Monday, October 08, 2007
JSH: When equals means equal, more logic
>From searching on Google I see there is a rather huge resonance with my latest postings on logic, so I thought I'd do a post that explains the simple concept of letting equals mean equal, as in like 1=1 is valid in mathematics, but 1=0 is not, as with the latter you have a direct contradiction.
If you think globally about it EVERY mathematical statement reduces to an identity, like
x^2 + y^2 = z^2, if x=3, y=4, and z=5, you have
9 + 16 = 25
25 = 25
and if that does not occur you do not have a valid mathematical statement. Another way to say it is that if you plug in all the numbers and do all the mathematical operations then at the end you just get an identity, which in logic is a tautology as it is always true.
A few years ago I was pondering something or other (not sure what) and I realized that every logical statement just maps a truth to itself, or as I said to myself then, every logical statement connects a truth to itself.
Like with mathematics, for a logical statement, you just get the same truth on both sides.
You must. There is no logical statement in existence that does not map a truth to itself.
When you have equals, you have equality, in logic, like in mathematics.
Once I realized that then I realized that anything else could NOT be a logical statement, just like 1=0 is not mathematically valid, as it breaks the equals I like to say.
So if you make sure the equals, as in the equals sign or a declaration of equality, actually mean equal, you must have a logical statement, and intriguingly you can then clean up ALL the supposed logical paradoxes.
I like the consideration of the set of all sets that exclude themselves, except itself, as that one is the easiest.
Harder I think for people to understand is considering a sentence that declares itself to be false.
e.g. This sentence is false.
In what I call 3 logic I declare that sentence to be false as it's negatably true and then I continue as if it's not a big deal, but one way to look at that sentence is as a direct declaration that true equals false, which is not letting the equals be equal.
Now in allowing in a system of logic false statements I relied on the idea that a false statement is negatably true, like if you say 1 is 3 three, yeah that's false, but if you say 1 is NOT 3, that is true, so the previous is negatably true.
But I had to have a third type—something other than true or false—as what if you have sjdfkj jumps ships?
What in the hell is a "sjdfkj"?
So if you negate that to get sjdfkj does not jump ships, you still have gobbledygook, so I call that the 0 type, neither true nor false.
Oh I'd like to emphasize that the set of all sets that exclude themselves, except itself, is a complete solution and gives you a single set, as there can be only one in all of infinity.
The use of the exception class also gives you the way to resolve the so-called "Barber's Paradox", simply by using an exception.
The Barber shaves every man in the village who doesn't shave himself, except himself.
Now to me using exceptions is natural and rather neat as I have a background in computer programming.
Now then if a logical statement connects a truth to itself then a false statement is NOT a logical statement, nor is one with a truth value of 0, where I say false statement for the false one and for the 0 one I say malformed.
So no logical statement can contradict itself; therefore, there can be no logical paradox.
And in a little post you have a lot of logical issue wrapped up neatly, and notice that there is no problem with a sentence referring to itself!
I solve problems compactly in sensible and easily explainable ways which does not work well for the people writing tomes and abstruse papers on the same subjects (and managing to do so for DECADES) so I think part of the reason for blocks to acceptance of my research is jobs.
Jobs for academics who need to write tomes and abstruse papers, even if there is a simple solution available, so they ignore the simple so they can keep writing tomes and abstruse papers in a system that rewards them for doing so.
Wouldn't you? If your job depended on it?
If you think globally about it EVERY mathematical statement reduces to an identity, like
x^2 + y^2 = z^2, if x=3, y=4, and z=5, you have
9 + 16 = 25
25 = 25
and if that does not occur you do not have a valid mathematical statement. Another way to say it is that if you plug in all the numbers and do all the mathematical operations then at the end you just get an identity, which in logic is a tautology as it is always true.
A few years ago I was pondering something or other (not sure what) and I realized that every logical statement just maps a truth to itself, or as I said to myself then, every logical statement connects a truth to itself.
Like with mathematics, for a logical statement, you just get the same truth on both sides.
You must. There is no logical statement in existence that does not map a truth to itself.
When you have equals, you have equality, in logic, like in mathematics.
Once I realized that then I realized that anything else could NOT be a logical statement, just like 1=0 is not mathematically valid, as it breaks the equals I like to say.
So if you make sure the equals, as in the equals sign or a declaration of equality, actually mean equal, you must have a logical statement, and intriguingly you can then clean up ALL the supposed logical paradoxes.
I like the consideration of the set of all sets that exclude themselves, except itself, as that one is the easiest.
Harder I think for people to understand is considering a sentence that declares itself to be false.
e.g. This sentence is false.
In what I call 3 logic I declare that sentence to be false as it's negatably true and then I continue as if it's not a big deal, but one way to look at that sentence is as a direct declaration that true equals false, which is not letting the equals be equal.
Now in allowing in a system of logic false statements I relied on the idea that a false statement is negatably true, like if you say 1 is 3 three, yeah that's false, but if you say 1 is NOT 3, that is true, so the previous is negatably true.
But I had to have a third type—something other than true or false—as what if you have sjdfkj jumps ships?
What in the hell is a "sjdfkj"?
So if you negate that to get sjdfkj does not jump ships, you still have gobbledygook, so I call that the 0 type, neither true nor false.
Oh I'd like to emphasize that the set of all sets that exclude themselves, except itself, is a complete solution and gives you a single set, as there can be only one in all of infinity.
The use of the exception class also gives you the way to resolve the so-called "Barber's Paradox", simply by using an exception.
The Barber shaves every man in the village who doesn't shave himself, except himself.
Now to me using exceptions is natural and rather neat as I have a background in computer programming.
Now then if a logical statement connects a truth to itself then a false statement is NOT a logical statement, nor is one with a truth value of 0, where I say false statement for the false one and for the 0 one I say malformed.
So no logical statement can contradict itself; therefore, there can be no logical paradox.
And in a little post you have a lot of logical issue wrapped up neatly, and notice that there is no problem with a sentence referring to itself!
I solve problems compactly in sensible and easily explainable ways which does not work well for the people writing tomes and abstruse papers on the same subjects (and managing to do so for DECADES) so I think part of the reason for blocks to acceptance of my research is jobs.
Jobs for academics who need to write tomes and abstruse papers, even if there is a simple solution available, so they ignore the simple so they can keep writing tomes and abstruse papers in a system that rewards them for doing so.
Wouldn't you? If your job depended on it?
# posted by James Harris @ 20:27 
