Abstract This paper concerns the characterization of paradoxical reasoning in terms of structures of proofs. The starting point is the observation that many paradoxes use self-reference to give a statement a double meaning and that this double meaning results in a contradiction. Continuing by constraining the concept of meaning by the inferences of a derivation “self-contradictory reasoning” is formalized as reasoning with statements that have a double meaning, or equivalently, cannot be given any meaning. The “meanings” derived this way are global for the argument as a whole. That is, they are not only constraints for each separate inference step of the argument. It is shown that the basic examples of paradoxes, the liar paradox and Russell's paradox, are self-contradictory. Self-contradiction is not only a structure of paradoxes but is found also in proofs using self-reference. Self-contradiction is formalized in natural deduction systems for naïve set theory, and it is shown that self-contradiction is related to normalization. Non-normalizable deductions are self-contradictory.
Let us consider Russell's paradox:
Let t be the set of all sets not containing themselves. Assume that t contains itself. Hence, by the definition of t , t does not contain itself. This contradicts the assumption that t contains itself and hence t does not contain itself. Since t does not contain itself, it follows from the definition of t that t contains itself. This is a contradiction.
Let us take a closer look at the part of Russell's paradox that proves that t does not contain itself. Let ER be this part of Russell's paradox. We observe that the assumption that t contains itself is used twice in ER . We shall now distinguish the use of an assumption from how it is used. Let us therefore, to express that an assumption is used in an argument, say that the assumption occurs in an argument. Thus there are two occurrences of the assumption that t contains itself in ER . One of these two occurrences of the assumption that t contains itself is used together with the definition of t to derive that t does not contain itself. To contradict this last proposition the other occurrence of the assumption that t contains itself is used. Hence, there are two occurrences of the assumption that t contains itself in ER , and they are used in such a way that they contradict each other. In the last step of ER , the conclusion that t does not contain itself is drawn from the contradiction that the assumption that t contains itself leads to. In a sense the two occurrences of the assumption that t contains itself are identified in this step. Considering the two occurrences of the assumption that t contains itself as one and the same proposition, we have that there in ER is a proposition which is used in two ways and that the two ways of using the proposition are incompatible.
A self-contradictory argument is, informally, an argument, as ER above, in which there is a proposition which is used in two or more ways such that not all of the ways of using the proposition are compatible. In this article we aim to make those ideas more precise and formally express the notion of self-contradictory reasoning in some formal systems.