I propose a novel metaphysical explanation of identity and distinctness facts called the Modal Proposal. According to the Modal Proposal, for each identity fact – that is, each fact of the form a=b – that fact is metaphysically explained by the fact that it is necessary that the entities involved are indiscernible, and for each distinctness fact –that is, each fact of the form a≠b – that fact is metaphysically explained by the fact that it is possible for the entities (...) involved to be discernible. I argue that the Modal Proposal has greater payoffs at less cost than any of its competitors. It gives simple, uniform, and intuitive explanations of identity and distinctness that conserve longstanding philosophical insights about identity that go back to Leibniz. It does this while making our fundamental base more parsimonious, determining whether controversial cases of identity or distinctness are possible, and expanding our understanding of these central philosophical relations. It does all this while avoiding controversial ontological commitments. (shrink)

Should we believe our controversial philosophical views? Recently, several authors have argued from broadly conciliationist premises that we should not. If they are right, we philosophers face a dilemma: If we believe our views, we are irrational. If we do not, we are not sincere in holding them. This paper offers a way out, proposing an attitude we can rationally take toward our views that can support sincerity of the appropriate sort. We should arrive at our views via a certain (...) sort of ‘insulated’ reasoning – that is, reasoning that involves setting aside certain higher-order worries, such as those provided by disagreement – when we investigate philosophical questions. (shrink)

Prevailing opinion—defended by Jason Brennan and others—is that voting to change the outcome is irrational, since although the payoffs of tipping an election can be quite large, the probability of doing so is extraordinarily small. This paper argues that prevailing opinion is incorrect. Voting is shown to be rational so long as two conditions are satisfied: First, the average social benefit of electing the better candidate must be at least twice as great as the individual cost of voting, and second, (...) the chance of casting the decisive vote must be at least 1/N, where N stands for the number of citizens. It is argued that both of these conditions are often true in the real world. (shrink)

There is a well-known moral quandary concerning how to account for the rightness or wrongness of acts that clearly contribute to some morally significant outcome – but which each seem too small, individually, to make any meaningful difference. One consequentialist-friendly response to this problem is to deny that there could ever be a case of this type. This paper pursues this general strategy, but in an unusual way. Existing arguments for the consequentialist-friendly position are sorites-style arguments. Such arguments imagine varying (...) a subject’s predicament bit by bit until it is clear that a relevant difference has been achieved. The arguments offered in this paper are structurally different, and do not rely on any sorites series. For this reason, they are not vulnerable to objections that have been leveled against the sorites-style arguments. (shrink)

What should a person do when, through no fault of her own, she ends up believing a false moral theory? Some suggest that she should act against what the false theory recommends; others argue that she should follow her rationally held moral beliefs. While the former view better accords with intuitions about cases, the latter one seems to enjoy a critical advantage: It seems better able to render moral requirements ‘followable’ or ‘action-guiding.’ But this tempting thought proves difficult to justify. (...) Indeed, whether it can be justified turns out to depend importantly on the rational status of epistemic akrasia. Furthermore, it can be argued, from premises all parties to the moral ignorance debate should accept, that rational epistemic akrasia is possible. If the argument proves successful, it follows that a person should sometimes act against her rationally held moral convictions. (shrink)

Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...) primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic ("Post-") and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged. (shrink)

This paper is about how to aggregate outside opinion. If two experts are on one side of an issue, while three experts are on the other side, what should a non-expert believe? Certainly, the non-expert should take into account more than just the numbers. But which other factors are relevant, and why? According to the view developed here, one important factor is whether the experts should have been expected, in advance, to reach the same conclusion. When the agreement of two (...) (or of twenty) thinkers can be predicted with certainty in advance, their shared belief is worth only as much as one of their beliefs would be worth alone. This expectational model of belief dependence can be applied whether we think in terms of credences or in terms of all-or-nothing beliefs. (shrink)

Initially, you judge that p. You then learn that most experts disagree. All things considered, you believe that the experts are probably right. Still, p continues to seem right to you, in some sense. You don’t yet see what, if anything, is wrong with your original reasoning. In such a case, we’ll say that you are ‘inclined’ toward p. This paper explores various roles that this state of inclination can play, both within epistemology and more broadly. Specifically, it will be (...) argued that: (i) inclinations can promote the accuracy of inquiring groups; (ii) they can support rational participation within philosophy despite pervasive disagreement; (iii) they allow us to make sense of an important way in which two people can continue to disagree even after they ‘conciliate’; (iv) inclinations carry information about individuals’ independent judgments and for this reason must be accounted for when updating on the opinions of others; (v) inclinations are connected to understanding in a way that belief is not; (vi) and awareness of the inclination-belief distinction enables us to respond to a provocative challenge purporting to show that critical thinking, or ‘thinking for oneself’, typically reduces expected accuracy and hence should be discouraged. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...) successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.

Conciliationism faces a challenge that has not been satisfactorily addressed. There are clear cases of epistemically significant merely possible disagreement, but there are also clear cases where merely possible disagreement is epistemically irrelevant. Conciliationists have not yet accounted for this asymmetry. In this paper, we propose that the asymmetry can be explained by positing a selection constraint on all cases of peer disagreement—whether actual or merely possible. If a peer’s opinion was not selected in accordance with the proposed constraint, then (...) it lacks epistemic significance. This allows us to distinguish the epistemically significant cases of merely possible disagreement from the insignificant ones. (shrink)

Six people are in trouble. We can save five of them or just the sixth. What should we do? John Taurek (1977) defends a radical view: We are not required to save the greater number. Taurek's paper has persuaded some. But even the unpersuaded agree that Taurek poses a deep and important challenge: From where does the priority of the many derive? It seems difficult, or even impossible, to convince someone who denies the importance of the numbers... to care about (...) the numbers. That’s what this paper aims to do. It will argue that the priority of the many follows, with minimal other assumptions, from something all should accept: the modest premise that if we can improve one person's chance of survival—without affecting anyone else—we should. (shrink)

David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...) little attention, and on a number of philosophical reconstructions of the viewpoint (in particular, by Hand, Kitcher, and Tait). (shrink)

In The Boundary Stones of Thought, Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

Roger White (2015) sketches an ingenious new solution to the problem of induction. He argues from the principle of indifference for the conclusion that the world is more likely to be induction- friendly than induction-unfriendly. But there is reason to be skeptical about the proposed indifference-based vindication of induction. It can be shown that, in the crucial test cases White concentrates on, the assumption of indifference renders induction no more accurate than random guessing. After discussing this result, the paper explains (...) why the indifference-based argument seemed so compelling, despite ultimately being unsound. (shrink)

In the Symposium, Plato has Socrates claim that the priestess Diotima once claimed that Eros is a lover of wisdom or someone who is “in between wisdom and ignorance. In fact, you see, none of the gods loves wisdom or wants to become wise—for they are wise—and no one else who is wise loves wisdom.” Perhaps the best starting point for understanding the philosophy of Aristotle is that in principle, he rejects Diotima’s etymological wordplay that claims that philosophy implies an (...) absence of wisdom. At the heart of Aristotle’s philosophy is the claim not only that all humans desire to know, but also that human animals (and indeed some non-human animals) are capable of understanding the world and its contents in a broad array of epistemic capacities, including veridical perception, artisanal expertise, theoretical science, practical wisdom, and, indeed, even σοφία itself of the highest objects of cognition, namely the nature of the gods (Metaph. 1.1.980a22–981b7). My chapter explores the philosophy of Aristotle in five different parts. Part I examines the “organon” or linguistic and logical tool-kit that analyzes human reasoning from the most basic components of a proposition up through syllogistic presentation of the causes that explain what research has uncovered. Parts II and III of the chapter explore the main contours of Aristotle’s “theoretical sciences”—what today we call natural science, metaphysics, and theology. Part IV of the chapter examines what the Nicomachean Ethics calls “the philosophy of human things,” namely the study of ethics and politics. Finally, Part V looks at how Aristotle and his school systemized two different fields of human study, namely that of artistic mimesis or the art of poetry, and deliberative, judicial, and epidictic oratory or the art of rhetoric. (shrink)

On some accounts of vagueness, predicates like “is a heap” are tolerant. That is, their correct application tolerates sufficiently small changes in the objects to which they are applied. Of course, such views face the sorites paradox, and various solutions have been proposed. One proposed solution involves banning repeated appeals to tolerance, while affirming tolerance in any individual case. In effect, this solution rejects the reasoning of the sorites argument. This paper discusses a thorny problem afflicting this approach to vagueness. (...) In particular, it is shown that, on the foregoing view, whether an object is a heap will sometimes depend on factors extrinsic to that object, such as whether its components came from other heaps. More generally, the paper raises the issue of how to count heaps in a tolerance-friendly framework. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of (...) the connectives in question. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

A textbook for modal and other intensional logics based on the Open Logic Project. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals.

This chapter describes Kurt Gödel's paper on the incompleteness theorems. Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Gödel's first incompleteness theorem showed that this assumption was false: it states that there are sentences of number theory that are (...) neither provable nor refutable. The first theorem is general in the sense that it applies to any axiomatic theory, which is ω-consistent, has an effective proof procedure, and is strong enough to represent basic arithmetic. Their importance lies in their generality: although proved specifically for extensions of system, the method Gödel used is applicable in a wide variety of circumstances. Gödel's results had a profound influence on the further development of the foundations of mathematics. It pointed the way to a reconceptualization of the view of axiomatic foundations. (shrink)

Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar (...) at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s. (shrink)

Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. Completeness of these tableaux, however, relies on the cut rule.

Eco-Rational Education proposes an educational response to climate change, environmental degradation, and desctructive human relations to ecology through the delivery of critical land-responsive environmental education. -/- The book argues that education is a powerful vehicle for both social change and cultural reproduction. It proposes that the prioritisation and integration of environmental education across the curriculum is essential to the development of ecologically rational citizens capable of responding to the environmental crisis and an increasingly changing world. Using philosophical analysis, particularly environmental (...) philosophy, pragmatism, and ecofeminism, the book develops an understanding of contemporary issues in education, especially inquiry-based learning as pedagogy, diversifying knowledge, environmental and epistemic justice, climate change education, and citizenship education. -/- Eco-Rational Education will be of interest to researchers and post-graduate students of social and political philosophy, educational philosophy, as well as environmental philosophy, ethics, and teacher education. (shrink)

Education as identity formation in Western-style liberal-democracies relies, in part, on neutrality as a justification for the reproduction of collective individual identity, including societal, cultural, institutional and political identities, many aspects of which are problematic in terms of the reproduction of environmentally harmful attitudes, beliefs and actions. Taking a position on an issue necessitates letting go of certain forms of neutrality, as does effectively teaching environmental education. We contend that to claim a stance of neutrality is to claim a position (...) beyond criticism. In the classroom this can also be an epistemically damaging position to hold. To further explore the problem of neutrality in the classroom, and to offer a potential solution, we will look to the philosophical community of inquiry pedagogy, and advocate for the addition of place-based education; a form of experiential education that promotes learning in local communities in which the school is situated, each with its own history, culture, economy and environment. However, how we understand ‘place’ is fundamental to understanding the potential of place-based education in giving students a ‘sense of place’—how they perceive a place, which includes place attachment and place meaning. To this end, we look to Indigenous understandings of Place and social reconstruction learning to inform place-based pedagogies. Doing so, we hold, opens a pathway to ethical education. (shrink)

At first glance, it looks like Aristotle can’t make up his mind about the ethical or moral status of non-human animals in his ethical treatises. Somewhat infamously, the Nicomachean Ethics claims that “there is neither friendship nor justice towards soulless things, nor is there towards an ox or a horse” (EN 8.11.1161b1–2). Since Aristotle thinks that friendship and justice are co-extensive (EN 8.9.1159b25–32), scholars have often read this passage to entail that humans have no ethical obligations to non-human animals. By (...) contrast, although the Eudemian Ethics denies that non-human animals can participate in “primary” friendships, EE 7.2 claims that “the other kinds of friendship are also found among animals; and it is evident that utility is present to some extent among them both in relation to humankind, in the case of tame animals, and in relation to each other” (EE 7.2.1236b3–11). Does the Nicomachean account of non-human animals contradict that of the Eudemian Ethics? Ultimately, I believe the Nicomachean account is consistent with the Eudemian account. Nonetheless, I argue that Aristotle’s treatment of non-human animals differs significantly in the two texts. My chapter explores this difference in greater detail and considers the ramifications of such a difference for our understanding of Aristotle’s place in the philosophical tradition concerning the ethical status of non-human animals. (shrink)

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.

Arguably the foundational text of Western political theory, Aristotle's Politics has become one of the most widely and carefully studied works in ethical and political philosophy. This volume of essays offers fresh interpretations of Aristotle's key work and opens new paths for students and scholars to explore. The contributors embrace a variety of methodological approaches that range across the disciplines of classics, political science, philosophy, and ancient history. Their essays illuminate perennial questions such as the relationship between individual and community, (...) the nature of democratic deliberation, and how to improve political institutions. Offering groundbreaking studies that both set Aristotle within the context of his own time and draw on contemporary discussion of his writings, this collection will provide researchers with an understanding of many of the major scholarly debates surrounding this key text. (shrink)

I have sought to keep a running tabulation of all books, edited collections, translations, and journal articles that are substantially devoted to Aristotle’s ethical and political writings (excluding their historical reception and neo-Aristotelian virtue ethics, both of which are subjects that require their own bibliographies). Criteria for inclusion in this bibliography are: (1) published after January 1, 2011 and before December 31, 2020 in a peer-reviewed journal or volume published by an academic press; (2) devoted to one of Aristotle’s ethical (...) or political works (e.g., Politics, Nicomachean Ethics, Eudemian Ethics, Magna Moralia, Oeconomica, Athenian Constitution, Protrepticus); and/or (3) devoted to ethical or political concepts examined elsewhere in Aristotle’s corpus (e.g., Rhetoric, Poetics, zoological treatises, etc.). Although this bibliography includes primarily Anglophone scholarship, it aims to include scholarship written in all modern languages. I encourage scholars to notify me about published works that I have omitted or listed incorrectly. (shrink)

This paper introduces Conceptual Intelligence (CI), a novel theory of cognition that positions the capacity for boundary creation and revision as the fundamental characteristic of intelligent systems. Unlike traditional computational models that treat intelligence as symbol manipulation within fixed frameworks, CI proposes that genuine intelligence lies in the ability to generate, interrogate, and evolve the conceptual boundaries that structure perception and reasoning. Drawing on insights from philosophy of mind, cognitive science, and artificial intelligence, this theory offers both a descriptive account (...) of human cognitive development and a prescriptive framework for designing adaptive artificial systems. The implications extend across education, scientific discovery, creative expression, and the development of artificial general intelligence, suggesting that boundary fluidity—rather than computational power—may be the key to understanding and creating truly intelligent systems. (shrink)

The opening words of the second book of Aristotle’s Nicomachean Ethics are as familiar as any in his corpus: Excellence of character results from habituation [ethos]—which is in fact the source of the name it has acquired [êthikê], the word for ‘character-trait’ [êthos] being a slight variation of that for ‘habituation’ [ethos]. This makes it quite clear that none of the excellences of character [êthikê aretê] comes about in us by nature; for no natural way of being is changed through (...) habituation [ethizetai]. Equally familiar, unfortunately, is the characterization of Aristotle’s notion of character formation as a form of habituation by means of the repetition of actions which results in a “habit” in the same way that a weight lifter produces muscles through the repetition of exertions. As a 19th century commentator remarked on the passage above, “[insert Grant].” From a Socratic perspective, such a view of becoming good seems hopelessly rigid and unconnected to the intellectual development which knowledge of the good requires. Habit and habituation in Aristotle seem eminently familiar and eminently unphilosophical. Such a view would be mistaken on at least three counts. First, the notion of character formation (to use the broadest possible term for the phenomenon of habituation) in Aristotle is significantly more complicated than the notion that through habituation one develops good habits which are what we mean by ethical virtue. Although character formation includes the development of proper emotional responses, such as taking pleasure in what is fine and being repulsed by what is shameful, it is equally concerned with cognitive development independent of the intellectual virtues. Second, although Aristotle’s terms for “ethics” (êthica), character-trait (êthos), and habituation (ethos, ethismos, or ethizetai) are linguistically and conceptually interrelated, his notion of “ethical state” (hexis) is both linguistically and conceptually quite distinct from the notion of “habit,” at least as we use that term today. As one Aristotle translator has put it, “A hexis is not only not the same thing as a habit, but is almost exactly its opposite.” For Aristotle, a hexis is a dynamic equilibrium which, although always productive of virtuous actions, is nonetheless the basis for being virtuous in varied circumstances. Thirdly, once Aristotle’s notion of a character state is retrieved from its false association with “habit” and repetitive habituation, one sees that its apparent divorce from practical reason is more a fixture of Aristotle’s analytical method and its connotations of inflexibility or fixedness are in fact antithetical to Aristotle’s description of ethical virtue. Rather than view ethical “character” in its Greek etymological sense as an indelibly fixed or engraved mark or stamp (charactêr) upon one’s soul, Aristotle’s notion of ethical character (êthos) or virtue (aretê) captures the notion of a virtuoso who is responsive in an excellent fashion to what reason perceives in particular and changing circumstances. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

One tradition in relevant and paraconsistent logics has been to develop sys- tems intended for applications to arithmetic and computability theory. The aspiration, as in Meyer [38] and others, is to recover enough working mathe- maticsforrealcomputation, butwithoutthelimitativeresultsofTuring, Gödel, etc.; or more cautiously, as in Dunn [22], to respect relevance and with that be insulated against the possibility of a genuine inconsistency. We distill these goals into guiding questions, and study the options for logics within a range of relevant systems. We (...) focus on strong truth functional logics RM3and PAC [6] and their expansions, with application to inconsistent arithmetics [62, 63]. We argue that this approach, while having many virtues, does not fully answer our guiding questions. This points to weak relevant logics like Routley/Sylvan’s DKQ [54], Brady’s MCQ [14], and Logan and Boccuni’s DL2Qt,f c [31]. The recurring theme is that paraconsistent computability struggles with functional- ity [17, 41, 43]. A method for advancing on the ‘function problem’ is sketched with Kleene’s theorem as a worked example. (shrink)

Place-based education is founded on the idea that the student’s local community is one of their primary learning resources. Place-based education’s underlying educational principle is that students need to first have an experiential understanding of the history, culture, and ecology of the environment in which they are situated before tackling broader national and global issues. Such attempts are a step in the right direction in dealing with controversial issues in a democracy by providing resources for synthesising curriculum though theory (curriculum (...) content, including traditional subjects) and practice (practical learning as experiential education). Nevertheless, many place-based accounts discount Indigenous conceptions of place, which are integral to Indigenous ways of knowing, being and doing, especially in relation to the ontological relationship to Land and Country. Indigenous philosophy has the potential to inform what place-based education means by students developing a ‘sense of place’ when engaging with local communities as their ‘own’ place. In this article we re-imagine the community of inquiry as a place-responsive pedagogy by learning from Indigenous philosophy, and recentring Indigenous notions of place in experiential learning. In addition to exploring the pedagogical potential of a place-responsive account of the community of inquiry, we look to the implications for teacher preparation. (shrink)

I argues that the claim made in the Aristotelian Oeconomica 1.2 (1343a28–31) that farming is just is a point about what Aristotle calls justice as the whole of virtue (EN 5.1.1129b11–19), namely a notion of justice that embodies what is lawful rather than a notion of particular justice within the framework of human exchanges. The first part of the chapter shows why it is problematic to interpret Oec. 1.2 as a claim about particular justice: whereas Aristotelian particular justice consists in (...) finding the mean between gain and loss in inter-personal (or pros heteron) exchanges between two or more parties, Oec. 1.2 juxtaposes farming with all human exchanges, including those that Aristotle clearly recognizes as just. Indeed, Oec. 1.2. seems to claim that farming is just precisely because it is not involved with other humans. The second part of the chapter considers whether one can illuminate Oec. 1.2 by means of Xenophon's account of the justice of farming in his Oeconomicus (e.g., 5.12); but careful examination of what “the earth teaches” the farmer for Xenophon is a function ultimately of care (epimeleia) rather than the exchange-based model of justice assumed in Oec. 1.2. The third part of the chapter proposes an alternative basis for understanding the text, namely that Oec. 1.2 ultimately assumes the framework of what Aristotle calls universal or complete justice, much like one finds in Hesiod’s account of divine justice in relationship to work and agriculture (e.g. WD 230–243). Such a solution views the farmer in an exchange with the earth, but Aristotle usually (for instance, in Politics 1.8–11) characterizes such an exchange as either in accord with nature or contrary to nature (e.g., Pol. 1.10.1258a35–b5), rather than in terms of what is just or unjust. The chapter suggests that the Aristotelian Oec. 1.2 thus shows that claims about “natural” and “unnatural” forms of property appropriation have some affinity with the theocentric framework of Hesiod’s account of justice. (shrink)

The second book of Aristotle’s Politics is generally taken to examine politeiai or constitutions that either exist in cities that are said to be well governed or were proposed by theoreticians and are thought to be well organized (II.1, 1260b30–32; II.12, 1274b26–28). Prominent are Aristotle’s examinations of Plato’s Republic and the constitution of Sparta; but Aristotle also devotes chapters to the examination of Plato’s Laws, the proposed constitutions of Phaleas of Chalcedon and Hippodamos of Miletus, and the existing constitutions of (...) Crete, Carthage, and Solon’s Athens. Prominent, also, is scholarly analysis of Aristotle’s remarks about the Republic and Sparta – whether his criticisms of Plato are insightful and fair or whether his remarks about Sparta are historically accurate and hit upon the causes of Sparta’s fourth century decline. Far less prominent, though, is scholarly discussion of Aristotle’s remarks about the other constitutions analyzed in Politics II, how Politics II as a whole is organized, or what light – if any – the book sheds on the Politics as a whole. Such scholarly neglect is unfortunate for at least two major reasons. First, Politics II claims that since the Politics proposes to study which political community is best of all for people who are most able to live in accord with one’s prayers (kat’ euchên), it is necessary to investigate other constitutions reputed to be best (II.1, 1260b27–29). Such a claim suggests a close link between Politics II and Politics VII/VIII but such a link needs elaboration. In what way does Politics II prepare the way for Politics VII/VIII? Secondly, Politics II contains not only criticism of other constitutions, but also substantive political theorizing – most noticeably about the organization of property in the best regime, the problem of political innovation, and the relationship between political theorizing and chance. Aristotle’s critiques of Plato’s Republic, the constitution of Hippodamos, and the constitution of Carthage generate a series of substantive theoretical reflections about the problematic nature of political innovation and political theorizing that rise above mere examinations of the views of his predecessors. Viewing Politics II solely as a repository of criticism loses sight of the theoretical insights Aristotle offers therein, self-conscious insights about the very nature of political theorizing. (shrink)

Aristotle's account of natural slavery appears to be internally inconsistent concerning whether slavery is advantageous to the natural slave. Whereas the Politics asserts that slavery is beneficial to the slave, the ethical treatises deny such a claim. Examination of Aristotle's arguments suggests a distinction which resolves the apparent contradiction. Aristotle distinguishes between the common benefit between two people who join together in an association And the same benefit which exists between a whole and its parts. Master and slave share no (...) common benefit, but instead the slave receives the same benefit a master does, albeit only through participation in the master as a part within a whole. Although Aristotle's distinction hardly justifies his doctrine of slavery, it saves Aristotle from one alleged internal inconsistency and sheds light on what Aristotle means by association and the common good. (shrink)

At least since Foster (2002), scholars interested in Aristotle’s views about environmental ethics have focused primarily upon his teleological account of non-human animals as the basis for an Aristotelian environmental virtue ethics. But although Aristotle’s scientific account of non-human animals can serve as the basis for a form of environmental ethics akin to “nature preservation,” one finds in his account of “household management” (or oikonomikē) a very different sort of environmental ethic, one that looks much more like a form of (...) “nature conservation.” My paper argues that for Aristotle, oikonomikē is indeed both “environmental” and an “ethic.” The first part of my chapter shows that oikonomikē is fundamentally environmental insofar as it concerns what Aristotle in Politics 1.8 calls “property acquisition in accord with nature” or “true wealth” (Pol. 1.8.1256b26–27, 30–31), namely the prudential use of natural resources in a way that recognizes the dependence of ways of life upon different habitats or environments. The second part of my chapter shows that oikonomikē is also an “ethic” insofar as it embeds three different kinds of limit on the use of animate and inanimate natural resources, namely that natural resource use is subordinate to expertise about the happiness of the political community (or what Aristotle calls “politics” [πολιτική]); that natural resource use, as an instrumental good, is limited and governed by the craft or expertise of oikonomikē; and that notions of unlimited commercial “wealth acquisition” (χρηματιστική) are based on ethically problematic ways of life. After showing that Aristotle’s account of oikonomikē is indeed an environmental ethic, the third part of my chapter considers the objection that oikonomikē is a weak environmental ethic insofar as it is exploitatively or axiological anthropocentric. I respond that Aristotle’s account of non-human animal domestication provides conceptual resources for thinking about non-human animals as subordinate members in one’s ecological-political community, especially with respect to his invocation of the craft analogy (in the example of shepherds) and inter-species utility friendships. Although Aristotle has multiple sources relevant for understanding environmental ethics, his account of oikonomikē is both a neglected source and a potentially fruitful one. (shrink)

Scholars have disagreed on whether Cicero’s De Amicitia is a philosophically serious or even coherent work. Such criticisms, I believe, can be met by an examination of the successive accounts of friendship that the character of Gaius Laelius provides in the dialogue. I argue that the dialogue offers three such accounts of friendship which taken together provide a comprehensive and coherent account of friendship. Further, I defend Cicero’s account against criticisms that Aulus Gellius had raised in the 2nd century CE (...) (criticisms that have been repeated to the modern day). The problem of defining friendship is a thread that brings unity to the entire dialogue and shows Cicero philosophizing about both the origins of friendship and its dissolution, especially when the bonds of friendship are in tension with obligations to the political community. (shrink)

Although arguments for and against competing theories of vagueness often appeal to claims about the use of vague predicates by ordinary speakers, such claims are rarely tested. An exception is Bonini et al. (1999), who report empirical results on the use of vague predicates by Italian speakers, and take the results to count in favor of epistemicism. Yet several methodological difficulties mar their experiments; we outline these problems and devise revised experiments that do not show the same results. We then (...) describe three additional empirical studies that investigate further claims in the literature on vagueness: the hypothesis that speakers confuse ‘P’ with ‘definitely P’, the relative persuasiveness of different formulations of the inductive premise of the Sorites, and the interaction of vague predicates with three different forms of negation. (shrink)

In Nicomachean Ethics V.6 Aristotle contrasts political justice with household justice, paternal justice, and despotic justice. My paper expands upon Aristotle’s sometimes enigmatic remarks about political justice through an examination of his account of justice within the oikia or ‘household’. Understanding political justice requires explicating the concepts of freedom and equality, but for Aristotle, the children and wife within the household are free people even if not citizens, and there exists proportionate equality between a husband and wife. Additionally, Aristotle’s articulation (...) and defence of political justice arises out of his examination of despotic justice in the first book of the Politics. Not only are the polis and the oikia similar insofar as they are associations, but Nicomachean Ethics VIII.9–11 suggests they are even isomorphic with respect to justice and friendship. Thus, in this paper I explore the relationships between father and son, husband and wife, master and slave, and between siblings in order to see what they tell us about Aristotle’s understanding of freedom, equality, and justice. (shrink)

Although there is much scholarship on Aristotle’s account of friendship (φιλία), almost all of it has focused on inter-personal relationships between human animals. Nonetheless, in both Aristotle’s ethical and zoological writings, he documents the intra- and inter-species friendship between many kinds of animals, including between human and non-human animals. Such non-human animal friendships establish both an indirect basis for establishing moral ties between humans and non-human animals (insofar as we respect their capacity to love and befriend others) and a direct (...) basis for establishing such ties (insofar as Aristotle provides a framework for thinking about utility and pleasure friendships between human and non-human animals). My paper defends Aristotle’s limitation of inter-species friendships to only utility and pleasure friendships and responds to scholars who claim that Aristotle recognizes no moral ties between human and non-human animals. (shrink)

It gives me great pleasure and honor to introduce myself as the incoming Editor-in-Chief of Polis: The Journal for Ancient Greek and Roman Political Thought. For the last decade I have served as an Associate Editor and the Book Review Editor of the journal. I am very excited about charting new paths for the journal, while continuing to publish first-rate scholarship in our area strengths. Although ‘polis’ is a Greek word that identifies a specific Greek historical political institution, in many (...) important ways the culture of Greek political thought interacted with the political culture of Regal, Republican, and Imperial Rome through Greek settlements in the Western Mediterranean, Roman military conquest, and cultural Hellenization. In the last year the editorial staff of Polis decided to expand the domain of our journal to include scholarship on Roman political thought and Greek-language political thought from the Hellenistic and Roman Imperial periods. Although Polis has been publishing reviews of books devoted to these new domains for several issues, it is quite gratifying to publish an entire special issue, edited by Grant Nelsestuen and Associate Editor Daniel Kapust, devoted to Roman political thought. Additional articles on Roman political thought are in the journal’s publication queue and we look forward to becoming an important venue for the publication of original scholarship and book reviews on Roman political thought and its subsequent reception, along with our original remit for Greek political thought and its reception. The editorial staff of Polis has undergone a number of changes in conjunction with my new role. First and foremost, my mentor and predecessor Kyriakos Demetriou has retired from the position of Editor-in-Chief and become a member of our Editorial Board. I have had the privilege of working with Kyriakos for over 15 years, during which time he transformed the newsletter of the Society for Ancient Greek Political Thought into a pre-eminent peer-reviewed academic journal. I may only hope 15 years from now, that Polis has continued to grow as a source for landmark scholarship on Greek and Roman political thought. During the editorial transition, Associate Editors Rosanna Lauriola and Essam Safty have left our editorial staff. We are grateful for their service to the journal and wish them well in their scholarly endeavors. Finally, I am pleased to announce several additions to our editorial staff. Dimitri El Murr, Professor of Ancient Philosophy at the Ecole Normale Supérieure (Université Paris Sciences et Lettres), has joined the journal as an Associate Editor responsible for the refereeing and editing of Francophone submissions to Polis. Demetra Kasimis, Assistant Professor of Political Science at the University of Chicago, has joined as an Associate Editor responsible for the refereeing of submissions on the political aspects of Greek literature (such as epic, tragedy, and comedy). Matthew Simonton, Associate Professor of History at Arizona State University, has joined the journal as an Associate Editor and Book Review Editor. Finally, Dr. Alexandra Wilding has joined our editorial staff as the journal’s copy-editor. Learn from and enjoy the scholarship of our current issue. Submit to Polis your scholarship on ancient Greek and Roman political thought. Be a part of the future of Polis. Thornton Lockwood Editor-in-Chief. (shrink)

The closing chapters of Aristotle’s Nicomachean Ethics x are regularly described as “puzzling,” “extremely abrupt,” “awkward,” or “surprising” to readers. Whereas the previous nine books described—sometimes in lavish detail—the multifold ethical virtues of an embodied person situated within communities of family, friends, and fellow-citizens, NE x 6-8 extol the rarified, god-like and solitary existence of a sophos or sage (1179a32). The ethical virtues that take up approximately the first half of the Ethics describe moral exempla who experience fear fighting for (...) their communities, are sensitive to the esteem and recognition of others, and feel love for a desire to live together with a wide variety of kinds of friends. Such good people take pleasure in prudently expending sums to improve their communities—communities in which they exchange goods and participate in ruling and being ruled in a cooperative fashion. The exemplum of x 7-8, by contrast, is a person whose activity consists almost entirely in exercising his or her mind (nous)—a part of one’s soul that Aristotle explicitly notes is disconnected from human emotions and that can be exercised, insofar as one as wise, in a wholly solitary fashion (1178a15-16, a19-20; 1177a33-34). Although Aristotle’s claim that a “life in accord with the mind” is best and most pleasant (1178a6-7) may jar the intuitions of many people—he himself endorses Anaxagoras’ claim that the happy person will appear as “someone who is absurd” (atopos, 1179a15) to most people—it is false to claim that his conclusions in NE x 6-8 are unexpected or unanticipated, or that the text is in any way discontinuous with what precedes it. The Nicomachean Ethics exhibits aspects of ring-composition both within the work as a whole and within book 10. In NE i 5 Aristotle introduces his rendition of a trope that he inherits (most immediately) from Plato, viz. that the problem of the good can be considered like a contest between three different kinds of life—the life of enjoyment (apolaustikos), the life of politics (politikos), and the life of contemplation (theôrêtikos). NE x 6-8 returns to that contest, rendering a verdict about which life takes first, second, and third place. As the opening lines of NE x 6 note, that verdict presupposes central claims articulated between the first and last books of the Ethics, specifically about the nature of the virtues, the nature of friendship, and the nature of pleasure. NE i 5 and x 6-8 thus serve as “bookends” which encase arguments and proposition located between them. Book 10 exhibits a similar ring-composition. NE x opens with methodological reflection on the gap between the theoretical positions that thinkers articulate and the way they live their lives (especially in the case of scolds who simultaneously criticize pleasure on a scientific level but seek it on a practical level [1172a33-b8]). Chapter 8 concludes with an explicit reiteration of the methodological point, a reiteration that underscores how Aristotle’s treatment of the contest of lives incorporates central aspects of the discussion of pleasure in NE x 1-5. Aristotle’s reiteration of the problem of a gap between theory and practice also draws attention to the centrality of his treatment of pleasure in his adjudication of the three lives. Although the life of pleasure takes a distant third in the contest of lives, the contemplative life wins the contest in part because it itself is the most pleasant form of life. The central philosophical problem looming behind Aristotle’s treatment of the contest of lives concerns the relationship between the notion of activity (energeia) and the notion of a way of life (bios). In the last four decades, much of the scholarship on NE x 6-8 has sought to address the question of what activities a certain way of life includes or excludes. “Monistic” or “dominant” end interpretations of the best life have viewed it as including only contemplative activities whereas “inclusivist” end interpretations of the best life have viewed it as including non-contemplative activities. Although this chapter largely side-steps this debate—partially because there are ample recent first-rate introductory treatments of the debate, partially because I have addressed the question elsewhere in my own writing —I argue that focusing solely on the contest between the two best ways of life overshadows the way that Aristotle incorporates insights from the third way of life into the best way of life. Although the notion of a contest among different kinds of lives presupposes that the lives are mutually exclusive, Aristotle has no problem saying that the contemplative life trumps the life of pleasure because the contemplative life is more pleasant. I proceed in two parts. In the first part I examine what I will call the “outer” ring of the Ethics, namely the relationship between the contest of lives proposed in NE i 5 and its overall resolution in x 6-8. I show how i 5 and x 6 establish the framework for the contest, how x 7 adjudicates that contest in large part by relying upon premises articulated in the interim between books 1 and 10, and how x 6 and x 8 determine the second and third place positions in the contest. In the second part I focus on what I will call the “inner” ring of NE x. More than half of NE x struggles with the nature of pleasure, its role in a happy life, and the methodological problems of examining pleasure within a practical science. I show how x 1 and x 8 are “bookends” that underscore the methodological problem of examining pleasure and how the contest of lives draws upon the proximate conclusions concerning the nature of pleasure in NE x 1-5. (shrink)

In numerous places in his Ethics and Politics, Aristotle associates political justice (or ruling in turns) and the regime of polity. I argue that there is a necessary connection between political justice and polity due to their origins in political mixing. Aristotle is the first to discover political justice and polity because his predecessors had thought that the elements which they combine -- excellence and equality in the case of political justice, and oligarchy and democracy in the case of polity (...) -- were antithetical. The novelty of Aristotle's 'discoveries' points to their connection, namely that both originate in the political mixing of elements. This article examines such political mixing in detail and shows how an institutional arrangement such as ruling in turns can be adapted to different regime-types. (shrink)

The idea of a paraconsistent computability theory has been proposed as a way to work effectively with inconsistent sets of numbers. The viability of such a theory, though—the very coherence of the idea of an ‘inconsistent recursive relation’—has been called into doubt, most recently in (Choi, Synthese 200(5):418, 2022). In this paper we remove some doubt, by setting out a simple model of (naïve) set theory in LP, showing how to compute inconsistent sets in terms of extensions and antiextensions, and (...) establishing further representability results. This suggests a way that a longstanding and apparently impossible-to-answer question—how can inconsistency be computed?—can be answered. (shrink)