malformed statement is one for which a conclusion does not follow given its structure.

  5. A false statement is one that while structurally correct is not true.
See: http://mymath.blogspot.com/2005/05/logical-formedness-axioms.html

It turns out that if you accept those axioms then necessarily you are accepting the equals means equal.

I think maybe part of the problem with people in the US is that equal can mean just about anything, like note that the Founding Fathers said "all men are created equal" and had slaves!

Sardonic humor aside I think that for most people the failure in understanding such trivial logic is what I call a two-step failure which has to do with how their brains process information, as it LOSES pieces of information in trying to move from noting that equals means equal, and realizing that as a necessity.

As consider the suppose paradoxical statement:

Consider a set of all sets that exclude themselves.

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.

The easy fix is, consider a set of all sets that exclude themselves, except itself.

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.





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