Procedural isomorphism, analytic information and β-conversion by value

Abstract

This article solves, in a logically rigorous manner, a problem originally advanced as a counterexample to Chomsky’s theory of binding and recently discussed in a 2004 paper by Stephen Neale. The example is this. John loves his wife, and so does Peter. Hence John and Peter share a property. But which one? (i) Loving John’s wife: then John and Peter love the same woman. (ii) Loving one’s own wife: then, unless they are married to the same woman, John loves one woman and Peter loves another woman. Since ‘John loves his wife’ is ambiguous between attributing (i) or (ii) to John, ‘So does Peter’ is also ambiguous between attributing (i) or (ii) to Peter. With unrestricted β-reduction, the lambda-term counterparts of the attributions of (i) and (ii) to John both β-reduce to (ii). Which, intuitively, they should not. With suitably restricted β-conversion, the two redexes do not reduce to the same contractum and can be reconstructed from their respective contracta. This article details how to apply this restricted rule of β-conversion to contexts containing anaphora such as ‘his’ and ‘so does’. The logical contribution of the article is a generally valid form of β-conversion ‘by value’ rather than ‘by name’. The philosophical application of β-conversion ‘by value’ to a context containing anaphora is another contribution of this article.