Can Sentences Self-Refer? : Gödel and the Liar

Abstract

In this article we discuss the issue of "self-reference";, i.e., the question whether (or in which sense) sentences may be said to refer to themselves. Following Wittgenstein, we suggest that the clearest thing to say is that sentences cannot of themselves "do" or "say" anything, but that it is human beings that "do" and "say". Consequently, instances of self-reference have to be considered as part of specific human practices. We illustrate these general remarks through the examination of the Liar Paradox and Gödel's Incompleteness Theorem (which uses a formally undecidable sentence, which is sometimes taken to "say" "I am not provable."). We emphasise that in the context of the "foundations of mathematics" it is important to separate technicalmathematical from philosophical questions, and argue (again following Wittgenstein) that Gödel's Incompleteness Theorem was more a contribution to the former than the latter. In other words, Gödel's result runs the risk of being over-interpreted, and of falling foul of Wittgensteinian philosophy, if it is interpreted philosophically to include focally a sentence that literally self-refers.