I patronize the local atheist boutique to buy bumper stickers, pens, pencils, lapel pins, and badges. This one sat on my desk for a couple of weeks. Then, I had a reply. "Watches only prove that beaches were not designed to be watches." The fallacy goes to the root of arguments for atheism.You cannot prove a negative assertion. When you try, you run into non-sequiturs.

One can't prove a negative assertion???? I think not.

Consider: There do not exist integers m, n n not = 0 and m, n relatively prime such that (m/n)^2 = 2. That is as negative a proposition one will find.

Well, suppose such an m, n exists. Then m^2/n^2 = 2 hence m^2 = 2 * n^2. m^2 is even hence m is even (I leave this to you to prove as an exercise).

Thus m = 2*k. Substituting we get (2*k)^2 = n^2 hence 4 * k^2 = 2 * n^2. Divide both sides by 2, to get 2*k^2 = n^2 . This implies that n is even. But we assumed the m and n have no prime factors in common so we have a contradiction. Thus the negative proposition is proved.

This proof was found during the time and Pythagoras and it wrecked his system of basing every quantity on integers and ratios of integers. The square rioot of 2 is irrational. One can generalize this to prove that the square root of an integer which is the product of primes raised to an odd power is not rational (i.e. the ratio of integers with no common factor except 1). Even more negatives that can be proved. And still more were proved. A circle cannot be squared using ruler and compass nor can the general angle be trisected using ruler and compass.

Yes, of course all of those are only the result of attempting to prove a positive assertion and running into a contradiction that falsifies it. Then, you turn it over and make it a disproof. That is just a subset of how proofs are done in algebra. Andrew Wyles's proof of Fermat's Conjecture is a great example.

Perhaps I need a better way of stating my hypothesis, but I do not want to separate the rational from the empirical.

Yes, of course all of those are only the result of attempting to prove a positive assertion and running into a contradiction that falsifies it. Then, you turn it over and make it a disproof. That is just a subset of how proofs are done in algebra. Andrew Wyles's proof of Fermat's Conjecture is a great example.

Perhaps I need a better way of stating my hypothesis, but I do not want to separate the rational from the empirical.

.

Bottom Line: A negative proposition is proved validly. a implies b if and only if -b implies -a. A proof by contradiction is a proof. Most of important mathematical results cannot be proved any other way.

I get the point, but it does not address the deeper issue. You can prove that 2 + 2 ≠ 6 because the statement is not isolated from all the rest of arithmetic. All you have done is take one truth (2 + 2 = 4) and misstate it into a falsehood.

And your objection does not tie the analytic to the synthetic. You cannot prove that the Moon has not been used as a base by aliens. You cannot prove that God does not exist. And all the rest. How are empirical observations, or the lack of them, different from logical truths?

I get the point, but it does not address the deeper issue. You can prove that 2 + 2 ≠ 6 because the statement is not isolated from all the rest of arithmetic. All you have done is take one truth (2 + 2 = 4) and misstate it into a falsehood.

And your objection does not tie the analytic to the synthetic. You cannot prove that the Moon has not been used as a base by aliens. You cannot prove that God does not exist. And all the rest. How are empirical observations, or the lack of them, different from logical truths?

you miss the technical logical point. Modus Tollens and Modus Ponens are equally valid.

A. a, a implies b yields b

B. -b, a implies b yields -a

they are equivalent and both valid. In indirect proof is as much a proof as a direct proof.

The word "prove" in this context means to yield by valid inference from the premises. A true premise always yields a true conclusion.

BTW, logic does not tell us which premises are true. Logic tells us only what premises imply or which conclusions can be inferred from the premises

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## BaalChatzaf

One can't prove a negative assertion???? I think not.

Consider: There do not exist integers m, n n not = 0 and m, n relatively prime such that (m/n)^2 = 2. That is as negative a proposition one will find.

Well, suppose such an m, n exists. Then m^2/n^2 = 2 hence m^2 = 2 * n^2. m^2 is even hence m is even (I leave this to you to prove as an exercise).

Thus m = 2*k. Substituting we get (2*k)^2 = n^2 hence 4 * k^2 = 2 * n^2. Divide both sides by 2, to get 2*k^2 = n^2 . This implies that n is even. But we assumed the m and n have no prime factors in common so we have a contradiction. Thus the negative proposition is proved.

This proof was found during the time and Pythagoras and it wrecked his system of basing every quantity on integers and ratios of integers. The square rioot of 2 is irrational. One can generalize this to prove that the square root of an integer which is the product of primes raised to an odd power is not rational (i.e. the ratio of integers with no common factor except 1). Even more negatives that can be proved. And still more were proved. A circle cannot be squared using ruler and compass nor can the general angle be trisected using ruler and compass.

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## syrakusos

AuthorYes, of course all of those are only the result of attempting to prove a positive assertion and running into a contradiction that falsifies it. Then, you turn it over and make it a disproof. That is just a subset of how proofs are done in algebra. Andrew Wyles's proof of Fermat's Conjecture is a great example.

Perhaps I need a better way of stating my hypothesis, but I do not want to separate the rational from the empirical.

.

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## BaalChatzaf

Bottom Line: A negative proposition is proved validly. a implies b if and only if -b implies -a. A proof by contradiction is a proof. Most of important mathematical results cannot be proved any other way.

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## syrakusos

AuthorI get the point, but it does not address the deeper issue. You can prove that 2 + 2 ≠ 6 because the statement is not isolated from all the rest of arithmetic. All you have done is take one truth (2 + 2 = 4) and misstate it into a falsehood.

And your objection does not tie the analytic to the synthetic. You cannot prove that the Moon has

notbeen used as a base by aliens. You cannot prove that God does not exist. And all the rest. How are empirical observations, or the lack of them, different from logical truths?## Link to comment

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## BaalChatzaf

you miss the technical logical point. Modus Tollens and Modus Ponens are equally valid.

A. a, a implies b yields b

B. -b, a implies b yields -a

they are equivalent and both valid. In indirect proof is as much a proof as a direct proof.

The word "prove" in this context means to yield by valid inference from the premises. A true premise always yields a true conclusion.

BTW, logic does not tell us which premises are true. Logic tells us only what premises imply or which conclusions can be inferred from the premises

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