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There are two types of negative statements I will discuss.

Type 1: denial of an universally quantified proposition:: Not For all x, P(x).

Type 2 denial of an existential proposition:: not there exists x such that P(x).

note: P(x) means the predicate or property P is true of individual x.

The way you prove a Type 1 denial is to find an individual r, such that Not P®. The assertion There exists r such that Not P® is precisely the negation of For all x P(x). They are logically equivalent. This is simply a formal way of producing a counter example to a general assertion. You show the general assertion is false by producing an example that negates the general assertion.

The way you prove a Type 2 denial is to show the assumption that there exists x such that P(x) leads to a contradiction.

The well known example is to show that the square root of 2 is not rational. How do we do this? First assume the square root of 2 -is- rational. Then by definition of rational there exists two integers m and n, such that n != 0 and such that

(m/n)^2 = 2. We can assume without loss of generality that m and n have no factors in common except 1. Any common factors can be divided out and still leave the ratio intact. Under this assumption m^2 = 2*n^2. Which means m^2 is even.

This implies that m is even. (note. The square of any odd integer is odd --- I leave that for you to show as an exercise).

O.K. if m is even then by definition of even, m = 2*k for some integer k.

m^2 = 2*n^2 implies 4*k^2 = 2*n^2. Divide both sides by 2 and get 2*k^2 = n^2. But that means n^2 is also even. When means 2 divides n. Bzzzzzt. That means both m and n are even. But that contradicts the assumption that m and n have no factor in common other than 1.

The assumption that the square root of 2 is rational has lead to the contradiction that a pair of numbers both have and do not have a common factor other than 1.

Q.E.D.

And that, my friends is how one proves negatives statements of a certain type (Type 1 and Type 2 above).

Which contradicts the general assertion that one cannot prove a negative. There do exist honest to goodness negative propositions which can be proven. (Note the assertion that no negatives can be proven has been refuted by a counter example).

Q.E.D. again.

ruveyn

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That is not what is meant by 'proving a negative' in regard to the onus of proof principle.

Really?. I tend to take most things literally and verbetim. If some one says to me "You can't prove a negative" I roll out the proof of a negative to wit the proof that the square root of 2 is not rational. My reflex reaction to any general statement either explicit or implied is to look for a counter example. My entire life is a long trail of counter examples found to ill founded generalities. It is a messy job, but someone has to do it.

ruveyn

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Not rational doesn't mean negative. To prove a negative means to prove that X doesn't exist which is a contradiction in terms. Proof means "the cogency of evidence that compels acceptance by the mind of a truth or a fact" (Free Merriam-Webster dictionary) or in Leonard Peikoff's words:

“Proof,” in the full sense, is the process of deriving a conclusion step by step from the evidence of the senses, each step being taken in accordance with the laws of logic." (Leonard Peikoff, Introduction to Logic, Lecture 1)

How one can derive a conclusion from non-existent evidence or compel acceptance of a non-existent fact?

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Not rational doesn't mean negative. To prove a negative means to prove that X doesn't exist which is a contradiction in terms. Proof means "the cogency of evidence that compels acceptance by the mind of a truth or a fact" (Free Merriam-Webster dictionary) or in Leonard Peikoff's words:

“Proof,” in the full sense, is the process of deriving a conclusion step by step from the evidence of the senses, each step being taken in accordance with the laws of logic." (Leonard Peikoff, Introduction to Logic, Lecture 1)

How one can derive a conclusion from non-existent evidence or compel acceptance of a non-existent fact?

Negative means a logical negation, a denial. The basic negatives are denying the existence of an object with some specified property or denying a universally quantified proposition. It is a logical term in this context, not the additive inverse of a positive number.

ruveyn

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There is a difference between "A is not B" and "A is non-B".

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There is a difference between "A is not B" and "A is non-B".

x has the property P (usually written P(x)) is the positive form

not (x has the property P) (usually written ~P(x)) is the negative form.

For existential statements:

there exists x such that x has the property P (usually written Ex[P(x)]) This is the positive form

there does not exist x such that x has the property P or it is false that there exists x such that x has the property P (usually written ~(Ex[P(x)) ) : this is the negative form.

I use standard first order logic rather than the Aristotelian formulation. Because it is more flexible.

My usual challenge to people who like the traditional categorical syllogistic version of logic is this: Prove Prop 47 Euclid Book I (better known as Pythagoras' Theorem) using only categorical syllogisms and the traditional square of opposition. Do you care to have a go at it? Most mathematical proofs use conditional logic which is readily formulated as natural deduction in the context of first order logic. If you look at Euclid's Elements you will not see a single proof using categorical syllogisms.

ruveyn

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That is not what is meant by 'proving a negative' in regard to the onus of proof principle.

Really?. I tend to take most things literally and verbetim. If some one says to me "You can't prove a negative" I roll out the proof of a negative to wit the proof that the square root of 2 is not rational. My reflex reaction to any general statement either explicit or implied is to look for a counter example. My entire life is a long trail of counter examples found to ill founded generalities. It is a messy job, but someone has to do it.

You aren't taking it literally. You are ignoring the whole issue of the meaning and reason for the onus of proof principle, and are using the term "negative" in an equivocal way, out of context. It has nothing to do with an analysis of quantifiers and their negation. You haven't said anything at all about the onus of proof principle itself, what you think it means, and why you think it is an invalid generalization. Dropping that context and picking up on the word "negative" under your own interpretation in an entirely different discussion is not 'taking things literally'.