Counterpart semantics for a second-order mu-calculus

We propose a novel approach to the semantics of quantified μ-calculi, considering models where states are algebras; the evolution relation is given by a counterpart relation (a family of partial homomorphisms), allowing for the creation, deletion, and merging of components; and formulas are interpreted over sets of state assignments (families of substitutions, associating formula variables to state components). Our proposal avoids the limitations of existing approaches, usually enforcing restrictions of the evolution relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of

On Time chez Dummett

I discuss three connections between Dummett's writings about time and philosophical aspects of physics. The first connection (Section 2) arises from remarks of Dummett's about the different relations of observation to time and to space. The main point is uncontroversial and applies equally to classical and quantum physics. It concerns the fact that perceptual processing is so rapid, compared with the typical time-scale on which macroscopic objects change their observable properties, that it engenders the idea of a 'common now', spread across space. The other two connections are specific to quantum theory, as interpreted along the lines of Everett. So for these two connections, the physics side is controversial, just as the philosophical side is. In Section 3, I connect the subjective uncertainty before an Everettian 'splitting' of the multiverse to Dummett's suggestion, inspired by McTaggart, that a complete, i.e. indexical-free description of a temporal reality is impossible. And in Section 4, I connect Barbour's denial that time is real---a denial along the lines of Everett, rather than McTaggart---to Dummett's suggestion that statements about the past are not determinately true or false, because they are not effectively decidable.Comment: 25 pages; no figure

The actual future is open

Open futurism is the indeterministic position according to which the future is 'open,' i.e., there is now no fact of the matter as to what future contingent events will actually obtain. Many open futurists hold a branching conception of time, in which a variety of possible futures exist. This paper introduces two challenges to (branching-time) open futurism, which are similar in spirit to a challenge posed by Kit Fine to (standard) tense realism. The paper argues that, to address the new challenges, open futurists must (i) adopt an objective, non-perspectival notion of actuality and (ii) subscribe to an A-theoretic, dynamic conception of reality. Moreover, given a natural understanding of "actual future," (iii) open futurism is naturally coupled with the view that a unique, objectively actual future exists, contrary to a common assumption in the current debate. The paper also contends that recognising the existence of a unique actual future helps open futurists to avoid potential misconceptions

The Modal Future Hypothesis Debugged

This note identifies and corrects some problems in developments of the thesis that predictive expressions, such as English "will", are modals. I contribute a new argument supporting Cariani and Santorio's recent claim that predictive expressions are non-quantificational modals. At the same time, I improve on their selectional semantics by fixing an important bug. Finally, I show that there are benefits to be reaped by integrating the selection semantics framework with standard ideas about the future orientation of modals

A Temporal Logic for Hyperproperties

Hyperproperties, as introduced by Clarkson and Schneider, characterize the correctness of a computer program as a condition on its set of computation paths. Standard temporal logics can only refer to a single path at a time, and therefore cannot express many hyperproperties of interest, including noninterference and other important properties in security and coding theory. In this paper, we investigate an extension of temporal logic with explicit path variables. We show that the quantification over paths naturally subsumes other extensions of temporal logic with operators for information flow and knowledge. The model checking problem for temporal logic with path quantification is decidable. For alternation depth 1, the complexity is PSPACE in the length of the formula and NLOGSPACE in the size of the system, as for linear-time temporal logic

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Refinement Modal Logic

In this paper we present {\em refinement modal logic}. A refinement is like a bisimulation, except that from the three relational requirements only `atoms' and `back' need to be satisfied. Our logic contains a new operator 'all' in addition to the standard modalities 'box' for each agent. The operator 'all' acts as a quantifier over the set of all refinements of a given model. As a variation on a bisimulation quantifier, this refinement operator or refinement quantifier 'all' can be seen as quantifying over a variable not occurring in the formula bound by it. The logic combines the simplicity of multi-agent modal logic with some powers of monadic second-order quantification. We present a sound and complete axiomatization of multi-agent refinement modal logic. We also present an extension of the logic to the modal mu-calculus, and an axiomatization for the single-agent version of this logic. Examples and applications are also discussed: to software verification and design (the set of agents can also be seen as a set of actions), and to dynamic epistemic logic. We further give detailed results on the complexity of satisfiability, and on succinctness

An Everettian Account of Modality

In this thesis I propose that if Everettian Quantum Mechanics (EQM) is correct, then ordinary-objects contained within Everettian worlds ground the truth of nomic de re modal statements in a desirable way. Guided by desiderata set out following a brief assessment of notable modal accounts, I outline one way in which an Everettian account of objective de re modality can be formulated. By applying Eternalism and a formulation of Worm Theory to branching EQM with overlapping worlds, I arrive at an Everettian account of modality whereby concrete ordinary-objects – perduring ‘Branching-Worms’ – ground the truth of de re modal statements, in virtue of having parts which exemplify properties that the modal statement asserts of the ordinary-objects. I conclude that the Everettian modal account I have outlined requires further development in certain areas but hopefully shows some promise as a contending account of modality