The outcome set of a belief change operator is the set of outcomes that can be obtained with it. Axiomatic characterizations are reported for the outcome sets of the standard AGM contraction operators and eight types of base-generated contraction. These results throw new light on the properties of some of these operators.

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics.

We analyze the behaviour of declarations of independence between existential quantifiers in quantifier prefixes of Independence-Friendly (IF) sentences; we give a syntactical criterion to decide whether a sentence beginning with such prefix exists, such that its truth values may be affected by removal of the declaration of independence. We extend the result also to equilibrium semantics values for undetermined IF sentences.

The main theorem defines a schema of sound and recursive inference rules; we show more explicitly what happens for some simple special classes of sentences.

In the last section, we extend the main result beyond the scope of prenex sentences, in order to give a proof of the fact that the fragment of IF sentences with knowledge memory has only first-order expressive power.

Primal infon logic was introduced in 2009 in connection with access control. In addition to traditional logic constructs, it contains unary connectives p said indispensable in the intended access control applications. Propositional primal infon logic is decidable in linear time, yet suffices for many common access control scenarios. The most obvious limitation on its expressivity is the failure of the transitivity law for implication: $x \to y$ and $y \to z$ do not necessarily yield $x \to z$. Here we introduce and investigate equiexpressive “transitive” extensions TPIL and TPIL* of propositional primal infon logic as well as their quote-free fragments TPIL0 and TPIL0* respectively. We prove the subformula property for TPIL0* and a similar property for TPIL*; we define Kripke models for the four logics and prove the corresponding soundness-and-completeness theorems; we show that, in all these logics, satisfiable formulas have small models; but our main result is a quadratic-time derivation algorithm for TPIL*.

Galles & Pearl (l998) claimed that “for recursive models, the causal model framework does not add any restrictions to counterfactuals, beyond those imposed by Lewis’s [possible-worlds] framework.” This claim is examined carefully, with the goal of clarifying the exact relationship between causal models and Lewis’s framework. Recursive models are shown to correspond precisely to a subclass of (possible-world) counterfactual structures. On the other hand, a slight generalization of recursive models, models where all equations have unique solutions, is shown to be incomparable in expressive power to counterfactual structures, despite the fact that the Galles and Pearl arguments should apply to them as well. The problem with the Galles and Pearl argument is identified: an axiom that they viewed as irrelevant, because it involved disjunction (which was not in their language), is not irrelevant at all.

In the present paper we propose a system of propositional logic for reasoning about justification, truthmaking, and the connection between justifiers and truthmakers. The logic of justification and truthmaking is developed according to the fundamental ideas introduced by Artemov. Justifiers and truthmakers are treated in a similar way, exploiting the intuition that justifiers provide epistemic grounds for propositions to be considered true, while truthmakers provide ontological grounds for propositions to be true. This system of logic is then applied both for interpreting the notorious definition of knowledge as justified true belief and for advancing a new solution to Gettier counterexamples to this standard definition.

In response to the paper by Cotnoir and Bacon published in RSL 2/2012, we would like to add some remarks regarding supplementation principles. It is known that in a classical mereology, the Strong Supplementation Principle (SSP) together with antisymmetry enforces the Weak Supplementation Principle (WSP). Instead, in the nonwellfounded mereology, the failure of extensionality causes the failure of antisymmetry (Cotnoir, 2010), hence the investigated model is also nonantisymmetric. Cotnoir supposes that the failure of antisymmetry implies the failure of (WSP) when (PP1) is applied, however gives no explicit argument, which we would like to supply in this paper. Additionally, when (PP2) is applied, (SSP) implies (WSP), hence the failure of antisymmetry does not necessarily imply the failure of (WSP).

Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula over the class of intuitionistic Kripke models iff it is invariant with respect to asimulations between intuitionistic models.