Please, smack me in the face if I miss the point here, but one of the rules of logic states that through falsification, we can prove something to be true.

But how, if we have proven it false, have we proven anything other than: this statement is false.

Unless by falsification the intent was to mean if the antithesis is proven, because, by law of nature if there is one force then there must be another in opposition, then the opposition is true. But thats not what I read, when I read “falsification.” What I read is: If A does not = A, then A = A.

This is why I hated logic. This one rule. To me, it makes NO sense.

For example, Euclid’s proof that there is an infinite number of primes:

Presume that there is not an infinite number of primes; that instead there is a largest prime number, call it N.

Now consider the number which is the product of all of the prime numbers through N. (If, for example, you think 11 is the largest prime, then consider the product of 2 x3 x5 x7 x 11 = 2310). Add 1 to that number; call the new number P. (2311 in my example.)

Either P is prime or not.

If P is prime, it is larger than N — it is the product of all primes through N, plus 1. Then N is not the largest Prime. Since this argument can be applied to any suppposed largest prime, the primes are infinite.

Suppose P is not prime, which is to say it is divisible by some prime. P is not integrally divisible by any prime 2…N, because it is the [(product of all primes 2…N) +1]. Then, if, by our supposition, P is not prime, then there is some other prime outside 2…N (it may be P itself) which divides P. Again N is not the largest prime, the argument can be applied to any N, and the primes are infinite. q.e.d.

PS: That the product of primes 2…N, plus 1 is not integrally divisible by any prime 2…N can be seen better symbolically:

(a x b x c x d + 1)/a = (b x c x d) + 1/a, and 1/a would never be an integral quotient

[size=150]PPS: I think the crux of such arguments is the Principal of the Excluded Middle: of two contradictory propositions one must be true and one must be false. If you prove one to be false, the other must be true. Above, the contradictory propositions were “The number of prime numbers is infinite.” and “The number of prime numbers is not infinite.”. Euclid proved that “The number of prime numbers is not infinite.” is false[/size].

The argument is that if there are only two possible outcomes, and it cannot be the first one, then the second one must be true.

Simple examples: I am not a woman, therefore I am a man. If I am alive, then I am not dead.

The argument is very powerful. t is often easier to prove a falsehood than a truth. So the usual way of proceeding is to assume that A is, in fact true, and see where it leads. If it leads, logically, to something that is patently absurd, then there are a limited number of reasons:

the logic applied was wrong or flawed;

the assumption was wrong.

If 1) is not the case - because the logic was clearly correct, then only 2) remains.

Bingo!

We do it all the time.

“Hey, I am not gilty, therefore I am innocent” (Not a very good argumetn, I agree, but it is widespread.)

“I had my book with me when I went to the bathroom and do not have it with me now. The book must be in the bathroom.” That is a deduction you have made even though you have not looked in the bathroom.

It is not the purpose of logic to state the nature of the universe, be it a system of unity or duality. However, logically speaking, the universe works itself out one way or another.

Afterall, take Einstein’s theory of universal expansion. Taking your law of nature into account, are we to theorize that there is a force outside the known universe that attempts to contain the boundries of space but fails as the force of the expanding universe is stronger?

But there could be more than two options. For example a formula could return a number >0, <0, or 0. If you can prove two of those are not the case, then you have proven that the formula returns the remaining value.

As Sherlock Holmes said, “when every possibility is explored, whatever remains, no matter how unlikely, must be the truth.”

In the case above, it is still possible to apply the rule though.

If we show that the assumption that the resuklt is 0 leads to contradictions, then we can say the the result is either > 0 or < 0. (Rather, ‘must be’.)