Varieties of Linear Calculi

A uniform calculus for linear logic is presented. The calculus has the form of a natural deduction system in sequent calculus style with general introduction and elimination rules. General elimination rules are motivated through an inversion principle, the dual form of which gives the general introduction rules. By restricting all the rules to their single-succedent versions, a uniform calculus for intuitionistic linear logic is obtained. The calculus encompasses both natural deduction and sequent calculus that are obtained as special instances from the uniform calculus. Other instances give all the invertibilities and partial invertibilities for the sequent calculus rules of linear logic. The calculus is normalizing and satisfies the subformula property for normal derivations.     

Pure Extensions, Proof Rules, and Hybrid Axiomatics

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic.A preliminary version of this paper was presented at the fifth conference on Advances in Modal Logic (AiML 2004) in Manchester. We would like to thank Maarten Marx for his comments on an early draft and Agnieszka Kisielewska for help with the proof reading.Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen     

Axiomatizations with Context Rules of Inference in Modal Logic

A certain type of inference rules in (multi-) modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.     

Refutations,Proofs, and Models in the Modal Logic K4

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk's logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.     

The many faces of counts-as: A formal analysis of constitutive rules

The paper proposes a logical systematization of the notion of counts-as which is grounded on a very simple intuition about what counts-as statements actually mean, i.e., forms of classification. Moving from this analytical thesis the paper disentangles three semantically different readings of statements of the type “X counts as Y in context c”, from the weaker notion of contextual classification to the stronger notion of constitutive rule. These many ways in which counts-as can be said are formally addressed by making use of modal logic techniques. The resulting framework allows for a formal characterization of all the involved notions and their reciprocal logical relationships.     

The Quantitative/Qualitative Watershed for Rules of Uncertain Inference

We chart the ways in which closure properties of consequence relations for uncertain inference take on different forms according to whether the relations are generated in a quantitative or a qualitative manner. Among the main themes are: the identification of watershed conditions between probabilistically and qualitatively sound rules; failsafe and classicality transforms of qualitatively sound rules; non-Horn conditions satisfied by probabilistic consequence; representation and completeness problems; and threshold-sensitive conditions such as ‘preface’ and ‘lottery’ rules. Special Issue Formal Epistemology I. Edited by Branden Fitelson     

Toward an Alliance Between the Issue-Processing Approach and Pragma-Dialectical Analysis

On the approach to discussions of policy choices that treats such discussions as instances of issue-processing, the joint use of the logic of questions and the logic of rules gives precise formulation to two sorts of issues. To one sort of issue belong issue-circumscribing questions; to another sort, issues-simplicter, which consist of disjunctions of policy proposals – so many proposed social rules – that are answers, in the case of each disjunction, to a given issue-circumscribing question. Work in pragma-dialectics can take over the issue-processing approach; and by doing so add to the pragma-dialectical repertory further dimensions in the analysis of issues and in protocol-narratives of discussion. The analysis and narratives would now include accounts of how issue-circumscribing questions generate initial standpoints and how discussions sometimes end with compromises between standpoints. Further research questions follow about transformations of issues and the comparison of successive rounds of discussion. A narrative of one period of discussion during `the War on Drugs' in the United States illustrates these points.     

Dynamic Topological Completeness for

Dynamic topological logic (DTL) combines topological and temporalmodalities to express asymptotic properties of dynamic systemson topological spaces. A dynamic topological model is a tripleX ,f , V , where X is a topological space, f : X X a continuousfunction and V a truth valuation assigning subsets of X to propositionalvariables. Valid formulas are those that are true in every model,independently of X or f. A natural problem that arises is toidentify the logics obtained on familiar spaces, such as . It [9] it was shown that any satisfiable formulacould be satisfied in some for n large enough, but the question of how the logic varieswith n remained open. In this paper we prove that any fragment of DTL that is completefor locally finite Kripke frames is complete for . This includes DTL; it also includes some largerfragments, such as DTL₁, where "henceforth" may not appear inthe scope of a topological operator. We show that satisfiabilityof any formula of our language in a locally finite Kripke frameimplies satisfiability in by constructing continuous, open maps from the plane intoarbitrary locally finite Kripke frames, which give us a typeof bisimulation. We also show that the results cannot be extendedto arbitrary formulas of DTL by exhibiting a formula which isvalid in but not in arbitrarytopological spaces.     

Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers p, p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most , the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel–Dummett logics with quantifiers over propositions.     

Reasoning with conditionals

This paper reviews the psychological investigation of reasoning with conditionals, putting an emphasis on recent work. In the first part, a few methodological remarks are presented. In the second part, the main theories of deductive reasoning (mental rules, mental models, and the probabilistic approach) are considered in turn; their content is summarised and the semantics they assume for if and the way they explain formal conditional reasoning are discussed, in particular in the light of experimental work on the probability of conditionals. The last part presents the recent shift of interest towards the study of conditional reasoning in context, that is, with large knowledge bases and uncertain premises.     

An Axiomatisation for the Multi-modal Logic of Knowledge and Linear Time LTK

The paper aims at providing the multi-modal propositional logicLTK with a sound and complete axiomatisation. This logic combinestemporal and epistemic operators and focuses on m odeling thebehaviour of a set of agents operating in a system on the backgroundof a temporal framework. Time is represented as linear and discrete,whereas knowledge is modeled as an S₅-like modality. A furthermodal operator intended to represent environment knowledge isadded to the system in order to achieve the expressive powersufficient to describe the piece of information available tothe agents at each moment in the flow of time.     

On the Algebraizability of Annotated Logics

Annotated logics were introduced by V.S. Subrahmanian as logical foundations for computer programming. One of the difficulties of these systems from the logical point of view is that they are not structural, i.e., their consequence relations are not closed under substitutions. In this paper we give systems of annotated logics that are equivalent to those of Subrahmanian in the sense that everything provable in one type of system has a translation that is provable in the other. Moreover these new systems are structural. We prove that these systems are weakly congruential, namely, they have an infinite system of congruence 1-formulas. Moreover, we prove that an annotated logic is algebraizable (i.e., it has a finite system of congruence formulas,) if and only if the lattice of annotation constants is finite.     

“Four-Valued” Semantics for the Relevant Logic R

This paper sets out two semantics for the relevant logic R based on Dunn's four-valued semantics for first-degree entailments. Unlike Routley's semantics for weak relevant logics, they do not use two ternary accessibility relations. Unlike Restall's semantics, they capture all of R. But there is a catch. Both of the present semantics are neighbourhood semantics, that is, they include sets of propositions in the specification of their frames.     

CE is Not a Conservative Extension of E

The logic CE (for Classical E) results from adding Boolean negation to Anderson and Belnap"s logic E. This paper shows that CE is not a conservative extension of E.     

Prima facie obligation

This paper presents a nonmonotonic deontic logic based on commonsense entailment. It establishes criteria a successful account of obligation should satisfy, and develops a theory that satisfies them. The theory includes two conditional notions of prima facie obligation. One is constitutive; the other is epistemic, and follows nonmonotonically from the constitutive notion. The paper defines unconditional notions of prima facie obligation in terms of the conditional notions.     

Don’t Ever Do That! Long-term Duties in <Emphasis Type="Italic">PD</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis></Subscript><Emphasis Type="Italic">L</Emphasis>

This paper studies long-term norms concerning actions. In Meyer’s Propositional Deontic Logic (PD _e L), only immediate duties can be expressed, however, often one has duties of longer durations such as: “Never do that”, or “Do this someday”. In this paper, we will investigate how to amend PD _e L so that such long-term duties can be expressed. This leads to the interesting and suprising consequence that the long-term prohibition and obligation are not interdefinable in our semantics, while there is a duality between these two notions. As a consequence, we have provided a new analysis of the long-term obligation by introducing a new atomic proposition I (indebtedness) to represent the condition that an agent has some unfulfilled obligation. Presented by Jacek Malinowski