Hume’s famous character Cleanthes claims that there is no difficulty in explaining the existence of causal chains with no first cause since in them each item is causally explained by its predecessor. Relying on logico-mathematical resources, we argue for two theses: (1) if the existence of Cleanthes’ chain can be explained at all, it must be explained by the fact that the causal law ruling it is in force, and (2) the fact that such a causal law is in force (...) cannot explain the occurrence of the events in the chain. In order to perform (1), we manage to express in mathematical terms the intuitive idea that indefinitely delayed explanation is ultimately no explanation. In order to achieve (2), we identify a logical relation we can prove to be as strong as the causal relation at issue in the Cleanthes passage, according to a precise notion of strength of relations. (shrink)

We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known (...) results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 is countable and distributive and it forms a Heyting algebra. (shrink)

A projective unifier for a modal formula A, over a modal logic L, is a unifier σ for A such that the equivalence of σ with the identity map is the consequence of A. Each projective unifier is a most general unifier for A. Let L be a normal modal logic containing S4. We show that every unifiable formula has a projective unifier in L iff L contains S4.3. The syntactic proof is effective. As a corollary, we conclude that all (...) normal modal logics L containing S4.3 are almost structurally complete, i.e. all admissible rules having unifiable premises are derivable in L. Moreover, L is structurally complete iff L contains McKinsey axiom M. (shrink)

Contact Logics provide a natural framework for representing and reasoning about regions in several areas of computer science. In this paper, we focus our attention on reasoning methods for Contact Logics and address the satisfiability problem and the unifiability problem. Firstly, we give sound and complete tableaux-based decision procedures in Contact Logics and we obtain new results about the decidability/complexity of the satisfiability problem in these logics. Secondly, we address the computability of the unifiability problem in Contact Logics and we (...) obtain new results about the unification type of the unifiability problem in these logics. (shrink)

We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$ . It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4 , continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely axiomatizable (...) by variable-free formulas over K4 that d-define interesting classes of topological spaces. Each of these logics has the finite model property and is decidable. Finally, we introduce quasi-scattered and semi-scattered spaces as generalizations of scattered spaces, develop their basic properties, axiomatize their corresponding modal logics, and show that they also arise as the d-logics of some subspaces of $\mathbb{Q}$. (shrink)

The problem of unification in a normal modal logic $L$ can be defined as follows: given a formula $\varphi$, determine whether there exists a substitution $\sigma$ such that $\sigma $ is in $L$. In this paper, we prove that for several non-symmetric non-transitive modal logics, there exists unifiable formulas that possess no minimal complete set of unifiers.

Epistemic logics are essential to the design of logical systems that capture elements of reasoning about knowledge. In this paper, we study the computability of unifiability and the unification types in several epistemic logics.

In the ordinary modal language, KD is the modal logic determined by the class of all serial frames. In this paper, we demonstrate that KD is nullary.