Self-contradictory Reasoning

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.