This article differentiates between linguistic and non-linguistic presumptions, and suggests that we should be interested in conversational presumptions, which could also be called speaker (-meant) presumptions. It also differentiates between potential and actual presumptions. It suggests that, in some cases, presumptions can be conversationally involved and cancellation is possible. It states what the hard cases are and attempts to clarify them via ontological considerations. It attempts to decrease the hard cases via (a) the notion that all actual presumptions are speaker-meant; (b) the distinction between ontological and linguistic presumptions; and (c) the uncontroversial notion that even ontological presumptions are susceptible to semantic under-determination that has to be addressed via pragmatic intrusion. Pragmatic intrusion for presuppositional cases, however, cannot really amount to completing or extending or in any case computing an explicature in an ordinary sense, as explicatures have to do with truth-conditions and try to engender propositionally complete forms that are truth-evaluable, unlike propositional fragments or schemata that are usually uttered in sentences (Carston, 2002; Wilson and Sperber, 2002, 2012). They are processes similar to explicatures, but since presumptions are only pre-conditions for an utterance's being true or false (Strawson, 1950; Macagno; 2016; Green, 1989; Marmaridou, 2000), one cannot say that these processes are completely like those of explicatures. Nevertheless, they are pragmatically-motivated, even if hard to cancel. (See Capone, 2009 on lack of cancellability or Jaszczolt, 2016 on entrenched meanings.) This more or less brings the author's approach in line with contextualism (even when accepting semantic invariantism, along the lines of Kratzer, 2012 or Cappelen and Lepore, 2005 (Invariantism is also shared by Carston (2002), in a form that is compatible with her contextualism.)).