In his response to ‘On Horn's Dilemma’ (Burton-Roberts, 1989a, henceforth OHD), Seuren makes it clear that he believes that the argument of OHD is vitiated by my failure to pay attention to his own logical system for presupposition. That system embodies (a) a pre-theoretical conception of P-failure (as in The king of France is wise) as being tied to a THIRD TRUTH-VALUE (‘Radical Falsity’) and (b) his view that a semantic ambiguity of negation (SAN) exists and should be welcomed. OHD, by contrast, was concerned with the connections between (a) the Strawson-Frege definition of P, (b) a pre-theoretical view of P in which P-failure is tied, not to a third truth-value, but to LACK of truth-value, and (c) Horn's view of negation as being semantically UNambiguous in the object language. So I disregarded Seuren's theory as being irrelevant to the argument of OHD. I shall make up for it here, though, showing that Seuren's system, and his own claims for it, provide ample corroboration of those arguments and, indeed, of the arguments of Burton-Roberts (1989b) against any formally trivalent approach to the logic of presupposition. I set the scene first with a brief account of the recent history of the distinction between trivalence and gapped bivalence (Section 2). Then I consider in Section 3 the argument of OHD and Seuren's response, and go on to examine Seuren's system (Section 4). Finally, and very summarily, I compare the pragmatic account of special negation outlined in OHD with the semantic account favoured by Seuren (Section 5).