An advanced-level logic textbook that presents proof construction on equal footing with model building. Potentially relevant to students of mathematics and computer science as well.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...) interval, the rational unit interval, the hyperreals , the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. (shrink)

This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the (...) exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. (shrink)

Provides comprehensive coverage of logical structure as well as the techniques of formal reasoning.

To improve the accuracy and timeliness of the trajectory tracking control of the driverless racing car during the race, this paper proposes a track tracking control method that integrates the rear wheel differential drive and the front wheel active steering based on optimal control theory and fuzzy logic method. The model of the lateral track tracking error of the racing car is established. The model is linearized and discretized, and the quadratic optimal steering control problem is constructed. Taking (...) advantage of the differential drive of dual-motor-driven racing car, the dual motors differential drive fuzzy controller is designed and integrated driving with active steering control. Simulation analysis and actual car verification show that this integrated control method can ensure that the car tracks different race tracks well and improve the track tracking control accuracy by nearly 30%. (shrink)

It is a commonplace of the history of analytic philosophy that Carnap swiftly adopted Tarskian semantics in the mid-1930s. There is no doubt that, in a very general sense, this is true. But to what extent are the innovative technical details characteristic of Tarski's method, specifically the handling of quantification by way of a satisfaction relation between formulas and variable assignments, reflected in Carnap's writings on semantics? Curiously enough, their essentials are in place just before Carnap took the purported (...) Tarskian turn, namely in the definition of analyticity he formulated some time between September 1932 and April 1933, but are missing from his publications on semantics after 1935 until the final chapter of Meaning and Necessity of 1947, at which point we find Carnap reverting to the use of specifically Tarskian techniques in the context of his novel semantics for quantified modal languages. (shrink)

Modal knowledge accounts like sensitivity or safety face a problem when it comes to knowing propositions that are necessarily true because the modal condition is always fulfilled no matter how random the belief forming method is. Pritchard models the anti-luck condition for knowledge in terms of the modal principle safety. Thus, his anti-luck epistemology faces the same problem when it comes to logical necessities. Any belief in a proposition that is necessarily true fulfills the anti-luck condition and, therefore, (...) qualifies as knowledge. Miščević shares Pritchard’s take on epistemic luck and acknowledges the resulting problem. In his intriguing article “Armchair Luck: Apriority, Intellection and Epistemic Luck” Miščević suggests solving the problem by supplementing safety with a virtue theoretic condition-“agent stability”-which he also spells out in modal terms. I will argue that Miščević is on the right track when he suggests adding a virtue-theoretic component to the safety condition. However, it should not be specified modally but rather in terms of performances that manifest competences. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

Jaśkowski’s discussive logic D2 was formulated with the help of the modal logic S5 as follows (see [7, 8]): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \in {D_{2}}}$$\end{document} iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}$$\end{document}, where (–)• is a translation of discussive formulae from Ford into the modal language. We say that a modal logic L defines D2 iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm D}_{2} = (...) \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}$$\end{document}. In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (†): have the same theses beginning with ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond}$$\end{document}’ as S5. Of course, all logics fulfilling the above condition, define D2. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D2 is equivalent to having the property (†). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (†). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D2. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of (...) intuition in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

This paper develops an account of Wittgenstein’s method of language-games as a method of logic that exhibits important continuities with Russell’s and the early Wittgenstein’s conceptions of logic and logical analysis as the method of philosophy. On the proposed interpretation, the method of language-games is a method for isolating and modeling aspects of the uses of linguistic expressions embedded in human activities that enables one to make perspicuous complex uses of expressions by gradually building (...) up the complexity of clarificatory models. Wittgenstein’s introduction of the language-game method constitutes an attempt to overcome certain limitations of calculus-based logical methods, and to respond in this way to problems with Russell’s and his own early philosophy of logic. The method is nevertheless compatible with the employment of calculus-based methods in logic and philosophy, and makes no exclusive claim to being the correct method. (shrink)

I argue that the analytic method has been circularly used to analyze the concept of “philosophy,” and that the result of this analysis has also been used to criticize African ethnophilosophy as nonphilosophical. I critically examine the criticism that ethnophilosophy implies cognitive relativism and the criticism that it implies authoritarianism. I defend ethnophilosophy against these criticisms, arguing that they are rooted in logical positivism, the view that philosophy essentially involves the use of the methods of science and (...) class='Hi'>logical analysis. I argue that such analysis and criticisms, given their pedigree, do not provide an adequate or accurate picture of the nature of philosophy. (shrink)

Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.

Graciela De Pierris presents a novel interpretation of the relationship between skepticism and naturalism in Hume's epistemology, and a new appraisal of Hume's place within early modern thought. Contrary to dominant readings, she argues that Hume does offer skeptical arguments concerning causation and induction in Book I, Part III of the Treatise, and presents a detailed reading of the skeptical argument she finds there and how this argument initiates a train of skeptical reasoning that begins in Part III and culminates (...) in Part IV. She goes on to demonstrate that Hume was committed to the Newtonian inductive method while rejecting the place of the supernatural in our understanding of nature. (shrink)

We examine a series of disputations which Luther participated in towards the end of his career: we argue that these disputations show that Luther was very familiar with the tools of medieval formal logic, and continued to make positive theological use of them until the end of his life.

Neuroscientists have long sought to study the dynamic activity of the human brain—what's happening in the brain, that is, while people are thinking, feeling, and acting. Ideally, an inside look at brain function would simultaneously and continuously measure the biochemical state of every cell in the central nervous system. While such a miraculous method is science fiction, a century of progress in neuroimaging technologies has made such simultaneous and continuous measurement a plausible fiction. Despite this progress, practitioners of modern (...) neuroimaging struggle with two kinds of limitations: those that attend the particular neuroimaging methods we have today and those that would limit any method of imaging neural activity, no matter how powerful.In this essay, I consider the liabilities and potential of techniques that measure human brain activity. I am concerned here only with methods that measure relevant physiologic states of the central nervous system and relate those measures to particular mental states. I will consider in particular the preeminent method of functional neuroimaging: BOLD fMRI. While there are several practical limits on the biological information that current technologies can measure, these limits—as important as they are—are minor in comparison to the fundamental logical restraints on the conclusions that can be drawn from brain imaging studies. (shrink)

Logical form and logical analysis as the search for it have been introduced during the development of logic and analytical philosophy and are still widely considered as key tools or methods for the solution of philosophical puzzles. It is instructive to have a look at a criticism of these presupositions and I present Wittgenstein as the author who provides such a criticism. I present a development of his view of logical form which went from the thesis of (...) the ineffability of logical form to the denial of the meaningfulness of the notion of logical form as such. This refusal is linked to Wittgenstein’s abandonment of the idea of the language of pure experience. The method of philosophical therapies is presented as an alternative to logical analysis and this methodology is linked with Wittgenstein’s consideration of game and family resemblance. (shrink)

Although "the Socratic method" is commonly understood as a style of pedagogy involving cross-questioning between teacher and student, there has long been debate among scholars of ancient philosophy about how this method as attributed to Socrates should be defined or, indeed, whether Socrates can be said to have used any single, uniform method at all distinctive to his way of philosophizing. This volume brings together essays by classicists and philosophers examining this controversy anew. The point of departure (...) for many of those engaged in the debate has been the identification of Socratic method with "the elenchus" as a technique of logical argumentation aimed at refuting an interlocutor, which Gregory Vlastos highlighted in an influential article in 1983. The essays in this volume look again at many of the issues to which Vlastos drew attention but also seek to broaden the discussion well beyond the limits of his formulation. Some contributors question the suitability of the elenchus as a general description of how Socrates engages his interlocutors; others trace the historical origins of the kinds of argumentation Socrates employs; others explore methods in addition to the elenchus that Socrates uses; several propose new ways of thinking about Socratic practices. Eight essays focus on specific dialogues, each examining why Plato has Socrates use the particular methods he does in the context defined by the dialogue. Overall, representing a wide range of approaches in Platonic scholarship, the volume aims to enliven and reorient the debate over Socratic method so as to set a new agenda for future research. Contributors are Hayden W. Ausland, Hugh H. Benson, Thomas C. Brickhouse, Michelle Carpenter, John M. Carvalho, Lloyd P. Gerson, Francisco J. Gonzalez, James H. Lesher, Mark McPherran, Ronald M. Polansky, Gerald A. Press, François Renaud, and W. Thomas Schmid, Nicholas D. Smith, P. Christopher Smith, Harold Tarrant, Joanne B. Waugh, and Charles M. Young. (shrink)

I distinguish two ways of developing anti-exceptionalist approaches to logical revision. The first emphasizes comparing the theoretical virtuousness of developed bodies of logical theories, such as classical and intuitionistic logic. I'll call this whole theory comparison. The second attempts local repairs to problematic bits of our logical theories, such as dropping excluded middle to deal with intuitions about vagueness. I'll call this the piecemeal approach. I then briefly discuss a problem I've developed elsewhere for comparisons of (...) class='Hi'>logical theories. Essentially, the problem is that a pair of logics may each evaluate the alternative as superior to themselves, resulting in oscillation between logical options. The piecemeal approach offers a way out of this problem andthereby might seem a preferable to whole theory comparisons. I go on to show that reflective equilibrium, the best known piecemeal method, has deep problems of its own when applied to logic. (shrink)

Merging logic and mathematics in deductive inference-an innovative, cutting-edge approach. Optimization methods for logical inference? Absolutely, say Vijay Chandru and John Hooker, two major contributors to this rapidly expanding field. And even though "solving logical inference problems with optimization methods may seem a bit like eating sauerkraut with chopsticks... it is the mathematical structure of a problem that determines whether an optimization model can help solve it, not the context in which the problem occurs." Presenting powerful, proven optimization (...) techniques for logic inference problems, Chandru and Hooker show how optimization models can be used not only to solve problems in artificial intelligence and mathematical programming, but also have tremendous application in complex systems in general. They survey most of the recent research from the past decade in logic/optimization interfaces, incorporate some of their own results, and emphasize the types of logic most receptive to optimization methods-propositional logic, first order predicate logic, probabilistic and related logics, logics that combine evidence such as Dempster-Shafer theory, rule systems with confidence factors, and constraint logic programming systems. Requiring no background in logic and clearly explaining all topics from the ground up, Optimization Methods for Logical Inference is an invaluable guide for scientists and students in diverse fields, including operations research, computer science, artificial intelligence, decision support systems, and engineering. (shrink)

This paper is dedicated to the first universities and mendicant schools, where thousands of students began to converge during the thirteenth century. Logic played an unpreceded role in basic and higher education. A “Parisian logical model” of education was shaped at the University of Paris, adopted by mendicant Orders in their schools of logic, diffused in all disciplines, and progressively spread in Southern Europe. Medieval education became heavily based upon logical, and even “logician” practices, with the “syllogization” of (...) exegetical, disputational, and evaluation practices.The notion of logical skill conveniently captures this unique situation for the discipline of logic, as well as the way medieval thinkers conceived of logic as a universal, transdisciplinary method, a natural operation of the mind, a modality of knowledge, the very form of teaching and graduating, a “habitus,” a technique and a science.The divisions of Aristotelian logic, the “artificial logic,” were for the first time naturalized and projected on the very structure of human mind, which was thereby “logicalized” and ascribed a “natural logic.” A strong anthropological dimension was bestowed on logic. The discipline of logic was deemed a necessary instrument in the philosophical “perfection” and the Christian “reparation” of man as an intellectual creature by a group of logically skilled, professional philosophers and theologians, whereas men deprived of logical education were described as “logically disabled,” and stuck into inferior forms of humanity.The world of medieval intellectual elites displayed a variety of social uses of logic, beyond academic circles, especially in the performance of pastoral duties. The possible historical records of the social usefulness of logic are explored: for students, the majority in medieval universities, who left university without a degree, but with a solid logical education, for ordinary mendicant friars, dedicated to preaching and confession, who frequented logical schools, and for members of the parish clergy sent to the faculties of arts. (shrink)

The aim of this paper is to present the method of Socratic proofs for seven modal propositional logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. This work is an extension of [10] where the method was presented for the most common modal propositional logics: K, D, T, KB, K4, S4 and S5.

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

It may sometimes be useful in philosophy to state a method of argument in very simple and even over-simplified terms; it may at a certain stage be useful to ignore for the moment the details and difficulties of particular philosophical arguments, and to try to state simply what is commonly assumed in a variety of different and even opposing arguments.

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In this Vol. I the author explains preferential structures and abstract size. In the associated Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author limits the (...) discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

Scholarship has grown increasingly nuanced in its grappling with the intersections of religion, science, and technology but requires a new paradigm. Contemporary approaches to specific technologies reveal a wide variety of perspectives but remain too often committed to typological classification. To be vigilant of our obligation to understand and reveal, scholars in the study of religion, science, and technology can adopt a hydra‐logical stance: we can recognize that there are cultural monsters possessing scientific, technological, and religious heads. These heads (...) may work with a common agenda or they might not. They might disagree, pulling their shared body back and forth in a public commotion that lays waste to their surroundings. They might see past one another or move in tandem—purposively or not. Evaluations of climate response and AI benefit from seeing how the various heads are inseparable: indeed, cutting one off simply promotes the growth of new heads. Methodological and analytical clarity, therefore, emerges in the transition from schemes of classification to the recognition of hydras. (shrink)

The thesis of this paper is that scientific method is to be thought of as a complex many-leveled regulatory hierarchy of principles, interacting with theory also viewed as a complex many-leveled hierarchy. This conception of method is illustrated in particular through one episode in the contemporary development of plasma physics, and related to others. It provides for method-theory interaction and for the development of method itself as science develops.

Adopting a central feature of Stoic epistemology, Descartes treated belief as action that might be undertaken wisely or rashly, and enunciated a method for avoiding false belief, a discipline of the will “to include nothing more in my judgments than what presented itself to my mind with such clarity and distinctness that I would have no occasion to put it in doubt”.1 He called such acts of the will “affirmations”, i.e., acts of accepting sentences or propositions as true.

The companion piece to this article, “Situating Moral Justification,” challenges the idea that moral epistemology's mission is to establish a single, all-purpose reasoning strategy for moral justification because no reasoning practice can be expected to deliver authoritative moral conclusions in all social contexts. The present article argues that rethinking the mission of moral epistemology requires rethinking its method as well. Philosophers cannot learn which reasoning practices are suitable to use in particular contexts exclusively by exploring logical relations among (...) concepts. Instead, in order to understand which reasoning practices are capable of justifying moral claims in different types of contexts, we need to study empirically the relationships between reasoning practices and the contexts in which they are used. The article proposes that philosophers investigate case studies of real-world moral disputes in which people lack shared cultural assumptions and/or are unequal in social power. It motivates and explains the proposed case study method and illustrates the philosophical value of this method through a case study. (shrink)

A new theoretical approach to Aristotelian Logic based on three axioms has been recently introduced. This formalization of the theory allowed for the unification of its uncommunicated traditional branches, thus restoring the theoretical unity of AL. In this brief paper, the applicability of the three AL axioms to Propositional Logic is explored. First, it is shown how the AL axioms can be applied to some simple PL arguments in a straightforward manner. Second, the development of a proof method for (...) PL inspired by the AL axioms is presented. This method mimics the underlying mechanics of the proof method from AL, and offers a complementary alternative to proof methods such as truth trees. (shrink)

There has recently been a good deal of controversy about Landauer's Principle, which is often stated as follows: The erasure of one bit of information in a computational device is necessarily accompanied by a generation of kTln2 heat. This is often generalised to the claim that any logically irreversible operation cannot be implemented in a thermodynamically reversible way. John Norton (2005) and Owen Maroney (2005) both argue that Landauer's Principle has not been shown to hold in general, and Maroney offers (...) a method that he claims instantiates the operation Reset in a thermodynamically reversible way. In this paper we defend the qualitative form of Landauer's Principle, and clarify its quantitative consequences (assuming the second law of thermodynamics). We analyse in detail what it means for a physical system to implement a logical transformation L, and we make this precise by defining the notion of an L-machine. Then we show that logical irreversibility of L implies thermodynamic irreversibility of every corresponding L-machine. We do this in two ways. First, by assuming the phenomenological validity of the Kelvin statement of the second law, and second, by using information-theoretic reasoning. We illustrate our results with the example of the logical transformation 'Reset', and thereby recover the quantitative form of Landauer's Principle. (shrink)

There is at present a certain dispute about counterfactuals taking place. What is at issue is whether counterfactuals with necessarily false antecedents are all true. Some hold that such counterfactuals are vacuously true, appearances notwithstanding. Let us call such people vacuists. Others hold that some counterfactuals with necessarily false antecedents are true; some are false: it just depends on their contents. Let us call such people non-vacuists. As a notable representative of the vacuists, I will take Tim Williamson. On the (...) other side, I will take the position defended by Berto, French, Priest, and Ripley. I will argue (unsurprisingly) that the better choice is Non-Vacuism. That, however, is a subsidiary aim of this paper. The main point is to illustrate the method of theory-choice. (shrink)