The IMPAQTS corpus of Italian political speeches annotated per implicit contents contains recurrent examples of NPs introduced by indefinite articles which are characterized by anaphoric specific reference: (Questa legge) hanno tentato di dichiararla anticostituzionale per non applicarla a una persona che si ritiene al di sopra della giustizia. (This law) they tried to declare it unconstitutional so as not to apply it to a person who thinks to be above justice. Such examples must be interpreted as anaphorically referring to a specific, identifiable referent, recoverable in the co-text or in the universe of discourse. Consequently, they activate a presupposition of existence for the mentioned entity, thus revealing that indefinite expressions can trigger existential presupposition in a similar way to definite descriptions. We analyze the formal characteristics of such “presuppositional indefinite descriptions”, which show a recurrent syntactic and semantic structure. As for their functions, we show that they allow a different distribution of the speaker's responsibility on the contents of the utterance, as compared to definite descriptions. This is particularly relevant for the syntactic expansions to the indefinite NPs, whose contents are typically derogatory and disputable, which – in persuasive discourse – makes it beneficial for them to be conveyed in an implicit way.