Succinctness in subsystems of the spatial mu-calculus

In this paper we systematically explore questions of succinctness in modal logics employed in spatial reasoning. We show that the closure operator, despite being less expressive, is exponentially more succinct than the limit-point operator, and that the $\mu$-calculus is exponentially more succinct than the equally-expressive tangled limit operator. These results hold for any class of spaces containing at least one crowded metric space or containing all spaces based on ordinals below $\omega^\omega$, with the usual limit operator. We also show that these results continue to hold even if we enrich the less succinct language with the universal modality

Tangled closure algebras

The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical ‘tangle modality’ connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation

The universal tangle for spatial reasoning

The topological $\mu$-calculus has gathered attention in recent years as a powerful framework for representation of spatial knowledge. In particular, spatial relations can be represented over finite structures in the guise of weakly transitive wK4 frames. In this paper we show that the topological $\mu$-calculus is equivalent to a simple fragment based on a variant of the `tangle' operator. Similar results were proven for transitive frames by Dawar and Otto, using modal characterisation theorems for the corresponding classes of frames. However, since these theorems are not available in our setting, which has the upshot of providing a more explicit translation and upper bounds on formula size.Comment: 20 page

Spatial logic of tangled closure operators and modal mu-calculus

There has been renewed interest in recent years in McKinsey and Tarski’s interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces. We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame. This yields completeness theorems over X for a number of languages: (i) the modal mucalculus with the closure operator ; (ii) and the tangled closure operators (in fact can express ); (iii) , ∀; (iv) , ∀, ; (v) the derivative operator ; (vi) and the associated tangled closure operators ; (vii) , ∀; (viii) , ∀,. Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with , X validates the well-known axiom G1. For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connecte

Untangled: A Complete Dynamic Topological Logic

Dynamic topological logic ($\mathbf{DTL}$) is a trimodal logic designed for reasoning about dynamic topological systems. It was shown by Fern\'andez-Duque that the natural set of axioms for $\mathbf{DTL}$ is incomplete, but he provided a complete axiomatisation in an extended language. In this paper, we consider dynamic topological logic over scattered spaces, which are topological spaces where every nonempty subspace has an isolated point. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original trimodal language. In particular, we show that the version of $\mathbf{DTL}$ based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete

Difference and Necessity: Dispositionalism, Deleuze, and the Finite Genesis of Transfinite Truths

Difference and Necessity: Dispositionalism, Deleuze, and the Finite Genesis of Transfinite Truth

An explanation of or-deletions and other paradoxical disjunctive inferences

Some inferences of the sort: A or B; therefore A, which are invalid in standard logics, are sensible in life: You can enter now or later; therefore, you can enter now. That these "or-deletions" follow necessarily or only possibly is a by-product of a theory of mental models. Its semantics for "or" refers to conjunctions of possibilities holding in default of knowledge to the contrary. It predicts new sorts of or-deletion, such as: He likes to drink red wine or white wine. So, he likes to drink red wine. and: You are permitted to do only one of the following: You can enter now. You can enter later. Therefore, you are permitted to enter now. They are invalid in standard logics, and neither previous pragmatic nor semantic theories predicted them. Four experiments corroborated their occurrence.Fundação para a Ciência e Tecnologia - FCTinfo:eu-repo/semantics/publishedVersio