(1) Atlantis is West of London.
(2) Atlantis is East of London.
(3) Atlantis and London have the same longitude.
But 'x is west of y' means the same as 'x is not east of y, and x does not have the same longitude as y', and now we have a problem.
If 'Atlantis is west of London' is false (as the free logician says), then at least one of 'Atlantis is east of London' or 'Atlantis and London the same longitude' has to be true -- but that contradicts the earlier assumption (of the negative free logician) that all of (1)-(3) are false.He asks "Does anyone see a good response to this objection on behalf of the proponent of negative free logic?" I answer: the objection relies, first, on the perfectly valid assumption that if p is equivalent to "~q and ~r", then ~p is equivalent to "q or r". Second, it relies on the invalid assumption that "Atlantis is West of London" is equivalent to a conjunction of two negatives. On the contrary, it is equivalent to the conjunction of two affirmatives. It is equivalent (according to our variety of NFL) to "something which is identical with Atlantis is West of London". This in turn is equivalent to "something which is identical with Atlantis is neither East of London nor has the same longitude as London". And the negation of that is
(4) It is not the case that something identical with Atlantis is not East of London, or it is not the case that something identical with Atlantis does not have the same longitude as London.
Now if (1) is false then (4) is true, and if Atlantis exists, it follows that Atlantis is East of London, or that Atlantis and London have the same longitude (I will omit the proof, which is elementary). But if Atlantis does not exist, both (2) - which is equivalent to "something which is identical with Atlantis is East of London" - and (3) - which is equivalent to "something which is identical with Atlantis has the same longtitude as London" - are false. For the presumption is that nothing is identical with Atlantis.
This requires the negative free logician to bite a bullet, but it is not the unpalatable one that Frost-Arnold suggests. We have to concede that sentences like "Atlantis is large" and "Atlantis is not large" are not true contradictories, but contraries. The second is not the negation of the first. The true negation of the first is "notthing is Atlantis or Atlantis is not large". But there cannot be simple solutions in philosophy. When we have exhausted everything that is wrong, what remains must be right.