CATEGORICAL ABSTRACT ALGEBRAIC LOGIC WEAKLY REFERENTIAL π-INSTITUTIONS

Wojcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wojcicki asserts that a logic has a referential semantics if and only if it is selfextensional. A second theorem of Wojcicki asserts that a logic has a weakly referential semantics if and only if it is weakly self-extensional. We formulate and prove an analog of this theorem in the categorical setting. We show that a pi-institution has a weakly referential semantics if and only if it is weakly self-extensional.