Conclusion

In this essay, we saw that the central concept of logic - that of logical consequence - can be formulated in different ways. Following a long­standing tradition, Tarski's model-theoretic formulation of the concept embodied some of the central features of logic: universality, formality, truth-preservation, and a priority.

Following the logical pluralist framework offered by Beall and Restall, we then explored what happens to the characterization of logical con­sequence if we change the emphasis from models to other cases, such as inconsistent and complete situations, or consistent, complete and non­distributive situations. We saw that different logics would then emerge - in particular, paraconsistent and quantum logics. But the logical pluralist framework in terms of cases is not the only one, and we saw that logical pluralism can also be characterized in terms of domains, with the advan­tage that the explicitly modal force of the notion of logical consequence is highlighted. In turn, the logical pluralist by domains challenges some of the received features of logic; in particular, the latter's alleged a priori and universal character. For some, this is a reductio ad absurdum of this form of logical pluralism; for others, this shows the close connection that logic has with all our forms of inquiry, particularly in science and mathematics.

This point naturally led to the issue of the application of logic. And we explored some of the connections between the application of logic and of mathematics. Both have pure and applied counterparts. Both are sometimes motivated by, and developed from, the needs that emerge in empirical contexts. Both have been successfully applied to areas for which they were not originally devised. Given the close connections that logic has with our forms of inquiry, particularly on the pluralist by domains conception, it is not surprising to find such connections between these fields. This is as it should be. There is no reason to leave logic hanging in the clouds.