 Home Philosophy Advances in Proof-Theoretic Semantics

## Introduction

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. Found a mistake? Please highlight the word and press Shift + Enter Related topics