I argue that recent defenses of the view that in 1936 Tarski required all interpretations of a language to share one same domain of quantification are based on misinterpretations of Tarski’s texts. In particular, I rebut some criticisms of my earlier attack on the fixed-domain exegesis and I offer a more detailed report of the textual evidence on the issue than in my earlier work. I also offer new considerations on subsisting issues of interpretation concerning Tarski’s views on the logical correctness of certain omega-arguments, on the Tarskian proof that Etchemendy took to be modal and fallacious, and on Tarski’s appeals to the “common concept of consequence”.

We examine the closure conditions of the probabilistic consequence relation of Hawthorne and Makinson, specifically the outstanding question of completeness in terms of Horn rules, of their proposed (finite) set of rules O. We show that on the contrary no such finite set of Horn rules exists, though we are able to specify an infinite set which is complete.

It is known that propositional relevant logics can be conservatively extended by the addition of a Heyting (intuitionistic) implication connective. We show that this same conservativity holds for a range of first-order relevant logics with strong identity axioms, using an adaptation of Fine’s stratified model theory. For systems without identity, the question of conservatively adding Heyting implication is thereby reduced to the question of conservatively adding the axioms for identity. Some results in this direction are also obtained. The conservative presence of Heyting implication allows the development of an alternative model theory for quantified relevant logics.

A number of people have proposed that we should be pluralists about logic, but there are several things this can mean. Are there versions of logical pluralism that are both high on the interest scale and also true? After discussing some forms of pluralism that seem either insufficiently interesting or quite unlikely to be true, the paper suggests a new form which might be both interesting and true; however, the scope of the pluralism that it allows logic is extremely narrow.

Towards NonBeing (Priest, 2005) gives a noneist account of the semantics of intentional operators and predicates. The semantics for intentional operators are modelled on those for the □ in normal modal logics. In this paper an alternative semantics, modelled on neighborhood semantics for □, is given and assessed.

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsov’s (2009) first-degree proof system ⊢B related to SIXTEEN3. The system GB is a standard Gentzen-type sequent calculus, FB is a four-place (horizontal) matrix sequent calculus, and QB is a quadruple (vertical) matrix sequent calculus. In contrast with GB, the calculus FB satisfies the subformula property, and the calculus QB reflects Odintsov’s coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of ⊢B are proved. In addition, it is shown how the sequent systems for ⊢B can be extended to cut-free sequent calculi for Odintsov’s LB, which is an extension of ⊢B by adding classical implication and negation connectives.

We show how to develop a multitude of rules of nonmonotonic logic from very simple and natural notions of size, using them as building blocks.

We introduce Information Bearing Relation Systems (IBRS) as an abstraction of many logical systems. These are networks with arrows recursively leading to other arrows etc. We then define a general semantics for IBRS, and show that a special case of IBRS generalizes in a very natural way preferential semantics and solves open representation problems for weak logical systems. This is possible, as we can “break” the strong coherence properties of preferential structures by higher arrows, that is, arrows, which do not go to points, but to arrows themselves.