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The relation between self awareness and intelligence is an open problem these days. Despite the fact that self awarness is usually related to Emotional Intelligence, this is not the case here. The problem described in this paper is how to model an agent which knows (Cognitive) Binary Logic and which is also able to pass (without any mistake) a certain family of Turing Tests designed to verify its knowledge and its discourse about the modal states of truth corresponding to well-formed formulae within the language of Propositional Binary Logic.