Beyond the representational-inferential opposition

I trace the twentieth-century development of representational and inferential theories of meaning to two semantic traditions, emphasizing either a world-directed or a reflexive (higher-order) intentionality, along with either a realist or a nominalist metaphysics. I reconstruct these two traditions using the resources of model theory and proof theory. On this basis, representational model-theoretic and inferential proof-theoretic semantic theorizing appear as progressive developments out of, or offer tools for more carefully thinking and talking about, the two pre-twentieth-century notions of denotational (world-directed) and connotational (reflexive or intraconceptual) meaning. As an illustration of the plausibility of this reconstruction, I outline a joint denotational and connotational semantics, and I show that some of the realist metaphysical questions raised in the so-called “hyperintensional revolution” can be given nominalist answers by treating connotational meaning in terms of the reflexive and intralinguistic categories of proof theory. This uncovers a rich domain for further historical and formal work, challenges currently dominant trends in semantic and metaphysical theorizing, and outlines a program for developing a more thorough understanding of the world-directed and reflexive aspects of language and mind.