Genuine Violations of Laws

Could laws of nature be violated, in the sense that some proposition is both a law and false? I argue that opponents of regularity theories of laws should accept the metaphysical possibility of such genuine violations. I begin with a clarification of this claim. The main argument is then developed in three steps. I first argue that opponents of regularity theory should endorse the modal-essence view: certain modal principles are essential to the laws of nature. Second, I argue that the modal-essence view entails sophisticated modal primitivism. Third, I argue from sophisticated primitivism to the metaphysical possibility of genuine violations of laws.     

A Modal Logic for Discretely Descending Chains of Sets

We present a modal logic for the class of subset spaces based on discretely descending chains of sets. Apart from the usual modalities for knowledge and effort the standard temporal connectives are included in the underlying language. Our main objective is to prove completeness of a corresponding axiomatization. Furthermore, we show that the system satisfies a certain finite model property and is decidable thus.     

Algebraic Semantics for Deductive Systems

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics.     

Game Logic - An Overview

Game Logic is a modal logic which extends Propositional Dynamic Logic by generalising its semantics and adding a new operator to the language. The logic can be used to reason about determined 2-player games. We present an overview of meta-theoretic results regarding this logic, also covering the algebraic version of the logic known as Game Algebra.     

The Russian Cards Problem

Suppose we have a stack of cards that is divided over some players. For certain distributions of cards it is possible to communicate your hand of cards to another player by public announcements, without yet another player learning any of your cards. A solution to this problem consists of some sequence of announcements and is called an exchange. It is called a direct exchange if it consists of (the minimum of) two announcements only. The announcements in an exchange have a special form: they are safe communications, an interesting new form of update. Certain unsafe communications turn out to be unsuccessful updates. A communication is a public announcement that is known to be true. Each communication may be about a set of alternative card deals only, and even about a set of alternatives to the communicating player's own hand only. We list the direct exchanges for a deal of seven cards where the two players holding three cards communicate their hands to each other. Our work may be applicable to the design of cryptographic protocols.     

Reduction and emergence in the physical sciences: Reply to Rueger

I analyse Rueger’s application of Kim’s model of functional reduction to the relation between the thermal conductivities of metal bars at macroscopic and atomic scales. 1) I show that it is a misunderstanding to accuse the functional reduction model of not accounting for the fact that there are causal powers at the micro-level which have no equivalent at the macro-level. The model not only allows but requires that the causal powers by virtue of which a functional predicate is defined, are only a subset of the causal powers of the properties filling the functional specification. 2) The fact that the micro-equation does not converge to the macro-equation in general but only under the constraint of a “solvability condition” does not show that reduction is impossible, as Rueger claims, but only that reduction requires inter-level constraints. 3) Rueger tries to analyse inter-level reduction with the conceptual means of intra-level reduction. This threatens the coherence of his analysis, given that it makes no sense to ascribe macroproperties such as thermal conductivity to entities at the atomic level. Ignoring the distinction between theses two senses of “reduction” is especially confusing because they have opposite directions: in intra-level reduction, the more detailed account reduces to the less detailed one, whereas in inter-level reduction, the less detailed theory is reduced to the more detailed one. 4) Finally I criticize Rueger’s way of using Wimsatt’s criteria for emergence in terms of non-aggregativity, to construct a concept of synchronic emergence. It is wrong to require, over and above non-aggregativity, irreducibility as a criterion for emergence.     

Cut Elimination inside a Deep Inference System for Classical Predicate Logic

Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivations.     

A Phenomenological Calculus for Anisotropic Systems

The phenomenological calculus is a relational paradigm for complex systems, closely related in substance and spirit to Robert Rosen’s own approach. Its mathematical language is multilinear algebra. The epistemological exploration continues in this paper, with the expansion of the phenomenological calculus into the realm of anisotropy.     

Loosely Guarded Fragment of First-Order Logic has the Finite Model Property

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.     

Subdirectly Irreducible Modal Algebras and Initial Frames

The duality between general frames and modal algebras allows to transfer a problem about the relational (Kripke) semantics into algebraic terms, and conversely. We here deal with the conjecture: the modal algebra A is subdirectly irreducible (s.i.) if and only if the dual frame A* is generated. We show that it is false in general, and that it becomes true under some mild assumptions, which include the finite case and the case of K4. We also prove that a Kripke frame F is generated if and only if the dual algebra F* is s.i. The technical result is that A is s.i. when the set of points which generate the dual frame A* is not of zero measure.