Towards a relational theory of meaning: allegorical foundations

The modern theory of meaning, as introduced by figures such as Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970), is fundamentally referential : the meaning of a syntanctic phrase is an extra-linguistic denotation (reference). The seminal work by Richard Montague (1930–1971)-in linguistics-William Law- vere (1937-2023)-in mathematics-and Joseph Goguen (1941–2006)-in computer science-later gave an algebraic foundation of this theory, whereby both syntax and semantics are algebras and meaning is assigned by algebra homomorphisms. In this thesis, we propose a different, relational approach to semantics, one grounded in the close connection between meaning as interpretation and as synonymy. While the former takes the form of (algebra) morphisms, a natural model of the latter is given by term congruences. We investigate this aspect within a categorical framework. Our key observation is that the theory of al- legories, introduced by Freyd and Scedrov, naturally induces a rich theory of relations on syntactic terms — term relations — which we can think about as ‘relational notions of meaning’ (linguists may say sense relations). Remarkably, these relations carry a rich structure, which takes the form of an algebra of term relations. Within this algebra, we develop an effective theory of synonymy as congruence. Our formulation leads to a theory of meaning that is not only expressive and structurally elegant, but also provides a formal setting in which to articulate a classic idea attributed to Noam Chomsky: that syntactic structure determines, and mirrors, semantic structure. We formalize this intuition by proving a rep- resentation theorem, which exhibits how meaning reflects back into syntax.