Following-from and Transitivity

A major part of the philosophical investigation of the
notion of logical consequence consists in an attempt at elucidating
its nature--what consequence consists in. Yet, consequence is also a
relation, and as such one can sensibly ask what its formal properties
are. Arguably, Tarski's most notorious contribution to the
philosophical investigation of the notion of consequence is
constituted by his theory of what consequence consists in: truth
preservation in every model. An at least equally important
contribution to such investigation is however represented by his
studies concerning an abstract theory of consequence relations, aimed
at determining the formal behaviour of any such relation. In such
studies, he mentions four properties a consequence relation worthy of
this name must have: reflexivity, monotonicity, transitivity and (less
centrally) compactness. I think all of these properties are at least
questionable. But in the talk I will focus on transitivity (roughly,
if A follows from X and C follows from Y together with A, then C
follows from X,Y), trying to make adequate sense of a position
according to which consequence is not transitive and assess what
impact its correctness would have on our understanding of consequence.
I have a particular investment in this issue, since I have proposed
elsewhere a solution to the sorites paradox which consists in placing
some principled restrictions on transitivity. To fix ideas, I will put
on the table a range of philosophically interesting non-transitive
consequence relations, introducing briefly their rationale. I will
then discuss and dispose of two very influential objections of
principle to the use of non-transitive consequence relations. On this
basis, I will explore some fine details of the logical and normative
structures generated by non-transitivity, and I will close by
remarking on the peculiar local character exhibited by spaces of
propositions whose entailment relations are not transitive.