Pointers to Propositions

The semantic paradoxes, whose paradigm is the Liar, played a crucial role at a crucial juncture in the development of modern logic. In his 1908 seminal paper, Russell outlined a system, soon to become that of the Principia Mathematicae, whose main goal was the solution of the logical paradoxes, both semantic and settheoretic. Russell did not distinguish between the two and his theory of types was designed to solve both kinds in the same uniform way. Set theoreticians, however, were content to treat only the set-theoretic paradoxes, putting aside the semantic ones as a non-mathematical concern. This separation was explicitly proposed, eighteen years after Russell’s paper, by Ramsey, though he, like Russell, advocated a system that addresses both kinds. Since then, the semantic paradoxes have been viewed within the perspective of the theory of truth, where they have occupied a respectable niche, but one of rather specialized interest. In this work I shall try to move the issues arising from the semantic paradoxes to a more central place within the philosophy of language. It is not so much the paradoxes themselves as what they reveal about mechanisms incorporated in natural language that is philosophically so significant. These are mechanisms that enable us to make within the same language statements that, in the usual order of things, would require semantic ascents to metalanguages. The implications of the emerging picture extend well beyond the specific concerns of the paradoxes. The work has philosophical and technical aspects. I shall try to make the philosophical points clear, without relying too much on the more technical parts that occupy most of section 4 and part of section 5. The emerging picture is roughly this. Statements are made, or can be made, by means of objects—the “pointers” of the title—which form a kind of network. The semantics consists of recursive rules, by which every pointer is either given a truth-value—signifying the truth or the falsity of the statement expressed through it—or is classified as a failure, i.e., as failing to express something true or false. Each pointer has an associated sentence (the sentence it “points” to), which, in turn, may refer, either by name or through quantification, to pointers. By using a pointer, one can therefore make, or try to make, a statement about statements. These cross references generate the network. In general, the associated sentence is not sufficient for determining the statement made through a pointer. The pointer’s place in the network enters as well. Modo grosso the place determines the metalevel at which the sentence is read; that is,