If a certain semantic relation (which we call 'local consequence') is allowed to guide expectations about which rules are derivable from other rules, these expectations will not always be fulfilled, as we illustrate. An alternative semantic criterion (based on a relation we call 'global consequence'), suggested by work of J.W. Garson, turns out to provide a much better - indeed a perfectly accurate - guide to derivability.

The paper generalizes Van McGee's well-known result that there are many maximal consistent sets of instances of Tarski's schema to a number of non-classical theories of truth. It is shown that if a non-classical theory rejects some classically valid principle in order to avoid the truth-theoretic paradoxes, then there will be many maximal non-trivial sets of instances of that principle that the non-classical theorist could in principle endorse. On the basis of this it is argued that the idea of classical (...) recapture, which plays such an important role for non-classical logicians, can only be pushed so far. (shrink)

In a recent paper, Barrio, Pailos and Szmuc (BPS) show that there are logics that have exactly the validities of classical logic up to arbitrarily high levels of inference. They suggest that a logic therefore must be identified by its valid inferences at every inferential level. However, Scambler shows that there are logics with all the validities of classical logic at every inferential level, but with no antivalidities at any inferential level. Scambler concludes that in order to identify a logic, (...) we at least need to look at the validities and the antivalidities of every inferential level. In this paper, I argue that this is still not enough to identify a logic. I apply BPS’s techniques in a super/sub-valuationist setting to construct a logic that has exactly the validities and antivalidities of classical logic at every inferential level. I argue that the resulting logic is nevertheless distinct from classical logic. (shrink)

Introduction to this special issue of The Australasian Journal of Logic.

In recent years, some theorists have argued that the clogics are not only defined by their inferences, but also by their metainferences. In this sense, logics that coincide in their inferences, but not in their metainferences were considered to be different. In this vein, some metainferential logics have been developed, as logics with metainferences of any level, built as hierarchies over known logics, such as \, and \. What is distinctive of these metainferential logics is that they are mixed, i.e. (...) the standard for the premises and the conclusion is not necessarily the same. However, so far, all of these systems have been presented following a semantical standpoint, in terms of valuations based on the Strong Kleene truth-tables. In this article, we provide sound and complete sequent-calculi for the valid inferences and the invalid inferences of the logics \ and \, and introduce an algorithm that allows obtaining sound and complete sequent-calculi for the global validities and the global invalidities of any metainferential logic of any level. (shrink)