Abstract

An exploratory approach is taken in order to better understand the generation of reflexive paradoxes. First to be explored is a standard theorem of elementary logic, the Diagonal Lemma, which holds that given a relation R defined on a set S, there can be no S-element which is R-related to all and only those S-elements which are not self R-related. It is demonstrated, via informal logic, that specific applications of the Diagonal Lemma cause two well known paradoxes and a newly constructed paradox. Moreover, these specific applications of the Diagonal Lemma are demonstrated to be hierarchically related to each other. ;The well known paradoxes thus explored are Russell's paradox, which is generated by the apparent fact that the set of all sets which are not self-containing paradoxically both contains and does not contain itself; and the Liar paradox, which is generated by the apparent fact that the truth claim "This truth claim is not true" is paradoxically both true and not true. The newly constructed paradox, the Description paradox, is generated by the apparent fact that an otherwise accurate self-description by the originating source of that self-description becomes paradoxically both accurate and inaccurate if the statement "This self-description is inaccurate" is added to the self-description. ;Standard solutions of Russell's paradox and the Liar paradox are explored and arguments are given to demonstrate that such solutions are inadequate in that they do not deal comprehensively with the underlying concept of "sethood" for Russell's paradox and of "truth" for the Liar paradox. Moreover, reason is given to believe that any proposable solution to either paradox will be inadequate in the same way, and any proposable solution to the Description paradox will fall short in analogous ways, namely that any such solution would not deal comprehensively with the underlying concept of "originating source". ;Finally, the Description paradox is re-examined as a modern version of Descartes' famous "I think, therefore I am" argument. It is argued that if Descartes' argument is so reinterpreted, then the "I am" is interpretable as Descartes' solution to the Description paradox that is the "I think."