Ray Jennings, Professor of Philosophy at Simon Fraser University for many years, was a clever and imaginative logician.  Among many projects and inquiries, he, together with the late Peter K. Schotch, founded a novel approach to paraconsistent logic which they called preservationism.  The motto for that project was straightforward:

Find something you like about your premises and preserve it. 

A second motto offered a blunter (and equally sound) piece of advice:

Don’t make things worse.

The broad point here is that there are many things we may well like about some inconsistent sets of sentences.  For example, each of their members might be supported by observations and/or calculations.  In such cases, identifying which sentences should be rejected can be very difficult; we may find all of them useful in various applications.  This raises a practical question: how should we reason with our best data and models when they include inconsistent information?

Jennings and Schotch’s approach weakened ^- intro, replacing it with a sequence of increasingly weak rules of aggregation:  for example, if we have inconsistent premises that be divided into two consistent sets, then instead of ^-intro, we can apply the 2/3 rule to infer the disjunction of pairwise conjunctions of any three sentences appearing in the premises.  They defined the level of a set of sentences S as the least number of consistent sets that cover S.  In a set of level two or higher, ^-intro produces contradictions and ‘explodes’, but a weakened aggregation rule (“two out of three” (which applies if the set’s level is two) can still allow us to infer the disjunctions of pairwise conjunctions of any 3 sentences in the set.  This rule preserves level 2 in the same sense that we preserve consistency when we close a consistent set of premises under the rules of classical logic. 

In general, Schotch and Jennings’s weaker aggregation rules preserved the least number of consistent sets required to cover an inconsistent set.  (If the inconsistent set includes a contradictory sentence, there is no such number—but in this case we know where the problem is, and how to “fix” it – the contradiction has to go…)  Inconsistencies persist, but reasoning with them this way won’t produce contradictions: the weakening of aggregation, replacing ^- intro with 2/n+1, the disjunction of pairwise conjunctions of any n+1 sentences in the set.    This preserves level n – closing a set of sentences under it won’t produce a more inconsistent set (one with a higher level than the set we began with).    

Among other linguistic explorations, Ray wrote about the linguistic history of “or,” focusing on it as an example of how “logical words” change over time.  Ray suggested that the use of the “logical” words resembles how we use punctuation more how we use lexical words: on his account, the “logical” words contribute very little to how we interpret the sentences they appear in.  Instead, we can treat them as juxtaposing the sentences they ‘join’ in ways that suggest a logical relation between them;  in everyday speech and writing, these juxtapositions enable us to grasp the intended relation, so

Ray’s grasp of language and his careful attention to how we really speak, write and interpret it were remarkable; I hope readers will continue find challenges, insights and treasures in his work.