A Meta-Logical Framework for the Equivalence of Syntactic and Semantic Theories

This paper introduces a meta-logical framework—based on the theory of institutions (a categorical version of abstract model theory)—to be used as a tool for the formalization of the two main views regarding the structure of scientific theories, namely the syntactic and the semantic views, as they have emerged from the relevant contemporary discussion. The formalization leads to a proof of the equivalence of the two views, which supports the claim that the two approaches are not really in tension. The proof is based on the Galois connection between classes of sentences and classes of models defined over some institution. First, the history of the syntactic–semantic debate is recalled and the theory of institutions formally introduced. Secondly, the notions of syntactic and semantic theories are formalized within the institution and their equivalence proved. Finally, the novelty of the proposed framework is highlighted with respect to existing formalizations.