We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the linearity rule, which (bottom-up) operates by linearising branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking (...) if data exists along certain paths within a nested sequent. Such rules require us to generalise our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalisation of the usual nested sequent formalism. Our calculi exhibit favourable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems. (shrink)

What determines whether an action is right or wrong? Morality, Rules, and Consequences: A Critical Reader explores for students and researchers the relationship between consequentialist theory and moral rules. Most of the chapters focus on rule consequentialism or on the distinction between act and rule versions of consequentialism. Contributors, among them the leading philosophers in the discipline, suggest ways of assessing whether rule consequentialism could be a satisfactory moral theory. These essays, all of which are previously unpublished, provide students in (...) moral philosophy with essential material and ask key questions on just what the criteria for an adequate moral theory might be. (shrink)

This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as (...) completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics. (shrink)

In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee's honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essentially three-dimensional structure of the bee's honeycomb.

Australia and New Zealand boast an active community of scholars working in the field of history, philosophy and social studies of science. Australasian Studies in History and Philosophy of Science aims to provide a distinctive publication outlet for their work. Each volume comprises a group of thematically-connected essays edited by scholars based in Australia or New Zealand with special expertise in that particular area. In each volume, a majority ofthe contributors are from Australia or New Zealand. Contributions from elsewhere are (...) by no means ruled out, however, and are actively encouraged wherever appropriate to the balance of the volume in question. Earlier volumes in the series have been welcomed for significantly advancing the discussion of the topics they have dealt with. I believe that the present volume will be greeted equally enthusiastically by readers in many parts of the world. R. W. Home General Editor Australasian Studies in History And Philosophy of Science viii ACKNOWLEDGEMENTS The majority of the papers in this collection had their origin in the 2001 Australasian Association for History, Philosophy, and Social Studies of Science annual conference, held at the University of Melbourne, where streams of papers on the themes of scientific realism and commonsense were organised. (shrink)

"The Hardest Logic Puzzle Ever" was first described by the late George Boolos in the Spring 1996 issue of the Harvard Review of Philosophy. Although not dissimilar in appearance from many other simpler puzzles involving gods (or tribesmen) who always tell the truth or always lie, this puzzle has several features that make the solution far from trivial. This paper examines the puzzle and describes a simpler solution than that originally proposed by Boolos.

To many observers, the moratorium on commercial whaling, which came into force under the aegis of the International Whaling Commission (IWC) in 1986, is both a moral and an environmental victory. Moreover, many governments have found it to be an advantageous, easy and costless policy to support. However, a critical analysis of the diverse viewpoints of IWC member states, especially those expressed by the delegations of the United Kingdom, Norway and Japan at the 1996 Annual Meeting of the IWC in (...) Aberdeen, raises doubts about the moral and practical coherence of the arguments supporting the continuation of the moratorium. In this paper, these doubts will be rehearsed, and an alternative policy to that of the moratorium—sustainable resource management—is put forward as founded on a more coherent moral and practicable basis. (shrink)

To many observers, the moratorium on commercial whaling, which came into force under the aegis of the International Whaling Commission in 1986, is both a moral and an environmental victory. Moreover, many governments have found it to be an advantageous, easy and costless policy to support. However, a critical analysis of the diverse viewpoints of IWC member states, especially those expressed by the delegations of the United Kingdom, Norway and Japan at the 1996 Annual Meeting of the IWC in Aberdeen, (...) raises doubts about the moral and practical coherence of the arguments supporting the continuation of the moratorium. In this paper, these doubts will be rehearsed, and an alternative policy to that of the moratorium—sustainable resource management—is put forward as founded on a more coherent moral and practicable basis. (shrink)

The claim that indigenous communities are entitled to have intellectual property rights (IPRs) to both their plant varieties and their botanical knowledge has been put forward by writers who wish to protect the plant genetic resources of indigenous communities from uncompensated use by biotechnological transnational corporations. We argue that while it is necessary for indigenous communities to have suchrights, the entitlement argument is an unsatisfactory justification for them. A more convincing foundation for indigenous community IPRs is the autonomy theory developed (...) by Will Kymlicka. (shrink)

This article presents results from a multidisciplinary research project on the integration and transfer of language knowledge into robots as an empirical paradigm for the study of language development in both humans and humanoid robots. Within the framework of human linguistic and cognitive development, we focus on how three central types of learning interact and co-develop: individual learning about one's own embodiment and the environment, social learning (learning from others), and learning of linguistic capability. Our primary concern is how these (...) capabilities can scaffold each other's development in a continuous feedback cycle as their interactions yield increasingly sophisticated competencies in the agent's capacity to interact with others and manipulate its world. Experimental results are summarized in relation to milestones in human linguistic and cognitive development and show that the mutual scaffolding of social learning, individual learning, and linguistic capabilities creates the context, conditions, and requisites for learning in each domain. Challenges and insights identified as a result of this research program are discussed with regard to possible and actual contributions to cognitive science and language ontogeny. In conclusion, directions for future work are suggested that continue to develop this approach toward an integrated framework for understanding these mutually scaffolding processes as a basis for language development in humans and robots. (shrink)

In this work we answer a long standing request for temporal embeddings of deontic STIT logics by introducing the multi-agent STIT logic TDS . The logic is based upon atemporal utilitarian STIT logic. Yet, the logic presented here will be neutral: instead of committing ourselves to utilitarian theories, we prove the logic TDS sound and complete with respect to relational frames not employing any utilitarian function. We demonstrate how these neutral frames can be transformed into utilitarian temporal frames, while preserving (...) validity. Last, we discuss problems that arise from employing binary utility functions in a temporal setting. (shrink)

We formally introduce a novel, yet ubiquitous, category of norms: norms of instrumentality. Norms of this category describe which actions are obligatory, or prohibited, as instruments for certain purposes. We propose the Logic of Agency and Norms (LAN) that enables reasoning about actions, instrumentality, and normative principles in a multi-agent setting. Leveraging LAN , we formalize norms of instrumentality and compare them to two prevalent norm categories: norms to be and norms to do. Last, we pose principles relating the three (...) categories and evaluate their validity vis-à-vis notions of deliberative acting. On a technical note, the logic will be shown decidable via the finite model property. (shrink)

We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that is translatable to (...) a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics. (shrink)

This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic paradigms: labelled and nested sequent calculi. The formalism of labelled sequents has been successful in that cut-free calculi in possession of desirable proof-theoretic properties can be automatically generated for large classes of logics. Despite these qualities, labelled systems make use of a complicated syntax that explicitly incorporates the (...) semantics of the associated logic, and such systems typically violate the subformula property to a high degree. By contrast, nested sequent calculi employ a simpler syntax and adhere to a strict reading of the subformula property, making such systems useful in the design of automated reasoning algorithms. However, the downside of the nested sequent paradigm is that a general theory concerning the automated construction of such calculi (as in the labelled setting) is essentially absent, meaning that the construction of nested systems and the confirmation of their properties is usually done on a case-by-case basis. The refinement method connects both paradigms in a fruitful way, by transforming labelled systems into nested (or, refined labelled) systems with the properties of the former preserved throughout the transformation process. To demonstrate the method of refinement and some of its applications, we consider grammar logics, first-order intuitionistic logics, and deontic STIT logics. The introduced refined labelled calculi will be used to provide the first proof-search algorithms for deontic STIT logics. Furthermore, we employ our refined labelled calculi for grammar logics to show that every logic in the class possesses the effective Lyndon interpolation property. (shrink)

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)

This work provides proof-search algorithms and automated counter-model extraction for a class of STIT logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for STIT logics. A new class of cut-free complete labelled sequent calculi G3LdmL^m_n, for multi-agent STIT with at most n-many choices, is introduced. We refine the calculi G3LdmL^m_n through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in auxiliary calculi Ldm^m_nL. In the single-agent case, we (...) show that the refined calculi Ldm^m_nL derive theorems within a restricted class of (forestlike) sequents, allowing us to provide proof-search algorithms that decide single-agent STIT logics. We prove that the proof-search algorithms are correct and terminate. (shrink)

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated (...) labelled calculus. (shrink)

We employ a recently developed methodology -- called "structural refinement" -- to extract nested sequent systems for a sizable class of intuitionistic modal logics from their respective labelled sequent systems. This method can be seen as a means by which labelled sequent systems can be transformed into nested sequent systems through the introduction of propagation rules and the elimination of structural rules, followed by a notational translation. The nested systems we obtain incorporate propagation rules that are parameterized with formal grammars, (...) and which encode certain frame conditions expressible as first-order Horn formulae that correspond to a subclass of the Scott-Lemmon axioms. We show that our nested systems are sound, cut-free complete, and admit hp-admissibility of typical structural rules. (shrink)

Standpoint logic is a recently proposed formalism in the context of knowledge integration, which advocates a multi-perspective approach permitting reasoning with a selection of diverse and possibly conflicting standpoints rather than forcing their unification. In this paper, we introduce nested sequent calculi for propositional standpoint logics---proof systems that manipulate trees whose nodes are multisets of formulae---and show how to automate standpoint reasoning by means of non-deterministic proof-search algorithms. To obtain worst-case complexity-optimal proof-search, we introduce a novel technique in the context (...) of nested sequents, referred to as "coloring," which consists of taking a formula as input, guessing a certain coloring of its subformulae, and then running proof-search in a nested sequent calculus on the colored input. Our technique lets us decide the validity of standpoint formulae in CoNP since proof-search only produces a partial proof relative to each permitted coloring of the input. We show how all partial proofs can be fused together to construct a complete proof when the input is valid, and how certain partial proofs can be transformed into a counter-model when the input is invalid. These "certificates" (i.e. proofs and counter-models) serve as explanations of the (in)validity of the input. (shrink)

We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call "intuitionistic path axioms". We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.

We introduce a framework that allows for the construction of sequent systems for expressive description logics extending ALC. Our framework not only covers a wide array of common description logics, but also allows for sequent systems to be obtained for extensions of description logics with special formulae that we call "role relational axioms." All sequent systems are sound, complete, and possess favorable properties such as height-preserving admissibility of common structural rules and height-preserving invertibility of rules.

We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm , Tstit and Xstit. All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi G3Ldm and G3Tstit are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also Xstit can be characterized through relational frames, omitting the use of BT+AC frames.

The aim of this paper is to introduce and explain display calculi for a variety of logics. We provide a survey of key results concerning such calculi, though we focus mainly on the global cut elimination theorem. Propositional, first-order, and modal display calculi are considered and their properties detailed.

Whistleblowers have usually been treated as outcasts by private-sector employers. But legal, ethical, and practical considerations increasingly compel companies to encourage employees to disclose suspected illegal and/or unethical activities throughinternal communication channels. Internal disclosure policies/procedures (IDPP''s) have been recommended as one way to encourage such communication.This study examined the relationship between IDPP''s and employee whistleblowing among private-sector employers. Almost 300 human resources executives provided data concerning their organizations'' experiences.

We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt.

This is a comment to Graham Harman’s 2019 response to an article by Þóra Pétursdóttir and Bjørnar Olsen (2018) in which they propose that a materially grounded, archaeological perspective might complement Harman’s historical approach in Immaterialism (2016). Harman responds that his book is indeed already more archaeological than historical, stipulating that history is the study of media with a high density of information, whereas archaeology studies media with a low density of information. History, Harman holds, ends up in too much (...) detail, while archaeology has the advantage of lending itself to the imagination. Hence, his reading of history had the aim of tempering the historical information overload, in effect making the book a work of archaeology. In this comment, I want to do three things: (1) critique the idea that archaeological and historical media are inherently different with regard to their densities of information, (2) discuss how archaeology and history approach their media, and (3) reflect on conceptualisations of “archaeology” outside the discipline itself. (shrink)

The purpose of this paper is to review and discuss Luciano Floridi’s 2019 book The Logic of Information: A Theory of Philosophy as Conceptual Design, the latest instalment in his philosophy of information (PI) tetralogy, particularly with respect to its implications for library and information studies (LIS) .

Formalised knowledge systems, including universities and research institutes, are important for contemporary societies. They are, however, also arguably failing humanity when their impact is measured against the level of progress being made in stimulating the societal changes needed to address challenges like climate change. In this research we used a novel futures-oriented and participatory approach that asked what future envisioned knowledge systems might need to look like and how we might get there. Findings suggest that envisioned future systems will need (...) to be much more collaborative, open, diverse, egalitarian, and able to work with values and systemic issues. They will also need to go beyond producing knowledge about our world to generating wisdom about how to act within it. To get to envisioned systems we will need to rapidly scale methodological innovations, connect innovators, and creatively accelerate learning about working with intractable challenges. We will also need to create new funding schemes, a global knowledge commons, and challenge deeply held assumptions. To genuinely be a creative force in supporting longevity of human and non-human life on our planet, the shift in knowledge systems will probably need to be at the scale of the enlightenment and speed of the scientific and technological revolution accompanying the second World War. This will require bold and strategic action from governments, scientists, civic society and sustained transformational intent. (shrink)

Since its conception nearly 20 years ago, Logic Programming - the idea of using logic as a programming language - has been developed to the point where it now plays an important role in areas such as database theory, artificial intelligence and software engineering. However, there are still many challenging research issues to be addressed and the UK branch of the Association for Logic Programming was set up to provide a forum where the flourishing research community could discuss important issues (...) of Logic Programming which were often by-passed at the large international conferences. This volume contains the twelve papers which were presented at the ALPUK's 3rd conference which was held in Edinburgh, 10-12 April 1991. The aim of the conference was to give a broad but detailed technical insight into the work currently being done in this field, both in the UK and by researchers as far afield as Canada and Bulgaria. The breadth of interest in this area of Computer Science is reflected in the range of the papers which cover - amongst other areas - massively parallel implementation, constraint logic programming, circuit modelling, algebraic proof of program properties, deductive databases, specialised editors and standardisation. The resulting volume gives a good overview of the current progress being made in the field and will be of interest to researchers and students of any aspects of logic programming, parallel computing or database techniques and management. (shrink)

This article endeavors to identify the strongest versions of the two primary arguments against epistemic scientific realism: the historical argument—generally dubbed “the pessimistic meta-induction”—and the argument from underdetermination. It is shown that, contrary to the literature, both can be understood as historically informed but logically validmodus tollensarguments. After specifying the question relevant to underdetermination and showing why empirical equivalence is unnecessary, two types of competitors to contemporary scientific theories are identified, both of which are informed by science itself. With the (...) content and structure of the two nonrealist arguments clarified, novel relations between them are uncovered, revealing the severity of their collective threat against epistemic realism and its “no-miracles” argument. The final section proposes, however, that the realist’s axiological tenet “science seeks truth” is not blocked. An attempt is made to indicate the promise for a nonepistemic, purely axiological scientific realism—here dubbed “Socratic scientific realism.”. (shrink)

Linguistic Semantics: An Introduction is the successor to Sir John Lyons's important textbook Language, Meaning and Context (1981).While preserving the general structure of the earlier book, the author has substantially expanded its scope to introduce several topics that were not previously discussed, and to take account of new developments in linguistic semantics over the past decade. The resulting work is an invaluable guide to the subject, offering clarifications of its specialised terms and explaining its relationship to formal and philosophical semantics (...) and to contemporary pragmatics. With its clear and accessible style it will appeal to a wide student readership. Sir John Lyons is one of the most important and internationally renowned contributors to the study of linguistics. His many publications include Introduction to Theoretical Linguistics (1968) and Semantics (1977). (shrink)

STIT logic is a prominent framework for the analysis of multi-agent choice-making. In the available deontic extensions of STIT, the principle of Ought-implies-Can (OiC) fulfills a central role. However, in the philosophical literature a variety of alternative OiC interpretations have been proposed and discussed. This paper provides a modular framework for deontic STIT that accounts for a multitude of OiC readings. In particular, we discuss, compare, and formalize ten such readings. We provide sound and complete sequent-style calculi for all of (...) the various STIT logics accommodating these OiC principles. We formally analyze the resulting logics and discuss how the different OiC principles are logically related. In particular, we propose an endorsement principle describing which OiC readings logically commit one to other OiC readings. (shrink)

Melanie presented at twenty weeks of gestation to an obstetrics clinic in a critical access hospital in rural Vermont. She was excited to undergo routine fetal ultrasonography, but her obstetrician gave her grave news: the ultrasound revealed hypoplastic left heart syndrome, a devastating congenital heart defect. Initially, Melanie agreed in general to pursue surgical care for her fetus—a three‐stage process that has somewhat uncertain results and could only be done in tertiary care facilities far from her home in Vermont. A (...) week later, while the maternal fetal medicine and pediatric cardiology units made arrangements with colleagues in Boston, Melanie began having second thoughts. An ethics meeting was called to discuss conflicting clinician reactions to Melanie's dilemma. Most of the clinicians were stunned that the patient would change her mind. What advice should the ethics consultant offer the team about caring for Melanie? (shrink)

Some have claimed that people with very different beliefs literally see the world differently. Thus Thomas Kuhn: ‘what a man sees depends both upon what he looks at and also upon what his previous visual—conceptual experience has taught him to see’ (Kuhn 1970, p. ll3). This view — call it ‘Perceptual Relativism’ — entails that a scientist and a child may look at a cathode ray tube and, in a sense, the first will see it while the second won’t. The (...) claim is not, of course, that the child’s experience is ‘empty’; but that, unlike the scientist, it does not see the tube as a cathode ray tube. One way of supporting this claim is to say that one cannot see something as an F unless one has the concept F. Since the child plainly lacks the concept of a cathode ray tube, it cannot see it as a cathode ray tube. Although Perceptual Relativism is hard to believe, this supporting suggestion is not so implausible. After all, when we see (and more generally, perceive) the world, the world is presented to us in a particular way; so how can we see it as being that way unless we have some idea or conception of the way it is presented? We need not be committed to a representative theory of perception to think that perceptions in some sense represent the world. We can express this by saying that perceptions have content. Now it is a commonplace that the contents of beliefs and the other propositional attitudes involve concepts. The belief that this thing is a cathode ray tube involves, in some sense, the concept cathode ray tube. So the line of thought behind Perceptual Relativism may be expressed thus: seeing an F as an F is a state with content. (shrink)

This volume collects David Lyons' well-known essays on Mill's moral theory and includes an introduction which relates the essays to prior and subsequent philosophical developments. Like the author's Forms and Limits of Utilitarianism (Oxford, 1965), the essays apply analytical methods to issues in normative ethics. The first essay defends a refined version of the beneficiary theory of rights against H.L.A. Hart's important criticisms. The central set of essays develops new interpretations of Mill's moral theory with the aim of determining how (...) far rights can be incorporated in a utilitarian framework. They Mill's analysis of moral concepts promises to accommodate the argumentative force of rights, and also provide a significant new reading of Mill's theory of liberty. The last essay argues that the promise of Mill's theory of justice cannot be fulfilled. Utilitarianism is unable to account for crucial features of moral rights, or even for the moral force of legal rights whose existence might be justified on utilitarian grounds. (shrink)

Some machine learning models, in particular deep neural networks, are not very well understood; nevertheless, they are frequently used in science. Does this lack of understanding pose a problem for using DNNs to understand empirical phenomena? Emily Sullivan has recently argued that understanding with DNNs is not limited by our lack of understanding of DNNs themselves. In the present paper, we will argue, contra Sullivan, that our current lack of understanding of DNNs does limit our ability to understand with DNNs. (...) Sullivan’s claim hinges on which notion of understanding is at play. If we employ a weak notion of understanding, then her claim is tenable, but rather weak. If, however, we employ a strong notion of understanding, particularly explanatory understanding, then her claim is not tenable. (shrink)

This paper argues that a notion of statistical explanation, based on Salmon’s statistical relevance model, can help us better understand deep neural networks. It is proved that homogeneous partitions, the core notion of Salmon’s model, are equivalent to minimal sufficient statistics, an important notion from statistical inference. This establishes a link to deep neural networks via the so-called Information Bottleneck method, an information-theoretic framework, according to which deep neural networks implicitly solve an optimization problem that generalizes minimal sufficient statistics. The (...) resulting notion of statistical explanation is general, mathematical, and subcausal. (shrink)

The present paper provides an analysis of Euler’s solutions to the Königsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present (...) case. I also propose computational complexity and logical strength as measures of relevant information. (shrink)