Preservationism

Preservationism is an approach to logic that takes a broader perspective on the sorts of properties of sentences & sets of sentences that can be preserved by various consequence relations. There is a substantial body of work in this area, as well as a very vigourous active program of research. The work already published touches on several topics:

1. The notion of a level, or degree of incoherence and related measures, whose preservation leads to weakly aggregative modal and paraconsistent logics (and a more general understanding of aggregation as a structural feature of logics & its relations to graph theory-- with implications for the lottery paradox and applications in the history and philosophy of science).

2. The notion of (constrained) ambiguous projections of consistent images of inconsistent sets (here the preservation of the constraints leads both to an alternative semantics for LP and to a semantics for first degree entailment).

3. Paraconsistent implication connectives, based on the notion that the implication connective may be required to preserve features other than truth; when such demands are imposed, it becomes more difficult (and in some cases impossible) to arrive at trivializing conditional theorems.