A Theory of Structured Propositions

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn’t arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the λ-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a diagrammatic representation and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification and demonstrate various equivalences between the diagrammatic and logical representations and a fragment of the λ-calculus.