Ioannis G. Ntagkas

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s :69–90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus second-order logic divide. The descriptive approach is illustrated by Dedekind’s ‘discovery’ of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Frege’s Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects.

This comprehensive account of the concept and practices of deduction is the first to bring together perspectives from philosophy, history, psychology and cognitive science, and mathematical practice. Catarina Dutilh Novaes draws on all of these perspectives to argue for an overarching conceptualization of deduction as a dialogical practice: deduction has dialogical roots, and these dialogical roots are still largely present both in theories and in practices of deduction. Dutilh Novaes' account also highlights the deeply human and in fact social nature of deduction, as embedded in actual human practices; as such, it presents a highly innovative account of deduction. The book will be of interest to a wide range of readers, from advanced students to senior scholars, and from philosophers to mathematicians and cognitive scientists.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized.

Epicurus (in)famously argued that death is not harmful and therefore our standard reactions to it (like deep fear of death and going to great lengths to postpone it) are not rational, inaugurating an ongoing debate about the harm of death. Those who wish to resist this conclusion must identify the harm of death. But not any old harm will do. In order to resist both the claim that death is not harmful and the claim that our standard reactions to it are irrational, we must identify a harm associated with death that rationalizes our standard reactions. We can divide the potential harm properties death might have along two axes: intrinsic/extrinsic and instrumental/final. This gives us four places to search for the harm of death. I argue that it is nowhere to be found.

Scientific consensus is widely deferred to in public debates as a social indicator of the existence of knowledge. However, it is far from clear that such deference to consensus is always justified. The existence of agreement in a community of researchers is a contingent fact, and researchers may reach a consensus for all kinds of reasons, such as fighting a common foe or sharing a common bias. Scientific consensus, by itself, does not necessarily indicate the existence of shared knowledge among the members of the consensus community. I address the question of under what conditions it is likely that a consensus is in fact knowledge based. I argue that a consensus is likely to be knowledge based when knowledge is the best explanation of the consensus, and I identify three conditions—social calibration, apparent consilience of evidence, and social diversity, for knowledge being the best explanation of a consensus

TItis book is the joint project of a philosopher, Lehrer, and a mathematician, Wagner. The book is, therefore, divided into a first part written by Lehrer, which is primarily philosophical, and a second part written by Wagner that is primarily formal. The authors were, however, influenced by each other throughout. Our book articulates a theory of rational consensus in science and society. The theory is applied to politics, ethics, science, and language. We begin our exposition with an elementary mathematical model of consensus developed by Lehrer in a series of articles [1976a, 1976b, 1977, 1978]. Chapter 3 contains material from [1978]. Lehrer formulated the elementary model when he was a Fellow of the Center for Advanced Study in the Be havioral Sciences, Stanford, in 1973 with the invaluable mathematical assist of Kit Fine, Gerald Kramer and Lionel McKenzie. In the summer of ance 1977, Lehrer and Wagner met at the Center in a Summer Seminar on Freedom and Causality supported by the Andrew W. Mellon Foundation. Wagner read the manuscript of Lehrer [1978] and subsequently solved some mathematical problems of the elementary model. After discussions of philosophical prob lems associated with that model, Wagner developed the foundations for the extended model. These results were reported in Wagner [1978, 1981a].

This paper develops a theory of analog representation. We first argue that the mark of the analog is to be found in the nature of a representational system’s interpretation function, rather than in its vehicles or contents alone. We then develop the rulebound structure theory of analog representation, according to which analog systems are those that use interpretive rules to map syntactic structural features onto semantic structural features. The theory involves three degree-theoretic measures that capture three independent ways in which a system can be more or less analog. We explain how our theory improves upon prior accounts of analog representation, provides plausible diagnoses for novel challenge cases, extends to hybrid systems that are partially analog and partially symbolic, and accounts for some of the advantages and disadvantages of representing analogically versus symbolically.

In the science of consciousness, it’s oftentimes assumed that some creatures (or mental states) are more conscious than others. But in recent years, a number of philosophers have argued that the notion of degrees of consciousness is conceptually confused. This paper (1) argues that the most prominent objections to degrees of consciousness are unsustainable, (2) examines the semantics of ‘more conscious than’ expressions, (3) develops an analysis of what it is for a degreed property to count as degrees of consciousness, and (4) applies the analysis to various theories of consciousness. I argue that whether consciousness comes in degrees ultimately depends on which theory of consciousness turns out to be correct. But I also argue that most theories of consciousness entail that consciousness comes in degrees.

This paper examines the idea of "objective phenomenology," or a way of understanding the phenomenal character of conscious experiences that doesn’t require one to have had the kinds of experiences under consideration. My central thesis is that structural facts about experience—facts that characterize purely how conscious experiences are structured—are objective phenomenal facts. I begin by precisifying the idea of objective phenomenology and diagnosing what makes any given phenomenal fact subjective. Then I defend the view that structural facts about experience are objective. I also argue that structural facts about experience, despite being objective, nevertheless still give rise to an explanatory gap.

Sadi Carnot’s 1824 Reflections on the Motive Power of Fire created the new science of thermodynamics. It succeeded in its audacious goal of finding a very general theory of the efficiency of heat engines, by introducing and exploiting the strange and unexpected notion of a thermodynamically reversible process. The notion is internally contradictory. It requires the states of these processes to be both in unchanging equilibrium, with a perfect balance of driving forces, while also changing. The work of Sadi’s father, Lazare Carnot, on the efficiency of ordinary machines provided Sadi with a template of a very general theory of the efficiency of ordinary machines; and a characterization of the most efficient processes in them as those that minimize differences of driving forces and can be run in reverse. Lazare’s work could provide these resources because of its choice of a dissipative ontology of inelastic collisions among hard bodies. This historically ill-fated choice meant that Lazare’s machines were analogous to Sadi’s heat engines in their key aspects: they are built from essentially dissipative processes. Lazare’s strategies for controlling dissipation and optimizing his machines were transferrable to the analogous problems Sadi found in heat engines. The unanswerable historical question is whether Sadi would have sought a general theory of heat engines at all or found these general theoretical devices without the template provided in analogy by the prior work of Lazare.

Separatists about grounding take explanations to be separate from their corresponding grounding-facts. Grounding-facts are supposed to underlie, or back, such explanations. However, the backing relation hasn’t received much attention in the literature. The aim of this paper is to provide an informative definition of backing. First, I examine two prominent proposals: backing as explaining (Kovacs 2017; 2019a) and backing as grounding (see Sjölin Wirling 2020). Finally, I put forward my own proposal. I argue that under plausible assumptions about the role of backing and the nature of explanation, backing should be understood as a form of truthmaking, minimally construed.

This paper reviews the contemporary discussion on the epistemological and ontological effects of Big Data within social science, observing an increased focus on relationality and complexity, and a tendency to naturalize social phenomena. The epistemic limits of this emerging computational paradigm are outlined through a comparison with the discussions in the early days of digitalization, when digital technology was primarily seen through the lens of dematerialization, and as part of the larger processes of “postmodernity”. Since then, the online landscape has become increasingly centralized, and the “liquidity” of dematerialized technology has come to empower online platforms in shaping the conditions for human behavior. This contrast between the contemporary epistemological currents and the previous philosophical discussions brings to the fore contradictions within the study of digital social life: While qualitative change has become increasingly dominant, the focus has gone towards quantitative methods; while the platforms have become empowered to shape social behavior, the focus has gone from social context to naturalizing social patterns; while meaning is increasingly contested and fragmented, the role of hermeneutics has diminished; while platforms have become power hubs pursuing their interests through sophisticated data manipulation, the data they provide is increasingly trusted to hold the keys to understanding social life. These contradictions, we argue, are partially the result of a lack of philosophical discussion on the nature of social reality in the digital era; only from a firm metatheoretical perspective can we avoid forgetting the reality of the system under study as we are affected by the powerful social life of Big Data.

In the wake of the paleobiological revolution of the 1970s and 1980s, paleontologists continue to investigate far-reaching questions about how evolution works. Many of those questions have a philosophical dimension. How is macroevolution related to evolutionary changes within populations? Is evolutionary history contingent? How much can we know about the causes of evolutionary trends? How do paleontologists read the patterns in the fossil record to learn about the underlying evolutionary processes? Derek Turner explores these and other questions, introducing the reader to exciting recent work in the philosophy of paleontology and to theoretical issues including punctuated equilibria and species selection. He also critically examines some of the major accomplishments and arguments of paleontologists of the last 40 years.

Derek Turner, Professor of Philosophy, has written an introductory logic textbook that students at Connecticut College, or anywhere, can access for free. The book differs from other standard logic textbooks in its reliance on fun, low-stakes examples involving dinosaurs, a dog and his friends, etc. This work is published in 2020 under a Creative Commons AttributionNonCommercial-NoDerivatives 4.0 International License. You may share this text in any format or medium. You may not use it for commercial purposes. If you share it, you must give appropriate credit. If you remix, transform, add to, or modify the text in any way, you may not then redistribute the modified text.

This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if supertask computers are possible, this implies that arithmetical truth is determinate. In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks are of no help in explaining arithmetical determinacy.

Evolutionary Debunking Arguments argue that, if naturalism and evolution are true, then our moral beliefs are merely human constructs nature selected because they increased our prospects for survival and reproduction. Atheist Erik Wielenberg disagrees; he has recently argued that morality could be objectively real, and that we could have moral knowledge, even if naturalism and evolution are true. I argue that Wielenberg is unsuccessful in his attempt to deflect a major concern raised by EDA proponents, namely, that moral knowledge would be extremely unlikely given naturalism and evolution because it would involve a vast amount of unexplained lucky coincidences.

_Logic: The Basics_ is a hands-on introduction to the philosophically alive field of logical inquiry. Covering both classical and non-classical theories, it presents some of the core notions of logic such as validity, basic connectives, identity, ‘free logic’ and more. This book: introduces some basic ideas of logic from a semantic and philosophical perspective uses logical consequence as the focal concept throughout considers some of the controversies and rival logics that make for such a lively field This accessible guide includes chapter summaries and suggestions for further reading as well as exercises and sample answers throughout. It is an ideal introduction for those new to the study of logic as well as those seeking to gain the competence and skills needed to move to more advanced work in logic.

Dennett has maintained that a careful examination of our intuitive notion of qualia reveals that it is a confused notion, that it is advisable to accept that experience does not have the properties designated by it and that it is best to eliminate it. Because most scientists share this notion of qualia, the major line of attack of his project becomes that of raising objections against the ability of science to answer some basic questions about qualia. I try to show that science appeals to qualia and that it in fact adheres to a notion of qualia different from the one that Dennett has attributed to it. It is argued that qualia are amenable to scientific investigation and that this is the reason why science contributes toward the clarification of the notion of qualia. I also try to show that Dennett's skepticism about the abilities of science in answering questions posited by one of his thought experiments is unwarranted. I conclude that we need not accept Dennett's eliminativism about qualia.

There are researchers in cognitive science who use clinical and experimental evidence to draw some rather skeptical conclusions about a central feature of our conscious experience, its unity. They maintain that the examination of clinical phenomena reveals that human consciousness has a much more fragmentary character than the one we normally attribute to it. In the article, these claims are questioned by examining some of the clinical studies on the deficit of anosognosia. I try to show that these studies support a moderate sense of the unity of reflexive consciousness

Heisenberg’s uncertainty principle is a milestone of twentieth-century physics. We sketch the history that led to the formulation of the principle, and we recall the objections of Grete Hermann and Niels Bohr. Then we explain that there are in fact two uncertainty principles. One was published by Heisenberg in the Zeitschrift für Physik of March 1927 and subsequently targeted by Bohr and Hermann. The other one was introduced by Earle Kennard in the same journal a couple of months later. While Kennard’s principle remains untarnished, the principle of Heisenberg has recently been criticized in a way that is very different from the objections by Bohr and Hermann: there are reasons to believe that Heisenberg’s formula is not valid. Experimental results seem to support this claim.

This article focuses on the conceptual relationship between awe and the experience of the sublime. I argue that the experience of the sublime is best conceived as a species of awe, namely, as aesthetic awe. I support this conclusion by considering the prominent conceptual relations between awe and the experience of the sublime, showing that all of the options except the proposed one suffer from serious shortcomings. In maintaining that the experience of the sublime is best conceived as aesthetic awe, I draw from historical theories of the sublime as well as recent work in empirical psychology.