Completeness results for intuitionistic and modal logic in a categorical setting

• Published in: Annals of pure and applied logic Vol. 72; no. 1; pp. 25 - 101
• Main Authors: Makkai, M, Reyes, G.E
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 10.03.1995; North-Holland
• ISSN: 0168-0072
• DOI: 10.1016/0168-0072(93)00085-4

Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere (1992). The completeness theorems are used to conclude results asserting that certain logics are conservatively interpretable in others.