This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.

p-consequence is intended as a formalization of non-deductive reasoning. So far semantical or general properties have been presented more thoroughly ([2]–[5]). In the present paper we would like to focus on its syntactic properties.

We investigate two different broad traditions in the abstract valuational model theory for nontransitive and nonreflexive logics. The first of these traditions makes heavy use of the natural Galois connection between sets of valuations and sets of arguments. The other, originating with work by Grzegorz Malinowski on nonreflexive logics, and best systematized in Blasio et al. : 233–262, 2017), lets sets of arguments determine a more restricted set of valuations. After giving a systematic discussion of these two different traditions in (...) the valuational model theory for substructural logics, we turn to looking at the ways in which we might try to compare two sets of valuations determining the same set of arguments. (shrink)

p-consequence (plausible consequence; see [2]) allows for a formulation of non-deductive reasonings, i.e., such where the conclusion has weaker justification then assumptions and thus when added to the set of assumptions results in its extension. But theoretical modesty of p-consequence operation does not tell the difference between “good” and “worse” conclusions. Therefore the bisconsequence is introduced.