Post. Emil L. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 264–268.

~ / x ) ' , though still deducible of course from ' (3x) /x ' and from '(3x)<~/x', ceases to be deducible in general from 'fa' and from ' ~ / a ' . Note though that Nelson's version of logical t ru th (whatever it is) need not bar him from the usual convenient techniques which allow proof of ' ( 3x ) ( / x v -~/x) ' and inference of '(Bx)fx' from '(x)fx'. He can still accept those techniques as a semi-logical amalgam, comprising pure logic plus a true extra-logical premise to the effect tha t there is something. At the beginning of the discussion I showed that there is no need to allow inference of 'a exists' from 'fa' and from ' ~ / a ' . Now there is a curious line of thought, tangent a t tha t point, which merits passing mention in conclusion. Viz.: Even if 'o exists' cannot be inferred from 'fa' and from ' ~ / a ' , still ' 'a' is meaningful' can, and doesn't this revive the original problem in another form? One possible rejoinder is t h a t ' 'a' is meaningful,' if t rue, is analytic, so that 'fa' and ' ~ / a ' can still be contradictories; but before resting content with this rejoinder I should like to see a satisfactory analysis of meaningfulness. Another possible rejoinder is t h a t ' 'a' is meaningful' cannot be inferred from 'fa', bu t only from ' 'fa' is meaningful.' But then there is the counter-rejoinder t h a t ' 'fa' is meaningful' follows from ' 'fa' is t rue , ' and ' 'fa' is t rue ' follows from 'fa'. Paradoxes involving the word ' t rue, ' however, are no novelty. W. V. Q U I N E