We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, (...) semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)

In 1983, Valentini presented a syntactic proof of cut elimination for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for “Valentini”. The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are ‘hidden’ in these set based proofs of cut elimination. There is (...) often an underly ing assumption that the move to a proof of cut elimination for sequents built from multisets is easy. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of GLSV with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut elimination for GL in a multiset setting. Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut elimination for sequents built from multisets. The formal setting is particularly important for sequents built from multisets, in order to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is diffi cult to verify the correctness of his induction argument based on “width”. In our formulation however, verification of the induction argument is straight forward. Finally, the multiset setting also introduces a new complication in the the case of contractions above cut when the cut formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments. Finally, we show that the algorithm purporting to show the non termi nation of Valentini’s arguments is not a faithful representation of the original arguments, but is instead a transformation already known to be insufficient. (shrink)

We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automaton-labelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cut-free and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar (...) logics using our calculus. (shrink)

The Stoic, David Hume’s “man of action and virtue,” is often considered the forerunner and foundation of Adam Smith’s market man of morals (Hume 1985, 146–54). Ian Simpson Ross notes Smith’s enthusiasm for Stoic philosophers such as Cicero and Marcus Aurelius and the way Stoic philosophy informs Smith’s arguments on various topics such as self-command, self-love, and suicide (Ross 1995, 172, 384). Pierre Force confirms the influence of Stoicism in tracing Smith’s moral system as a contrast with the Epicurean/Augustinian tradition, (...) showing that for Smith and Stoics self-love merely points the way for an agent to choose rationally a course of action (Force 2003, 103–5). However, Deirdre McCloskey and others .. (shrink)

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.

We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring (...) a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here. (shrink)

We propose a new sequent calculus for bi intuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cut elimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cut elimination proof. We then present the derived calculus, and then present a proof (...) search strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for bi intuitionistic logic. As far as we know, our new calculus is the first sequent calculus for bi intuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cut elimination proof, and which can be used naturally for backwards proof search. Keywords: Bi-intuitionistic logic, display calculi, proof search. (shrink)

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. When the modality is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points.Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulaeX the superformulae involved (...) are always bounded by a finite set of formulaeX* L depending only onX and the logicL. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. (shrink)

We present a cut-free tableau calculus with histories and variables for the EXPTIME-complete multi-modal logic of common knowledge. Our calculus constructs the tableau using only one pass, so proof-search for testing theoremhood of ϕ does not exhibit the worst-case EXPTIME-behaviour for all ϕ as in two-pass methods. Our calculus also does not contain a “finitized ω-rule” so that it detects cyclic branches as soon as they arise rather than by worst-case exponential branching with respect to the size of ϕ. Moreover, (...) by retaining the rooted-tree form from traditional tableaux, our calculus becomes amenable to the vast array of optimisation techniques which have proved essential for “practical” automated reasoning in very expressive description logics. Our calculus forms the basis for developing a uniform framework for the family of all fix-point logics of common knowledge. However, there is still no “free lunch” as, in the worst case, our method exhibits 2EXPTIME-behaviour. A prototype implementation can be found at twb.rsise.anu.edu.au which allows users to test formulae via a simple graphical interface. (shrink)

We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machine-checked proof of cut-admissibility for RA. Unlike other “implementations”, we explicitly formalise the structural induction in Isabelle/HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.

Abstract:Dwelling near Lake Superior in the Anthropocene, we uncover a greater intimacy and acquaintance with our earthly responsibilities. Thoughts wash over us like waves as our thinking ebbs and flows between the fact that we must learn to dwell here while also coming to terms with the planetary implications of our very being. That ebb and flow is presented here in a series of waves, which can be read in or out of order, in an orderly or disorderly fashion. These (...) waves crash into one another as we reflect on place, dwelling, hospitality, deep history, enchantment, wildness, and thinking itself. We discover in this mixture an invitation to think more deeply about our responsibilities by contemplating one of the other bodies with which we cohabit the earth: the deep, blue body of Lake Superior. (shrink)

This paper provides practitioner and academic insights into the importance of examining non-technical skills in a multiteam system emergency response. The case of public health professionals is highlighted, illustrated with unique qualitative field data which focused upon the use of non-technical skills at a meso level of analysis. Results reflected the importance of context upon the multiteam system and highlighted seven non-technical skills used by public health professionals to support an effective response. Recommendations for future research and implications for practice (...) are noted for this hard to access professional group, located within emerging advances in the scientific inquiry of complex and increasingly evident, multi-team systems. (shrink)

We define cut-free display calculi for knowledge logics wherean indiscernibility relation is associated to each set of agents, andwhere agents decide the membership of objects using thisindiscernibility relation. To do so, we first translate the knowledgelogics into polymodal logics axiomatised by primitive axioms and thenuse Kracht's results on properly displayable logics to define thedisplay calculi. Apart from these technical results, we argue thatDisplay Logic is a natural framework to define cut-free calculi for manyother logics with relative accessibility relations.

We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme a ϕ → a ϕ. In the intended semantics, the plain..

The main technical errors are in the literature survey. On pages 44, 93-94, 131 and 133 I claim that Fitting's and/or Rautenberg's systems are incomplete because they omit contraction. The claim is wrong because contraction is implicit in their set notation. Their systems are complete because they allow contraction on any formula whereas the systems in this technical report explicitly build contraction into certain rules, allowing contraction only on certain types of formulae. Please accept my apologies for any confusion this (...) causes you. (shrink)

The ‘reflexive turn’ transcended disciplinary boundaries within the social sciences. Feminist scholars in particular have taken up its core concerns, establishing a wide-ranging literature on reflexivity in feminist theory and practice. In this paper, I contribute to this scholarship by deconstructing the ‘story’ of my own research as a white, genderqueer, masculine-presenting researcher in Ghana. This deconstruction is based on thirteen months of field research exploring LGBT activism in the capital city of Accra. Using a series of ethnographic vignettes, I (...) examine questions of queer subjectivity, embodiment and self/other dynamics in the research encounter. Specifically, I interrogate what a reflexive concern for power relations means when researchers share moments of commonality and difference with research participants, here in relation to axes of gender, sexuality, race and class. Finally, I explore the challenge of theorising resistance in light of feminist postcolonial critiques of the politics of representation. I conclude that it is only by locating these tensions and dissonances in the foreground of our inquiries that reflexivity becomes meaningful as a way of rendering knowledge production more accountable and transparent, of practising feminist solidarity, and of excavating our own queer research journeys. (shrink)