Substructural logics are logics obtained from a sequent formulation of intuitionistic or classical logic by rejecting some structural rules. The substructural logics considered here are linear logic, relevant logic and BCK logic. It is proved that first-order variants of these logics with an intuitionistic negation can be embedded by modal translations into S4-type extensions of these logics with a classical, involutive, negation. Related embeddings via translations like the double-negation translation are also considered. Embeddings into analogues of S4 are obtained with (...) the help of cut elimination for sequent formulations of our substructural logics and their modal extensions. These results are proved for systems with equality too. (shrink)

Current books on evolutionary theory all seem to take for granted the fact that students find evolution easy to understand when actually, from a psychological perspective, it is a rather counterintuitive idea. Evolutionary theory, like all scientific theories, is a means to understanding the natural world. Understanding Evolution is intended for undergraduate students in the life sciences, biology teachers or anyone wanting a basic introduction to evolutionary theory. Covering core concepts and the structure of evolutionary explanations, it clarifies both what (...) evolution is about and why so many people find it difficult to grasp. The book provides an introduction to the major concepts and conceptual obstacles to understanding evolution, including the development of Darwin's theory, and a detailed presentation of the most important evolutionary concepts. Bridging the gap between the concepts and conceptual obstacles, Understanding Evolution presents evolutionary theory with a clarity and vision students will quickly appreciate. (shrink)

In Neither Physics Nor Chemistry, Kostas Gavroglu and Ana Simoes examine the evolution of quantum chemistry into an autonomous discipline, tracing its development from the publication of early papers in the 1920s to the dramatic changes ...

This paper presents duality results between categories of neighbourhood frames for modal logic and categories of modal algebras (i.e. Boolean algebras with an additional unary operation). These results extend results of Goldblatt and Thomason about categories of relational frames for modal logic.

Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally (...) the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic. (shrink)

"Originally published in French in 1961, this book is one of the standard works on the question of alienation in Marx. In his study of Marx and the role of technology in the modern world, Kostas Axelos interprets Marx from his own distinctive, thought-provoking, philosophical position. Made available now in the translation by Ronald Bruzina, the book provides a meaningful interpretation of Marx and an introduction to Axelos's own philosophical thought" -- Book jacket.

The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural (...) rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models. (shrink)

The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one (...) finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are. (shrink)

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 1 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right (...) of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. (shrink)

This paper, a sequel to Models for normal intuitionistic modal logics by M. Boi and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of (...) formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated. (shrink)

Scientific knowledge is the most solid and robust kind of knowledge that humans have because of its inherent self-correcting character. Nevertheless, anti-evolutionists, climate denialists, and anti-vaxxers, among others, question some of the best-established scientific findings, making claims unsupported by empirical evidence. A common aspect of these claims is reference to the uncertainties of science concerning evolution, climate change, vaccination, and so on. This is inaccurate: whereas the broad picture is clear, there will always exist uncertainties about the details of the (...) respective phenomena. This book shows that uncertainty is an inherent feature of science that does not devalue it. In contrast, uncertainty advances science because it motivates further research. This is the first book on this topic that draws on philosophy of science to explain what uncertainty in science is and how it makes science advance. It contrasts evolution, climate change, and vaccination, where the uncertainties are exaggerated, and genetic testing and forensic science, where the uncertainties are usually overlooked. The goal is to discuss the scientific, psychological, and philosophical aspects of uncertainty in order to explain what it really is, what kinds of problems it actually poses, and why in the end it makes science advance. Contrary to public representations of scientific findings and conclusions that produce an intuitive but distorted view of science as certain, people need to understand and learn to live with uncertainty in science. This book is intended for anyone who wants to get a clear view of the nature of science. (shrink)

This book presents analyses of philosophical topics of importance to biology education. It is intended foremost for biology educators and teachers, and aims to show how philosophy of science in general, and philosophy of biology in particular, ...

This is an examination, a commentary, of links between some philosophical views ascribed to Gödel and general proof theory. In these views deduction is of central concern not only in predicate logic, but in set theory too, understood from an infinitistic ideal perspective. It is inquired whether this centrality of deduction could also be kept in the intensional logic of concepts whose building Gödel seems to have taken as the main task of logic for the future.

This is a review, with historical and critical comments, of a paper by I. E. Orlov from 1928, which gives the oldest known axiomatization of the implication-negation fragment of the relevant logic R. Orlov's paper also foreshadows the modal translation of systems with an intuitionistic negation into S4-type extensions of systems with a classical, involutive, negation. Orlov introduces the modal postulates of S4 before Becker, Lewis and Gödel. Orlov's work, which seems to be nearly completely ignored, is related to the (...) contemporaneous work on the axiomatization of intuitionistic logic. (shrink)

This paper presents an emerging aspect of intentionality through recent Artificial Intelligence (AI) developments in art and design. Our main thesis is that, if we focus just on the outcome of the artistic process, the intentionality of the artist does not have any relevance. Intention is measured as a result of actions regardless of whether they are human-based or not as long as there is an esthetical value intersubjectively acknowledged. In other words, what matters is the ‘intentio’ embedded in the (...) work of art (the output) rather than the ‘intendere’ (the process). By considering the former aspect, intentionality becomes free of human intervention, therefore 'unintentional'. This thesis is supported through a range of examples related to AI in general and AI art in particular. However, in the case of a self-driving vehicle, the macro-goal of the system is fixed by humans, even though the AI system itself will take care of decomposing the macro-goal into a set of micro-goals and take the corresponding decisions. In the case of art, the system does not need to have any exterior goal to the extent that art is pure freedom of expression, as many philosophers have acknowledged. In this sense, the decisions of any AI system generating art or design are independent of humans and must be judged only on the ground of the final outcome. A work of art will be judged ‘artistic’ to the extent that humans will recognize an artistic intent in the work itself. In particular, the introduction of random processes makes an AI-generated artwork fully independent of human creativity. (shrink)

The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference. This paper examines in the (...) setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category whose arrows are equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of logic, with the associated map into the category of equivalence relations of generality. This category is isomorphically represented in the category whose arrows are binary relations between finite ordinals, where composition is the usual simple composition of relations. This representation is related to a classical representation result of Richard Brauer. (shrink)

This is an exposition of Lambek's strengthening and generalization of the deduction theorem in categories related to intuitionistic propositional logic. Essential notions of category theory are introduced so as to yield a simple reformulation of Lambek's Functional Completeness Theorem, from which its main consequences can be readily drawn. The connections of the theorem with combinatory logic, and with modal and substructural logics, are briefly considered at the end.

What are genes? What do genes do? These seemingly simple questions are in fact challenging to answer accurately. As a result, there are widespread misunderstandings and over-simplistic answers, which lead to common conceptions widely portrayed in the media, such as the existence of a gene 'for' a particular characteristic or disease. In reality, the DNA we inherit interacts continuously with the environment and functions differently as we age. What our parents hand down to us is just the beginning of our (...) life story. This comprehensive book analyses and explains the gene concept, combining philosophical, historical, psychological and educational perspectives with current research in genetics and genomics. It summarises what we currently know and do not know about genes and the potential impact of genetics on all our lives. Making Sense of Genes is an accessible but rigorous introduction to contemporary genetics concepts for non-experts, undergraduate students, teachers and healthcare professionals. (shrink)

Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It (...) is found that with the simplest understanding of coherence this is the best one can hope for in bicartesian categories. The coherence for categories with binary products and sums provides an easy decision procedure for equality of arrows. It is also used to demonstrate that the categories in question are maximal, in the sense that in any such category that is not a preorder all the equations between arrows involving only binary products and sums are the same. This shows that the usual notion of equivalence of proofs in nondistributive conjunctive-disjunctive logic is optimally defined: further assumptions would make this notion collapse into triviality. (shrink)

This is a companion to a paper by the authors entitled “Gödel on deduction”, which examined the links between some philosophical views ascribed to Gödel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Gödel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Gödel’s unpublished notes for the elementary logic course he gave in 1939 (...) at the University of Notre Dame. Here one finds a presentation of this system of Gödel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Gödel’s Notre Dame course, and is followed by comments concerning Gödel’s natural deduction system. (shrink)

Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It (...) is found that with the simplest understanding of coherence this is the best one can hope for in bicartesian categories. The coherence for categories with binary products and sums provides an easy decision procedure for equality of arrows. It is also used to demonstrate that the categories in question are maximal, in the sense that in any such category that is not a preorder all the equations between arrows involving only binary products and sums are the same. This shows that the usual notion of equivalence of proofs in nondistributive conjunctive-disjunctive logic is optimally defined: further assumptions would make this notion collapse into triviality. (shrink)

Biologists rely on theories, apply models and construct explanations, but rarely reflect on their nature and structure. This book introduces key topics in philosophy of science to provide the required philosophical background for this kind of reflection, which is an important part of all aspects of research and communication in biology. It concisely and accessibly addresses fundamental questions such as: Why should biologists care about philosophy of science? How do concepts contribute to scientific advancement? What is the nature of scientific (...) controversies in the biological sciences? Chapters draw on contemporary examples and case studies from across biology, making the discussion relevant and insightful. Written for researchers and advanced undergraduate and graduate students across the life sciences, its aim is to encourage readers to become more philosophically minded and informed to enable better scientific practice. It is also an interesting and pertinent read for philosophers of science. (shrink)

(1984). Some methodological and historical considerations in low temperature physics: The case of superconductivity 1911–57. Annals of Science: Vol. 41, No. 2, pp. 135-149.

Ambient Intelligence provides the potential for vast and varied applications, bringing with it both promise and peril. The development of Ambient Intelligence applications poses a number of ethical and legal concerns. Mobile devices are increasingly evolving into tools to orientate in and interact with the environment, thus introducing a user-centric approach to Ambient Intelligence. The MINAmI (Micro-Nano integrated platform for transverse Ambient Intelligence applications) FP6 research project aims at creating core technologies for mobile device based Ambient Intelligence services. In this (...) paper we assess five scenarios that demonstrate forthcoming MINAmI-based applications focusing on healthcare, assistive technology, homecare, and everyday life in general. A legal and ethical analysis of the scenarios is conducted, which reveals various conflicting interests. The paper concludes with some thoughts on drafting ethical guidelines for Ambient Intelligence applications. (shrink)

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general (...) adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful. (shrink)