Non-deterministic semantics for first-order Logics of Formal (In)consistency

A paraconsistent logic is a logic which allows non-trivial inconsistent
theories. One of the oldest and best known approaches to the problem of
designing useful paraconsistent logics is da Costa's approach, which seeks
to allow the use of classical logic whenever it is safe to do so, but
behaves completely differently when contradictions are involved.
da Costa's approach has led to the family of Logics of Formal
(In)consistency (LFIs). Avron (2005,2006) provides
a natural semantic framework for propositional LFIs was provided using
many-valued structures, called {\em non-deterministic matrices (Nmatrices)}.
This framework has a number of attractive
properties: it is simple, modular, effective and (in the case of finite
Nmatrices) enjoys the benefits of decidability and compactness.

In this talk we extend the non-deterministic semantic framework to the full
first-order level, thereby providing non-deterministic semantics for a large
family of first-order LFIs, including da Costa's C*1. We show that the
semantic framework remains modular and effective, and apply its
effectiveness to prove an important proof-theoretical property of the
studied LFIs.