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I know Russell and Wittgenstein argued about negative truths. It is easy to prove the existence of some property provided there is considerable empirical evidence for its existence, but what if we are concerned with negative properties? Although we could provide some evidence for absence of a particular properties, we cannot prove the nonexistence. Have they reached any conclusions? What is it in the case of negative truths that makes them true? How does one prove that there does not exist some property?

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up vote 6 down vote accepted

A brief answer - It is possible to prove that something "can't exist", which in turn implies that it doesn't exist. But trying to prove that something doesn't exist directly is a futile effort.

iphigenie's links will give you more detail than I have time to re-hash at the moment though :)

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As Russell was a mathematician as well as being a logician & philosopher, I'm going to take to tackle this question in the realm of mathematics/logic, rather than of empirical science or in a general theory of knowledge.

As Ryno points out its eminently possible to prove that something cannot exist. Usually it involves a reductio, where one assumes that this object does exist and then derive a contradiction.

Brouwer argued for a constructive philosophy of mathematics where to show something exists one must construct it. It turns out that for this to be consistent with the the standard move above one must generalise classical logic to intuitionistic logic which is essentially the classical one without the law of the excluded middle, in fact Brouwer viewed this law as abstracted from finite experience, and then applied to the infinite without justification. (note that in set theory ZF one is allowed to make finite choices but not infinite ones, that requires an additional axiom - the axiom of choice C - to make the standard set theory ZFC, this is seen as very useful though still controversial).

This is already useful in calculus, where the intuitive idea of the tangent to a curve being an infinitesimal vector couldn't be made rigorous in-itself, it required the rigorous definition of a limit; however using intuitionistic logic, it turns out one can, in what is now called smooth infinitesimal analysis.

(Notably Deleuze writes in Difference and Repetition, “it is a mistake to tie the value of the symbol dx to the existence of infinitesimals; but it is also a mistake to refuse it any ontological ... value,...in fact, there is a treasure buried within the old so-called barbaric or pre-scientific interpretations of the differential calculus...a great deal of heart and a great deal of truly philosophical naivety is needed in order to take the symbol dx seriously...")

So Russell & Wittgenstein were not arguing over unsubstantial issue. I'd be interested to know who took which side...

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Wittgenstein thought that the world consists of facts, not objects. Russell did not like this as this does not allow proving that, for instance, there is no chair in a room where we cannot find it. Searching the room and not finding the chair, does not prove it is not in the room. –  glebovg Dec 22 '12 at 22:06
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