Field-Programmable Gate Arrays (FPGAs) have emerged as an attractive means of implementing logic circuits, providing instant manufacturing turnaround and negligible prototype costs. They hold the promise of replacing much of the VLSI market now held by mask-programmed gate arrays. FPGAs offer an affordable solution for customized VLSI, over a wide variety of applications, and have also opened up new possibilities in designing reconfigurable digital systems. Field-Programmable Gate Arrays discusses the most important aspects of FPGAs in a textbook (...) manner. It provides the reader with a focused view of the key issues, using a consistent notation and style of presentation. It provides detailed descriptions of commercially available FPGAs and an in-depth treatment of the FPGA architecture and CAD issues that are the subjects of current research. The material presented is of interest to a variety of readers, including those who are not familiar with FPGA technology, but wish to be introduced to it, as well as those who already have an understanding of FPGAs, but who are interested in learning about the research directions that are of current interest. (shrink)

Rice Combinational and Sequential Logic This text provides a guide to combinational and sequential logic, introducing students to the basic concepts then progressing to HND/first year undergraduate level. It has a highly practical emphasis, enabling the reader to simulate, build and test the circuits discussed in the text. Electronics Workbench files are provided for simulation, while easy-to-use XPLA software and specially designed hardware encourage the reader to build and test the circuits using programmable logic. The text (...) itself contains many self-assessment questions, complete with detailed answers, allowing students to test themselves as they learn. No previous experience of the topic is required and an abundance of practical examples helps students to apply the basic principles. The accompanying CD-ROM provides the XPLA development system software and Electronics Workbench files, as well as the Xilinx Webpack development system and 'walkthrough'. Hardware for Programmable Logic experiments can either be purchased or built following instructions from the text. Altogether, the text, software and hardware provide an interactive and stimulating teaching and learning resource. F. (shrink)

ABSTRACT I argue that this examination and appreciation for the shift to abductive reasoning should be extended to the intersection of neuroscience and novel brain-computer interfaces too. This paper highlights the implications of applying abductive reasoning to personalized implantable neurotechnologies. Then, it explores whether abductive reasoning is sufficient to justify insurance coverage for devices absent widespread clinical trials, which are better applied to one-size-fits-all treatments. INTRODUCTION In contrast to the classic model of randomized-control trials, often with a large number of (...) subjects enrolled, precision medicine attempts to optimize therapeutic outcomes by focusing on the individual.[i] A recent publication highlights the strengths and weakness of both traditional evidence-based medicine and precision medicine.[ii] Plus, it outlines a tension in the shift from evidence-based medicine’s inductive reasoning style (the collection of data to postulate general theories) to precision medicine’s abductive reasoning style (the generation of an idea from the limited data available).[iii] The paper’s main example is the application of precision medicine for the treatment of cancer.[iv] I argue that this examination and appreciation for the shift to abductive reasoning should be extended to the intersection of neuroscience and novel brain-computer interfaces too. As the name suggests, brain-computer interfaces are a significant advancement in neurotechnology that directly connects someone’s brain to external or implanted devices.[v] Among the various kinds of brain-computer interfaces, adaptive deep brain stimulation devices require numerous personalized adjustments to their settings during the implantation and computation stages in order to provide adequate relief to patients with treatment-resistant disorders. What makes these devices unique is how adaptive deep brain stimulation integrates a sensory component to initiate the stimulation. While not commonly at the level of sophistication as self-supervising or generative large language models,[vi] they currently allow for a semi-autonomous form of neuromodulation. This paper highlights the implications of applying abductive reasoning to personalized implantable neurotechnologies. Then, it explores whether abductive reasoning is sufficient to justify insurance coverage for devices absent widespread clinical trials, which are better applied to one-size-fits-all treatments.[vii] ANALYSIS I. The State of Precision Medicine in Oncology and the Epistemological Shift While a thorough overview of precision medicine for the treatment of cancer is beyond the scope of this article, its practice can be roughly summarized as identifying clinically significant characteristics a patient possesses (e.g., genetic traits) to land on a specialized treatment option that, theoretically, should benefit the patient the most.[viii] However, in such a practice of stratification patients fall into smaller and smaller populations and the quality of evidence that can be applied to anyone outside these decreases in turn.[ix] As inductive logic helps to articulate, the greater the number of patients that respond to a particular therapy the higher the probability of its efficacy. By straying from this logical framework, precision medicine opens the treatment of cancer to more uncertainty about the validity of these approaches to the resulting disease subcategories.[x] Thus, while contemporary medical practices explicitly describe some treatments as “personalized”, they ought not be viewed as inherently better founded than other therapies.[xi] A relevant contemporary case of precision medicine out of Norway focuses on the care of a patient with cancer between the ventricles of the heart and esophagus, which had failed to respond to the standard regimen of therapies over four years.[xii] In a last-ditch effort, the patient elected to pay out-of-pocket for an experimental immunotherapy (nivolumab) at a private hospital. He experienced marked improvements and a reduction in the size of the tumor. Understandably, the patient tried to pursue further rounds of nivolumab at a public hospital. However, the hospital initially declined to pay for it given the “lack of evidence from randomised clinical trials for this drug relating to this [patient’s] condition.”[xiii] In rebuttal to this claim, the patient countered that he was actually similar to a subpopulation of patients who responded in “open‐label, single arm, phase 2 studies on another immune therapy drug” (pembrolizumab).[xiv] Given this interpretation of the prior studies and the patient’s response, further rounds of nivolumab were approved. Had the patient not had improvements in the tumor’s size following a round of nivolumab, then pembrolizumab’s prior empirical evidence in isolation would have been insufficient, inductively speaking, to justify his continued use of nivolumab.[xv] The case demonstrates a shift in reasoning from the traditional induction to abduction. The phenomenon of ‘cancer improvement’ is considered causally linked to nivolumab and its underlying physiological mechanisms.[xvi] However, “the weakness of abductions is that there may always be some other better, unknown explanation for an effect. The patient may for example belong to a special subgroup that spontaneously improves, or the change may be a placebo effect. This does not mean, however, that abductive inferences cannot be strong or reasonable, in the sense that they can make a conclusion probable.”[xvii] To demonstrate the limitations of relying on the abductive standard in isolation, commentators have pointed out that side effects in precision medicine are hard to rule out as being related to the initial intervention itself unless trends from a group of patients are taken into consideration.[xviii] As artificial intelligence (AI) assists the development of precision medicine for oncology, this uncertainty ought to be taken into consideration. The implementation of AI has been crucial to the development of precision medicine by providing a way to combine large patient datasets or a single patient with a large number of unique variables with machine learning to recommend matches based on statistics and probability of success upon which practitioners can base medical recommendations.[xix] The AI is usually not establishing a causal relationship[xx] – it is predicting. So, as AI bleeds into medical devices, like brain-computer interfaces, the same cautions about using abductive reasoning alone should be carried over. II. Responsive Neurostimulation, AI, and Personalized Medicine Like precision medicine in cancer treatment, computer-brain interface technology similarly focuses on the individual patient through personalized settings. In order to properly expose the intersection of AI, precision medicine, abductive reasoning, and implantable neurotechnologies, the descriptions of adaptive deep brain stimulation systems need to deepen.[xxi] As a broad summary of adaptive deep brain stimulation, to provide a patient with the therapeutic stimulation, a neural signal, typically referred to as a local field potential,[xxii] must first be detected and then interpreted by the device. The main adaptive deep brain stimulation device with premarket approval, the NeuroPace Responsive Neurostimulation system, is used to treat epilepsy by detecting and storing “programmer-defined phenomena.”[xxiii] Providers can optimize the detection settings of the device to align with the patient’s unique electrographic seizures as well as personalize the reacting stimulation’s parameters.[xxiv] The provider adjusts the technology based on trial and error. One day machine learning algorithms will be able to regularly aid this process in myriad ways, such as by identifying the specific stimulation settings a patient may respond to ahead of time based on their electrophysiological signatures.[xxv] Either way, with AI or programmers, adaptive neurostimulation technologies are individualized and therefore operate in line with precision medicine rather than standard treatments based on large clinical trials. Contemporary neurostimulation devices are not usually sophisticated enough to be prominent in AI discussions where the topics of neural networks, deep learning, generative models, and self-attention dominate the conversation. However, implantable high-density electrocorticography arrays (a much more sensitive version than adaptive deep brain stimulation systems use) have been used in combination with neural networks to help patients with neurologic deficits from a prior stroke “speak” through a virtual avatar.[xxvi] In some experimental situations, algorithms are optimizing stimulation parameters with increasing levels of independence.[xxvii] An example of neurostimulation that is analogous to the use of nivolumab in Norway surrounds a patient in the United States who was experiencing both treatment-resistant OCD and temporal lobe epilepsy.[xxviii]Given the refractory nature of her epilepsy, implantation of an adaptive deep brain stimulation system was indicated. As a form of experimental therapy, her treatment-resistant OCD was also indicated for the off-label use of an adaptive deep brain stimulation set-up. Another deep brain stimulation lead, other than the one implanted for epilepsy, was placed in the patient’s right nucleus accumbens and ventral pallidum region given the correlation these nuclei had with OCD symptoms in prior research. Following this, the patient underwent “1) ambulatory, patient-initiated magnet-swipe storage of data during moments of obsessive thoughts; (2) lab-based, naturalistic provocation of OCD-related distress (naturalistic provocation task); and (3) lab-based, VR [virtual reality] provocation of OCD-related distress (VR provocation task).”[xxix] Such signals were used to identify when to deliver the therapeutic stimulation in order to counter the OCD symptoms. Thankfully, following the procedure and calibration the patient exhibited marked improvements in their OCD symptoms and recently shared her results publicly.[xxx] In both cases, there is a similar level of abductive justification for the efficacy of the delivered therapy. In the case study in which the patient was treated with adaptive deep brain stimulation, they at least had their neural activity tested in various settings to determine the optimum parameters for treatment to avoid them being based on guesswork. Additionally, the adaptive deep brain stimulation lead was already placed before the calibration trials were conducted, meaning that the patient had already taken on the bulk of the procedural risk before the efficacy could be determined. Such an efficacy test could have been replicated in the first patient’s cancer treatment, had it been biopsied and tested against the remaining immunotherapies in vitro. Yet, in the case of cancer with few options, one previous dose of a drug that appeared to work on the patient may justify further doses. However, as the Norwegian case presents, corroboration with known responses to a similar drug (from a clinical trial) could be helpful to validate the treatment strategy. (It should be noted that both patients were resigned to these last resort options regardless of the efficacy of treatment.) There are some elements of inductive logic seen with adaptive deep brain stimulation research in general. For example, abductively the focus could be that patient X’s stimulation parameters are different from patient Y’s and patient Z’s. In contrast, when grouped as subjects who obtained personalized stimulation, patients X, Y, and Z demonstrate an inductive aspect to this approach’s safety and/or efficacy. The OCD case holds plenty of abductive characteristics in line with precision medicine’s approach to treating cancer and as more individuals try the method, there will be additional data. With the gradual integration of AI into brain-computer interfaces in the name of efficacy, this reliance on abduction will continue, if not grow, over time. Moving forward, if a responsive deep brain stimulation treatment is novel and individualized (like the dose of nivolumab) and there is some other suggestion of efficacy (like clinical similarities to other patients in the literature), then it may justify insurance coverage for the investigative intervention, absent other unrelated reasons to deny it. III. Ethical Implications and Next Steps While AI’s use in oncology and neurology is not yet as prominent as its use in other fields (e.g., radiology), it appears to be on the horizon for both.[xxxi] AI can be found in both the functioning of the neurotechnologies as well as the implementation of precision medicine. The increasing use of AI may serve to further individualize both oncologic and neurological therapies. Given these implications and the handful of publications cited in this article, it is important to have a nuanced evaluation of how these treatments, which heavily rely on abductive justification, ought to be managed. The just use an abductive approach may be difficult as AI infused precision medicine is further pursued. At baseline, such technology relies on a level of advanced technology literacy among the general public and could exclude populations who lack access to basic technological infrastructure or know-how from participation.[xxxii] Even among nations with adequate infrastructure, as more patients seek out implantable neurotechnologies, which require robust healthcare resources, the market will favor patient populations that can afford this complex care.[xxxiii] If patients already have the means to pay for an initial dose/use of a precision medicine product out of pocket, should insurance providers be required to cover subsequent treatments?[xxxiv] That is, if a first dose of a cancer drug or a deep brain stimulator over its initial battery life is successful, patients may feel justified in having the costs of further treatments covered. The Norwegian patient’s experience implies there is a precedent for the idea that some public insurance companies ought to cover successful cancer therapies, however, insurance companies may not all see themselves as obligated to cover neurotechnologies that rely on personalized settings or that are based on precision/abductive research more than on clinical trials. CONCLUSION The fact that the cases outlined above rely on abductive style of reasoning implies that there may not be as strong a justification for coverage by insurance, as they are both experimental and individualized, when compared to the more traditional large clinical trials in which groups have the same or a standardized protocol (settings/doses). If a study is examining the efficacy of a treatment with a large cohort of patients or with different experimental groups/phases, insurance companies may conclude that the resulting symptom improvements are more likely to be coming from the devices themselves. A preference for inductive justification may take priority when ruling in favor of funding someone’s continued use of an implantable neurostimulator. There are further nuances to this discussion surrounding the classifications of these interventions as research versus clinical care that warrant future exploration, since such a distinction is more of a scale[xxxv] than binary and could have significant impacts on the “right-to-try” approach to experimental therapies in the United States.[xxxvi] Namely, given the inherent limitations of conducting large cohort trials for deep brain stimulation interventions on patients with neuropsychiatric disorders, surgically innovative frameworks that blend abductive and inductive methodologies, like with sham stimulation phases, have traditionally been used.[xxxvii] Similarly, for adaptive brain-computer interface systems, if there are no large clinical trials and instead only publications that demonstrate that something similar worked for someone else, then, in addition to the evidence that the first treatment/dose worked for the patient in question, the balance of reasoning would be valid and arguably justify insurance coverage. As precision approaches to neurotechnology become more common, frameworks for evaluating efficacy will be crucial both for insurance coverage and for clinical decision making. ACKNOWLEDGEMENT This article was originally written as an assignment for Dr. Francis Shen’s “Bioethics & AI” course at Harvard’s Center for Bioethics. I would like to thank Dr. Shen for his comments as well as my colleagues in the Lázaro-Muñoz Lab for their feedback. - [i] Jonathan Kimmelman and Ian Tannock, “The Paradox of Precision Medicine,” Nature Reviews. Clinical Oncology 15, no. 6 (June 2018): 341–42, https://doi.org/10.1038/s41571-018-0016-0. [ii] Henrik Vogt and Bjørn Hofmann, “How Precision Medicine Changes Medical Epistemology: A Formative Case from Norway,” Journal of Evaluation in Clinical Practice 28, no. 6 (December 2022): 1205–12, https://doi.org/10.1111/jep.13649. [iii] David Barrett and Ahtisham Younas, “Induction, Deduction and Abduction,” Evidence-Based Nursing 27, no. 1 (January 1, 2024): 6–7, https://doi.org/10.1136/ebnurs-2023-103873. [iv] Vogt and Hofmann, “How Precision Medicine Changes Medical Epistemology,” 1208. [v] Wireko Andrew Awuah et al., “Bridging Minds and Machines: The Recent Advances of Brain-Computer Interfaces in Neurological and Neurosurgical Applications,” World Neurosurgery, May 22, 2024, S1878-8750(24)00867-2, https://doi.org/10.1016/j.wneu.2024.05.104. [vi] Mark Riedl, “A Very Gentle Introduction to Large Language Models without the Hype,” Medium (blog), May 25, 2023, https://mark-riedl.medium.com/a-very-gentle-introduction-to-large-language-models-without-the-hype-5 f67941fa59e. [vii] David E. Burdette and Barbara E. Swartz, “Chapter 4 - Responsive Neurostimulation,” in Neurostimulation for Epilepsy, ed. Vikram R. Rao (Academic Press, 2023), 97–132, https://doi.org/10.1016/B978-0-323-91702-5.00002-5. [viii] Kimmelman and Tannock, 2018. [ix] Kimmelman and Tannock, 2018. [x] Simon Lohse, “Mapping Uncertainty in Precision Medicine: A Systematic Scoping Review,” Journal of Evaluation in Clinical Practice 29, no. 3 (April 2023): 554–64, https://doi.org/10.1111/jep.13789. [xi] Kimmelman and Tannock, “The Paradox of Precision Medicine.” [xii] Vogt and Hofmann, 1206. [xiii] Vogt and Hofmann, 1206. [xiv] Vogt and Hofmann, 1206. [xv] Vogt and Hofmann, 1207. [xvi] Vogt and Hofmann, 1207. [xvii] Vogt and Hofmann, 1207. [xviii] Vogt and Hofmann, 1210. [xix] Mehar Sahu et al., “Chapter Three - Artificial Intelligence and Machine Learning in Precision Medicine: A Paradigm Shift in Big Data Analysis,” in Progress in Molecular Biology and Translational Science, ed. David B. Teplow, vol. 190, 1 vols., Precision Medicine (Academic Press, 2022), 57–100, https://doi.org/10.1016/bs.pmbts.2022.03.002. [xx] Stefan Feuerriegel et al., “Causal Machine Learning for Predicting Treatment Outcomes,” Nature Medicine 30, no. 4 (April 2024): 958–68, https://doi.org/10.1038/s41591-024-02902-1. [xxi] Sunderland Baker et al., “Ethical Considerations in Closed Loop Deep Brain Stimulation,” Deep Brain Stimulation 3 (October 1, 2023): 8–15, https://doi.org/10.1016/j.jdbs.2023.11.001. [xxii] David Haslacher et al., “AI for Brain-Computer Interfaces,” 2024, 7, https://doi.org/10.1016/bs.dnb.2024.02.003. [xxiii] Burdette and Swartz, “Chapter 4 - Responsive Neurostimulation,” 103–4; “Premarket Approval (PMA),” https://www.accessdata.fda.gov/scripts/cdrh/cfdocs/cfpma/pma.cfm?id=P100026. [xxiv] Burdette and Swartz, “Chapter 4 - Responsive Neurostimulation,” 104. [xxv] Burdette and Swartz, 126. [xxvi] Sean L. Metzger et al., “A High-Performance Neuroprosthesis for Speech Decoding and Avatar Control,” Nature 620, no. 7976 (August 2023): 1037–46, https://doi.org/10.1038/s41586-023-06443-4. [xxvii] Hao Fang and Yuxiao Yang, “Predictive Neuromodulation of Cingulo-Frontal Neural Dynamics in Major Depressive Disorder Using a Brain-Computer Interface System: A Simulation Study,” Frontiers in Computational Neuroscience 17 (March 6, 2023), https://doi.org/10.3389/fncom.2023.1119685; Mahsa Malekmohammadi et al., “Kinematic Adaptive Deep Brain Stimulation for Resting Tremor in Parkinson’s Disease,” Movement Disorders 31, no. 3 (2016): 426–28, https://doi.org/10.1002/mds.26482. [xxviii] Young-Hoon Nho et al., “Responsive Deep Brain Stimulation Guided by Ventral Striatal Electrophysiology of Obsession Durably Ameliorates Compulsion,” Neuron 0, no. 0 (October 20, 2023), https://doi.org/10.1016/j.neuron.2023.09.034. [xxix] Nho et al. [xxx] Nho et al.; Erik Robinson, “Brain Implant at OHSU Successfully Controls Both Seizures and OCD,” OHSU News, accessed March 3, 2024, https://news.ohsu.edu/2023/10/25/brain-implant-at-ohsu-successfully-controls-both-seizures-and-ocd. [xxxi] Awuah et al., “Bridging Minds and Machines”; Haslacher et al., “AI for Brain-Computer Interfaces.” [xxxii] Awuah et al., “Bridging Minds and Machines.” [xxxiii] Sara Green, Barbara Prainsack, and Maya Sabatello, “The Roots of (in)Equity in Precision Medicine: Gaps in the Discourse,” Personalized Medicine 21, no. 1 (January 2024): 5–9, https://doi.org/10.2217/pme-2023-0097. [xxxiv] Green, Prainsack, and Sabatello, 7. [xxxv] Robyn Bluhm and Kirstin Borgerson, “An Epistemic Argument for Research-Practice Integration in Medicine,” The Journal of Medicine and Philosophy: A Forum for Bioethics and Philosophy of Medicine 43, no. 4 (July 9, 2018): 469–84, https://doi.org/10.1093/jmp/jhy009. [xxxvi] Vijay Mahant, “‘Right-to-Try’ Experimental Drugs: An Overview,” Journal of Translational Medicine 18 (June 23, 2020): 253, https://doi.org/10.1186/s12967-020-02427-4. [xxxvii] Michael S. Okun et al., “Deep Brain Stimulation in the Internal Capsule and Nucleus Accumbens Region: Responses Observed during Active and Sham Programming,” Journal of Neurology, Neurosurgery & Psychiatry 78, no. 3 (March 1, 2007): 310–14, https://doi.org/10.1136/jnnp.2006.095315. (shrink)

Galit Wellner’s exploration of new kinds of digital technologies employing AI algorithms that simulate features and functions of the human imagination leads her to propose a conceptual analysis of the imagination as a composite of perception and memory. Wellner poses the question of whether the output of such technological applications might be regarded as not merely simulating creative activity but as truly imaginative in their own right. Wellner concludes with a qualified “no.” The use of AI algorithms in conjunction with (...) human cognitive activity, conceived in terms of a layered architecture of the faculties in question, can in fact be understood as an essential component of imaginative, and thus creative, production, but humans are still needed in the mix. To the extent that the AI-algorithm-enhanced human system is capable of imagining and creating works of art, imagination can be extended to AI algorithms. But, the algorithms sans humans are not themselves imaginers. For Wellner, AI algorithms can augment and enhance human imaginative efforts, equipping us with a richer and vastly wider array of possibilities and options for aesthetic consideration, but ultimately, the human is the essential element. However, once the door is opened to accepting algorithmically determined alternatives as capable of successfully achieving desired results within a field of possible outcomes, it seems possible that the activities of connecting, coordinating, or meaningfully linking, combining, and establishing new imaginative layers that Wellner reserves as requiring humans to enact might also be programmable, algorithmically achievable tasks that instantiate genuine aesthetic decision making. (shrink)

Combinatory logic started as a programme in the foundation of mathematics and in an historical context at a time when such endeavours attracted the most gifted among the mathematicians. This small volume arose under quite differ ent circumstances, namely within the context of reworking the mathematical foundations of computer science. I have been very lucky in finding gifted students who agreed to work with me and chose, for their Ph. D. theses, subjects that arose from my own attempts 1 (...) to create a coherent mathematical view of these foundations. The result of this collaborative work is presented here in the hope that it does justice to the individual contributor and that the reader has a chance of judging the work as a whole. E. Engeler ETH Zurich, April 1994 lCollected in Chapter III, An Algebraization of Algorithmics, in Algorithmic Properties of Structures, Selected Papers of Erwin Engeler, World Scientific PubJ. Co. , Singapore, 1993, pp. 183-257. I Historical and Philosophical Background Erwin Engeler In the fall of 1928 a young American turned up at the Mathematical Institute of Gottingen, a mecca of mathematicians at the time; he was a young man with a dream and his name was H. B. Curry. He felt that he had the tools in hand with which to solve the problem of foundations of mathematics mice and for all. His was an approach that came to be called "formalist" and embodied that later became known as Combinatory Logic. (shrink)

Contents. Introduction. 1. Preliminaries. 2. Normal Form Games. 3. Extensive Games. 4. Applications of Game Theory. 5. The Methodology of Game Theory. Conclusion. Appendix. Bibliography. Index. Does game theory—the mathematical theory of strategic interaction—provide genuine explanations of human behaviour? Can game theory be used in economic consultancy or other normative contexts? Explaining Games: The Epistemic Programme in Game Theory—the first monograph on the philosophy of game theory—is an attempt to combine insights from epistemic logic and the philosophy of science (...) to investigate the applicability of game theory in such fields as economics, philosophy and strategic consultancy. I prove new mathematical theorems about the beliefs, desires and rationality principles of individual human beings, and explore in detail the logical form of game theory as it is used in explanatory and normative contexts. I argue that game theory reduces to rational choice theory if used as an explanatory device, and that game theory is nonsensical if used as a normative device. A provocative account of the history of game theory reveals that this is not bad news for all of game theory, though. Two central research programmes in game theory tried to find the ultimate characterisation of strategic interaction between rational agents. Yet, while the Nash Equilibrium Refinement Programme has done badly thanks to such research habits as overmathematisation, model-tinkering and introversion, the Epistemic Programme, I argue, has been rather successful in achieving this aim. "The 'epistemic' approach to game theory has emerged over the past twenty-five years. What is this approach? How does it differ from the conventional equilibrium-based approach to game theory? What have been its strengths and weaknesses to date? To find out, read this comprehensive and excellently written account". Adam Brandenburger, J. P. Valles Professor of Business Economics and Strategy, Stern School of Business, New York University "Reading Boudewijn de Bruin's book should be rewarding both for game theorists interested in the conceptual foundations of their discipline and for philosophers who want to learn more about formal analysis of strategic interaction. It provides an in-depth logical study of the currently dominant epistemic approaches to non-cooperative games, with an eye both to the attractions and to the serious challenges facing the Epistemic Programme". Wlodek Rabinowicz, Professor of Practical Philosophy, Department of Philosophy, Lund University . (shrink)

My working hypothesis is based on the project of a theory of science (Wissenschaftslehre) at the very beginning of the Prolegomena and it consists in conceiving this theory of science as the program which insures their cohesion to the whole of the Investigations in this work. In order to test this hypothesis, I will first examine the different steps which led to the project of a theory of science in the pre-phenomenological period. I will secondly expound the guidelines of the (...) theory of science, insisting in particular on its relation to pure logic. In the third section, I will attempt to define the function of the theory of knowledge and of phenomenology in such a programme. I will conclude with some remarks on the fate of this theory of science after the Logical Investigations. Most of Husserl’s allusions to the theory of science, be it in his lecture notes, the manuscripts that were written after 1901 or other works published during Husserl’s lifetime, seem to suggest that the theory of science kept on assuming the very role it was playing in Husserl’s seminal work: the theory of science provides its basis to Husserl’s philosophical agenda. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals). In (...) the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

This paper compares the roles classical and intuitionistic logic play in restricting the free use of truth principles in arithmetic. We consider fifteen of the most commonly used axiomatic principles of truth and classify every subset of them as either consistent or inconsistent over a weak purely intuitionistic theory of truth.

Abstract: One of the institutional highlights of the encounter between Austrian “wissen¬schaftliche Philosophie” and French “philosophie scientifique” in the first half of the 20th century was the “First International Congress for Unity of Science” that took place 1935 in Paris. In my contribution I deal with an episode of the philosophical mega-event whose protagonist was the American philosopher and semiotician Charles William Morris. At the Paris congress he presented his programme of a comprehensive, practice-oriented scientific philosophy and, in a more (...) elaborated version he published it two years later in Logical Positivism, Pragmatism and Scientific Empiricism (Morris 1937). Morris aimed at a synthesis of formalism, pragmatism, and traditional empiricism that combined the virtues of these accounts while avoided their shortocmings. The core of approach was a comprehensive theory of the concept of meaning. Through an analysis of the concept of meaning he sought to sort out the existing differences and the options for a possible future rapprochment between logical empiricism and pragmatism. Against the overly narrow logical empiricist understanding of philosophy as the syntax of the language of science Morris argued for a “scientific pragmatism” that comprised four levels: (1) Philosophy as Logic of Science, (2) Philosophy as Clarification of Meaning (Peirce), (3) Philosophy as Empirical Axiology (Dewey), and (4) Philosophy as Empirical Cosmology (Whitehead). (shrink)

Popper first developed his theory of scientific method – falsificationism – in his The Logic of Scientific Discovery, then generalized it to form critical rationalism, which he subsequently applied to social and political problems in The Open Society and Its Enemies. All this can be regarded as constituting a major development of the 18th century Enlightenment programme of learning from scientific progress how to achieve social progress towards a better world. Falsificationism is, however, defective. It misrepresents the real, problematic (...) aims of science. We need a new conception of scientific method, a meta-methodology which provides a framework for the improvement of the aims and methods of science as scientific knowledge improves. This aim-oriented empiricist idea can be generalized to form a conception of rationality – aim-oriented rationality – which helps us improve problematic aims and methods whatever we may be doing. In this way, Popper’s version of the Enlightenment programme can be much improved, indeed transformed. (shrink)

The young Wittgenstein called his conception of logic “New Logic” and opposed it to the “Old Logic”, i.e. Frege’s and Russell’s systems of logic. In this paper the basic objects of Wittgenstein’s conception of a New Logic are outlined in contrast to classical logic. The detailed elaboration of Wittgenstein’s conception depends on the realization of his ab-notation for first order logic.

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where (...) T is $\mathsf {ZFC}$ +Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework. (shrink)

Most of the Bolzano literature is exegetical, neglecting, unfortunately, the great potential of his logic as the beginning of a PROGRAMME. Specifically, his unorthodox construai of the consequence relation as triadic, and his account of logical form are promising beginnings which even as they stand shed light on question of relevance, the ancient problems of enthymemes and others. Instead of developing these suggestions, Bolzano scholars have been occupied with elucidating the ontology of sentences in themselves, and related topics. I (...) argue, and believe to be in agreement with Bolzano, that the nature of sentences is fully explained by the relations that hold between them, just as money has no nature or essence beyond the transactions it makes possible. It follows that the development of his logic would contribute at least as much to the understanding of sentences than any exegesis. (shrink)

Most of the Bolzano literature is exegetical, neglecting, unfortunately, the great potential of his logic as the beginning of a PROGRAMME. Specifically, his unorthodox construai of the consequence relation as triadic, and his account of logical form are promising beginnings which even as they stand shed light on question of relevance, the ancient problems of enthymemes and others. Instead of developing these suggestions, Bolzano scholars have been occupied with elucidating the ontology of sentences in themselves, and related topics. I (...) argue, and believe to be in agreement with Bolzano, that the nature of sentences is fully explained by the relations that hold between them, just as money has no nature or essence beyond the transactions it makes possible. It follows that the development of his logic would contribute at least as much to the understanding of sentences than any exegesis. (shrink)

The unity of science movement was itself far from unified. There may have been unity on the rallying call for a unity of science but that is as far as it went. Not only was there disagreement among the main protagonists on what was meant by the unity of science, but also on how to achieve it. In this paper I shall deal with Edgar Zilsel’s (1891-1944) conception. It represents an interesting break with the more programmatic approaches of Carnap, Neurath; (...) Zilsel emphasizes the need for an empirical demonstration of the unity of science, which has interesting meta-philosophical implications. The unity of science is not just central to his research programme; it is what constitutes the latter’s unifying principle, although this is far from evident if one takes a look at the historical essays he published in exile in the USA during the 1940s. (These important essays on the emergence of science, well known among historians of early modern science, have given rise to the so-called Zilsel Thesis, which holds that modern science came into being when, between 1300 and 1600, the social barriers between those who ‘labored with there minds and tongues’, i.e. the university scholars and the humanists, and those who ‘worked with their hands’, i.e the superior artisans, eroded because of the rise of free-enterprise capitalism.) But simultaneously, he also published a couple of smaller and far less known essays, directed against Southwest-German Neo-Kantianism (Rickert, Windelband), Dilthey’s philosophy of life, and interpretative sociology (Max Weber, W. Sombart, G. Simmel, R. Stammler, Alfred Weber, etc.). His main argumentwas that philosophers of cultural science and the humanities had a falseunderstanding of natural science. Because of this false understanding, they erroneously postulated a fundamental methodological difference between the natural sciences and the human sciences. According to Zilsel, this difference does not exist. In his view the historian is basically confronted with the same problems as the natural scientist. If the science of history is not compared with classical mechanics but with, for example, geophysics – that is, “the physics of earthquakes, sea-currents, volcanology, and meteorology” – one will arrive at the conclusion “that historical phenomena are hardly more difficult to predict than the weather, and certainly no more difficult than earthquakes and volcanic eruptions. What would scientists think of a geophysicist who abandoned the search for geophysical laws because of their inexactness? ” (SOMS: 202). (shrink)

Hilbert's plan for understanding the concept of infinity required the elimination of non‐finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy of progressively less elementary, but still constructive methods instead of finitist ones . A mathematical proof of this failure requires a definition of « finitist ».—The paper sketches the three principal methods for the syntactic analysis of non‐constructive mathematics, the resulting consistency proofs and constructive interpretations, modelled on Herbrand's theorem, and their (...) mathematical and logical consequences. A characterization of finitist proofs is sketched. A remark on the completeness of the predicate calculus concludes the paper. Throughout open problems and alternative approaches are emphasized.ZusammenfassungHilbert sah das Verstehen des Begriffs des Unendlichen in der Eliminierung nichtfiniter Methoden aus Beweisen finiter Sätze. Das Versagen dieses Unternehmens führt zu einer Hierarchic progressiv weniger elementarer, aber noch konstruktiver Methoden, statt der finiten . Ein Unmöglichkeitsbeweis des ursprünglichen Planes setzt eine Definition von « finit » voraus.—Die drei wichtigsten Methoden der syntaktischen Analyse nichtkonstruktiver mathematischer Theorien, die daraus gewonnenen Widerspruchsfreiheitsbeweise and konstruktiven Interpretationen and deren mathematische and logische Anwendungen werden besprochen. Eine Prazisierung des Begriffs « finit » wird skizziert. Eine Bemerkung zur Vollstandigkeit des engeren Funktionenkalküls beschliesst die Abhandlung.—Oflene Fragen and komplementare Fragestellungen werden betont. (shrink)

Turing's programme, the idea that intelligence can be modelled computationally, is set in the context of a parallel between certain elements from metamathematics and Popper's schema for the evolution of knowledge. The parallel is developed at both the formal level, where it hinges on the recursive structuring of Popper's schema, and at the contentual level, where a few key issues common to both epistemology and metamathematics are briefly discussed. In light of this connection Popper's principle of transference, akin to Turing's (...) belief in the relevance of the theory of computation for modelling psychological functions, is widened into the extended principle of transference. Thus Turing's programme gains a solid epistemological footing. *I am grateful to Claude Lamontagne and Jean-Pierre Delage for their comments on this paper. (shrink)

Transcendental philosophy was not born like Athena out of Zeus’s head, mature and in full armour from the very beginning. That is why in both prefaces to the Critique of Pure Reason (1781 and 1787) Kant introduces the concept of transcendental philosophy as an “idea.” The idea understood architectonically develops slowly and only gradually acquires a definite form. As witnessed by the works of Kant himself and of his predecessors and followers, the idea of transcendental philosophy has undergone a series (...) of changes and adjustments compared to the initial plan. In this context, my goal is not simply exegesis and historical investigation of transcendental philosophy, but also to look at it from a systematic and methodological perspective. I examine the concept of transcendental philosophy from the viewpoint of programmatic metaphilosophy. The first part discusses programmatics as a distinct subsection of metaphilosophy. I argue that Kant’s architectonic methodology and the methodology of Lakatos can be used to understand the inception, development and degradation of philosophical systems. In the second part I look at the project of transcendental philosophy and the stages of its development from the standpoint of architectonics. The third part shows that Lakatos’s methodology can provide a detailed insight into the elements of transcendental philosophy, a clear idea of its logic and identify the component parts that can be improved and developed. In spite of the different levels of detailing and epistemological prerequisites, the methodologies of Kant and Lakatos can be combined to achieve a metaphilosophically informed and progressive understanding of philosophical projects. (shrink)

It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style (...) axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics'' assuggested by Penelope Maddy. The answer is ``no'', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind'' andrelating it to the ``one and only axiom'' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper. (shrink)

By 'scientific theory' I shall mean a core of fundamental assumptions fleshed out by a suitable array of auxiliary assumptions. Let there be two or more competing scientific theories in a certain field, thrown up by rival research programmes in Lakatos's sense. Now consider these two questions: (1) Which of these theories should I accept? (2) Which of these theories should I work on? They are surely very different questions. The personal pronoun 'I' could be dropped from question (1), (...) which could be reformulated as 'Which of these theories should anyone accept as the best one, in the light of the presently available evidence?' In my (1984) I suggested that the best theory is the one that best fulfils what I depicted as the optimum aim for science; and this, I argued, will be the one that is best corroborated. Which means that we may turn question (1) into: (1 ') Which of these theories is best corroborated? (shrink)

Name der Zeitschrift: Studies in Logic, Grammar and Rhetoric Jahrgang: 45 Heft: 1 Seiten: 97-110.

In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows (...) how optimal principles, among those available, may be selected in a justifiable way. (shrink)

Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential and practical part of computer science. The idea of a formal system-a collection of rules and axioms, which define a universe of logical proofs-is what gives us programming languages and (...) modern-day programming. This book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses-natural deduction-is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts: Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words. Part II "Formal Syntactic Proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle. Part III "Formal Semantic Disproof" shows you how to construct mathematical counterexamples to shoe that proof is impossible. Jape can check the counterexamples you build. Part IV " Program Specification and Proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows. Aimed at undergraduates and graduates in computer science, logic, mathematics and philosophy, the text includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the web. (shrink)

This note is intended to address one particular issue in the relative status of Quantum Chemistry in comparison to both Chemistry and Physics. It has been suggested, in the context of the question of the reduction relations between Chemistry and Physics that Quantum Chemistry as a research programme is incapable of furnishing useful guidance to practising chemists. If true, this claim will let us qualify Quantum Chemistry as a degenerating research programme, which, due to its complexity has difficulty to be (...) applied to Chemistry. This claim is shown to be false. The replacement claim I wish to make is that Quantum Chemistry is perfectly capable of furnishing such guidance, but renders the ontological status of many models favored by chemists problematic. Quantum Chemistry, however, validates these models in an instrumental fashion. I will argue that Quantum Chemistry is a progressive research programme. (shrink)

Philippe Fournier | : Le point de départ de ma proposition est que la responsabilité est un terme plus approprié que la vertu pour désigner les exhortations au devoir civique dans l’ère contemporaine. De même, à défaut de voir l’implication citoyenne comme l’expression de la rationalité individuelle ou de la conscience morale dans la sphère publique, je propose de comprendre la responsabilité comme une matrice discursive et gouvernementale qui perpétue des modèles comportementaux bien spécifiques. J’entends ainsi démontrer que la responsabilité (...) est devenue une modalité indispensable du « gouvernement de la conduite » au sein d’une rationalité néolibérale. En plus de constituer une série de dispositions morales qui remédierait aux failles et aux lacunes de l’individualisme libéral, la responsabilité se matérialise dans les incitations à une citoyenneté active et participe de ce fait à la rationalisation du retranchement de l’État de la sphère sociale. Comme l’illustrent des courants aussi divers que le néoconservatisme, le communautarisme et l’économie sociale, la responsabilité se présente moins comme une solution de rechange aux créneaux du néolibéralisme qu’une série de techniques et de standards comportementaux visant à compléter et renforcer l’application élargie d’une logique micro-économique. Je montrerai que le recours à la notion de responsabilité est particulièrement visible dans les politiques d’aide sociale et au sein des nouveaux partenariats entre l’État, le milieu communautaire et les communautés locales. En effet, l’aide étatique est de plus en plus conditionnelle à la démonstration de certaines dispositions morales et psychologiques comme la volonté, la ténacité et la probité. L’article se divise en quatre parties. La première s’attarde à définir brièvement la « gouvernementalité », concept initialement développé par Michel Foucault. La deuxième se penche sur des travaux plus contemporains, notamment ceux du sociologue britannique Nikolas Rose qui se préoccupe de la responsabilité en tant que modalité gouvernementale au sein du néolibéralisme. La troisième partie met en relief les particularités du néolibéralisme américain. En vue de cerner les pratiques et discours associés à la responsabilité de façon plus précise, la dernière partie examine l’un des principaux programmes d’aide sociale aux États-Unis, l’Aid to Families with Dependent Children, devenu le Temporary Assistance for Needy Families en 1996, des années 1980 jusqu’à aujourd’hui. | : Our point of departure is that responsibility is a more appropriate term than virtue to designate the contemporary exhortations to civic duty. Instead of seeing civic engagement as a means to express individual rationality or moral conscience in the public sphere, this paper suggests that we should understand responsability as a discursive and governmental matrix that perpetuates specific behavioral standards. More specifically, the article aims to show that responsability has become an indispensable instrument of rule within a neoliberal rationality of government. As well as pointing to a range of moral dispositions that can offset neoliberalism’s shortcomings, the notion of responsibility is mobilised in the repeated public calls to active citizenship and plays a strategic role in the discursive justification of the state’s withdrawal from the social sphere. As illustrated by strands of thought as diverse as neoconservatism, communitarianism and social economy, responsibility is less an alternative to neoliberal tenets than a series of techniques and behavioral standards that complement and reinforce the extension of a microeconomic logic to all spheres of human activity. The appeals to responsibility are particularly visible in contemporary social policies and in the strategic partnerships between the state, community organisations and local communities. Indeed, state assistance has become increasingly conditional on the demonstration of aptitudes and moral qualities like earnestness, tenacity and willpower. The article proceeds in four parts. The first part briefly defines governmentality, a concept initially developed by Michel Foucault. The second part explores Nikolas Rose’s work on advanced liberalism and takes a look at contemporary articulations of responsibility. The third part examines the specificities of American neoliberalism. In order to identify the practices and discourses linked to the notion of responsibility, the final part of the article looks at one of the main social programs in the United States, namely the Aid to Families with Dependent Children, renamed the Temporary Assistance for Needy Families in 1996, from the early 1980s to our day. (shrink)

The iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the maximality of the universe of sets V in height and width. These principles have been heavily investigated by the first author and his collaborators within the Hyperuniverse Programme. The programme is based on two essential tools: the hyperuniverse, consisting of all countable transitive models (...) of ZFC, and V -logic, both of which are also fully discussed in the paper. (shrink)

A possible relevant meaning of Hilbert’s program is the following one: “give a constructive semantic for classical mathematics”. More precisely, give a systematic interpretation of classical abstract proofs about abstract objects, as constructive proofs about constructive versions of these objects.If this program is fulfilled we are able “at the end of the tale” to extract constructive proofs of concrete results from classical abstract proofs of these results.Dynamical algebraic structures or geometric theories seem to be a good tool for doing this (...) job. In this setting, classical abstract objects are interpreted through incomplete concrete specifications of these objects.The structure of axioms in geometric theories gives rise in a natural way to distributive lattices and pointfree topological spaces. Abstract objects correspond to classical points of these pointfree spaces.We shall insist on the Zariski spectrum of distributive lattices and commutative rings and give a constructive interpretation of the Krull dimension.We underline the fact that many abstract objects in classical mathematics can be viewed as points of spectral spaces corresponding to distributive lattices whose definition is concrete and natural.Two important facts are to be stressed.First, abstract proofs about points of these pointfree spaces can very often be reread as constructive proofs about constructible subsets of these spaces.Second, function spaces on these pointfree spaces are often explicitly used in elegant abstract theories. The structure of these function spaces is fully constructive: indeed by the compactness theorem, “all is finite”. The constructive rereading of the abstract proofs is in this setting is nothing but the simple constatation that abstract proofs use correctly axioms.RésuméUne manière pertinente de revisiter le Programme de Hilbert serait la suivante: « donner une sémantique constructive pour les mathématiques classiques ». Plus précisément donner une interprétation systématique des preuves classiques abstraites au sujet des objets abstraits, en terme de preuves constructives au sujet de contreparties constructives de ces objets abstraits.Si ce programme est rempli, nous sommes capables « à la fin de l’histoire » d’extraire des preuves constructives de résultats concrets à partir des preuves classiques abstraites de ces résultats.Les structures algébriques dynamiques, ou ce qui revient à peu près au même les théories géométriques, semblent être un bon outil pour réaliser ce travail. Dans cette optique, les objets abstraits des mathématiques classiques sont remplacés par des spécifications incomplètes mais concrètes de ces mêmes objets.La structure des théories géométriques donne naissance de manière naturelle à des treillis distributifs et à des espaces topologiques sans points. Les objets abstraits utilisés par les mathématiques classiques correspondent aux points classiques de ces espaces sans points.Dans cet article, nous illustrerons ce phénomène principalement avec le spectre de Zariski des treillis distributifs et celui des anneaux commutatifs, en indiquant notamment un équivalent constructif de la notion de dimension de Krull.Nous insistons sur le caractère extrêmement général de l’interpétation des objets abstraits idéaux des mathématiques classiques comme des points d’espaces spectraux associés à des treillis distributifs qui sont définis de façon naturelle et concrète.Souligons deux faits d’expérience importants.Tout d’abord, les preuves abstraites au sujet des points de ces espaces sans points peuvent en général être relues comme des preuves concernant les parties constructibles de ces espaces.Enfin, les espaces de fonctions continues sur ces espaces sans points sont souvent utilisés dans d’élégantes théories abstraites. Ces espaces de fonctions sont bien définis constructivement. Cela tient au « théorème de compacité » qui nous dit que dans le cadre en question « tout est fini ». La relecture constructive des preuves abstraites n’est alors rien d’autre que la constatation que les axiomes géométriques sont utilisés de manière correcte. (shrink)

In this paper, I argue that Dewey’s pragmatist aesthetics, and in particular, his concept of consummatory experience, should be engaged anew to rethink the merits of the Philosophy for Children programme, which arose in the 1970s in the US as an innovative educational programme that aims to use philosophy to help school children improve their ability to become more conscious of and make judgments about the aspects of their experience that have ethical, aesthetic, political, logical, or even metaphysical meaning. Although (...) an international success, the PFC programme has attracted many criticisms from a variety of directions. I claim that Deweyan concept of consummatory aesthetic experience is broad and flexible enough to provide a robust framework to make sense of the pedagogical horizon of PFC and therefore fruitfully engage the various critics of the movement coming from religious and social conservatives, educational psychologists, critical theorists, postmodernists/posthumanists, and professional philosophers themselves. The goal of this paper is to offer in a preliminary fashion the basic elements of Deweyan pragmatist aesthetics, which was principally elucidated in his Art as Experience, to defend PFC as a viable pedagogy. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

It is known that Hume’s Principle, adjoined to a suitable formulation of second-order logic, gives a theory which is almost certainly consistent4 and suffices for arithmetic in the sense that it yields the Dedekind-Peano axioms as theorems. While Hume’s Principle cannot be taken as a definition in any strict sense requiring that it provide for the eliminative paraphrase of its definiendum in every admissible type of occurrence, we hold that it can be viewed as an implicit definition of a (...) sortal concept of cardinal number and, accordingly, as being analytic of that concept. This, coupled with the fact that Hume’s Principle so conceived requires a prior understanding only of (second-order) logical vocabulary, is enough to justify regarding the resulting account of the foundations of arithmetic as a form of logicism. Whether this claim can ultimately be sustained is of course still very much a matter of controversy. But here, rather than add to the debate on that issue, I should like to assume that it can be, and on that basis, address some further questions. (shrink)

This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.