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)

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)

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)

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)

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.

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)

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)

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)

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential (...) theorems obtained by this elimination procedure. (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)

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)

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)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

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)

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)

In his account of the Persian Wars, the 5th century historian Herodotus reports an exchange between the Persian monarch Xerxes and a deposed Spartan king, Demaratus, who became what Lattimore later classified as a “tragic warner” to Xerxes. On the eve of the battle of Thermopylae, Xerxes asks how a small number of free Spartiates can stand up against the massive ranks of soldiers that Xerxes has assembled. Herodotus has Demaratus reply: So is it with the Lacedaemonians; fighting singly they (...) are as brave as any man living, and together they are the best warriors on earth. Free they are, yet not wholly free; for law is their master (ἐλεύθεροι γὰρ ἐόντες οὐ πάντα ἐλεύθεροι εἰσί· ἔπεστι γάρ σφι δεσπότης νόμος), whom they fear much more than your men fear you. This is my proof—what their law bids them, that they do; and its bidding is ever the same, that they must never flee from the battle before whatsoever odds, but abide at their post and there conquer or die. (Hdt. 7.104) As the outcome of Thermopylae makes clear, Demaratus is hardly guilty of boasting. But Greek political theorists were not only fascinated by the Spartans who fought at the battle of Thermopylae in 480; they also wondered how slightly more than a century later Sparta was ignobly defeated by Thebes at the battle of Leuctra in 371, after which Spartan hegemony collapsed. How did Sparta field citizen-soldiers in the 5th century who were without parallel in their bravery and military prowess? And why did Sparta, predominantly under the same political organization, collapse just over a century later? Such are the main questions that Xenophon, Plato, Aristotle, and Plutarch—what Cartledge (1999, 313–14) has characterized as “political-theoretical” laconophiles—seek to answer in their writings about Sparta. (shrink)

This booklet is a Korean adaptation and translation of Part VIII of forall x: Calgary (Fall 2021 edition), which is intended to be introductory material for modal logic. The original text is based on Robert Trueman's A Modal Logic Primer, which is revised and expanded by Richard Zach and Aaron Thomas-Bolduc in forall x: Calgary. (forall x: Calgary is based on forall x: Cambridge by Tim Button, which is in turn based on forall x by P. D. Magnus, and (...) also includes material from forall x: Lorain County Remix by Cathal Woods and J. Robert Loftis.) This booklet is licensed under a Creative Commons Attribution 4.0 license. (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.

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)

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)

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.

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)

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)

Numerous ancient sources attest that Artemisia of Halicarnassus, a fifth-century BCE tyrant whose polis came under Persian rule in 524 BCE, figures prominently in Xerxes’ naval campaign against Greece. At least since Pompeius Trogus’ first-century BCE Philippic History, interpretations of Artemisia have juxtaposed her “virile courage” (uirilem audaciam) with Xerxes’ “womanish fear” (muliebrem timorem) primarily as a means of belittling the effeminate non-Greeks. My paper argues that although Herodotus is aware of such interpretations of Artemisia, he depicts her primarily as (...) an excellent counsel, a woman who is not only brave in battle, but who is a wonder primarily because of her intellectual excellences in deliberative rhetoric and “geo-political” strategy in the Greco-Persian world. (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)

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.

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)

Aristotle’s Politics is a study of the political institutions of the 4th C. Mediterranean world, including both Greek communities (like Athens and Sparta) and non-Greek communities (like Persia and Carthage). The work is foundational for a number of modern scholarly disciplines such as political science, political theory, ancient history, and ancient philosophy; thus, the work annually is the subject of a robust number of scholarly studies (on average, about four monographs and 25 journal articles and book chapters per year). This (...) chapter reviews scholarship on the Politics, focusing on work produced since 2010. The chapter first reviews general introductions, editions, and interpretative strategies for understanding the Politics. It then reviews major subsections or conceptual issues within the Politics. The chapter concludes with a review of major trends in scholarship and suggestions for future scholarly directions in understanding Aristotle’s Politics. (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)

Martha Nussbaum has provided a sustained critique of Cicero’s De officiis (or On Duties), concerning what she claims is Cicero’s incoherent distinction between duties of justice, which are strict, cosmopolitan, and impartial, and duties of material aid, which are elastic, weighted towards those who are near and dear, and partial. No doubt, from Nussbaum’s cosmopolitan perspective, Cicero’s distinction between justice and beneficence seems problematic and lies at the root of modern moral failures to conceptualize adequately our obligations in situations of (...) famine and global inequality. And yet a careful reading of Cicero’s On Duties shows many duties that appear to be partial. It is hard to believe that Cicero’s discussion of these “partial duties” was a mere oversight or omission on his part. Rather, Cicero seemed to have no problem endorsing what I will call the “partial coherence” of duties in the work—namely, the claim that asserting that duties are exhaustively partial or impartial is a false dichotomy. With the help of conceptual analysis by Richard Kraut, my paper aims to establish and defend Cicero’s “partial coherence” against Nussbaum’s criticisms. (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 discussions of natural slavery, ‘barbarian kingship’, and the natural characteristics of barbarians or non-Greeks are usually read as calling into question the intellectual, ethical, and political accomplishments of non-Greeks. Such accounts of non-Greek inferiority or inability to self-govern also appear to presuppose a climatic or environmental account that on the whole would imply severe limitations on the possibility of political flourishing for peoples living outside the Greek Mediterranean basin. In light of such accounts, it is somewhat astounding to find (...) Aristotle praising the social and political institutions of Carthage within his discussion of the best constitution in the second book of the Politics. In the 4th century BCE, Carthage (modern day Tunis) was a Phoenician commercial empire that culturally and linguistically was Semitic and clearly non-Greek. That Aristotle contrasts Carthage, a non-Greek polis, favorably with both Sparta and Crete undermines both the claim that Aristotle embraced a categorical opposition between Greeks and non-Greeks and the claim that he was dismissive of the achievements of non-Greeks. (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)

In Thomas More’s Utopia, the character of Raphael Hythloday bestows upon the islanders of Utopia a library of Greek authors that includes Herodotus (alongside more traditional political thinkers such as Plato, Aristotle, and Thucydides). Herodotus’ inclusion on the Utopian reading list invites the question of whether his Histories is in any sense a work in utopian political theory. Although Herodotus is sometimes excluded from the canon of the Histories of political thought because of his lack of interest in political constitutions, (...) I argue: That Herodotus (like More) should be categorized as a political thinker because of his focus on political and social customs rather than constitutional arrangements. That Herodotus (similar to More) uses the otherness of political and social customs in distant societies as an opportunity to theorize and evaluate political institutions. And that Herodotus (like More) is a utopian political thinker in his evaluation of superlatively good political institutions and his embrace of incremental political reform. (shrink)

Cicero’s ethical and political writings present a detailed and sophisticated philosophy of just war, namely an account of when armed conflict is morally right or wrong. Several of the philosophical moves or arguments that he makes, such as a critique of “Roman realism” or his incorporation of the ius fetiale—a form of archaic international law—are remarkable similar to those of the contemporary just war philosopher Michael Walzer, even if Walzer is describing inter-state war and Cicero is describing imperial war. But (...) if it is clear that Walzer presents a detailed philosophy of just war, then I argue we should draw the same conclusion for Cicero. The result is a deeper appreciation of the insight and novelty of Cicero’s view of just war. The paper concludes by arguing against the claim that Cicero’s philosophy of just war is derivative from the Stoic philosopher Panaetius, whom Cicero drew upon in the organization of his On Duties. Just as Walzer’s Just and Unjust Wars was written in response to America’s war in Vietnam, Cicero’s just war philosophy was written in response to the wars (both civil and external) of Gaius Caesar. (shrink)

The purpose of Aristotle's discussion of political justice (τό πολιὸν[unrepresentable symbol]δν δί[unrepresentable symbol]αιον) in "EN" V.6-7 has been a matter of dispute. Although the notion of political justice which Aristotle seeks to elucidate is relatively clear, namely the notion of justice which obtains between free and equal citizens living within a community aiming at self-sufficiency under the rule of law, confusion arises when one asks how political justice relates to the other kinds of justice examined in "EN" V. Is political (...) justice a highly determinate subdivision of justice which Aristotle examines alongside the other varieties of particular justice analyzed in "EN" V.2-5? Or is political justice related to the analysis of ethical agency which follows in "EN" V.8-11? The question is complicated by the fact that the passage in question -- "EN" V 1134a17-1135a15 -- has occasioned much speculation about textual dislocations and has been incorporated into chapter divisions differently according to the two prevalent modern editorial divisions of the Ethics. To resolve these problems, I argue that Aristotle's account of political justice is situated within an extended aporetic analysis which begins in "EN" V.6 and extends through "EN" V.8. Aristotle introduces the notion of political justice within the extended analysis concerning the ascription of character states because calling someone just or unjust presupposes that the person is a fully mature ethical agent, but anyone capable of political justice possesses such agency. Once the extended argument in the second half of "EN" V is properly understood, it appears that the received text is not in need of emendation. To further support my claim that Aristotle's account of political justice introduces a new inquiry which is not analogous to the analyses of particular justice in the first half of "EN" V, I compare political justice to the other species of justice. (shrink)

In anticipation of updating annotated bibliographies on Aristotle’s Ethics and Politics for Oxford Bibliography Online, I have sought to keep a running tabulation of all books, edited collections, translations, and journal articles which are primarily devoted to Aristotle’s ethical and political writings (including their historical reception but excluding neo–Aristotelian virtue ethics). In general, criteria for inclusion in this bibliography are that the work be: (1) publication in a peer–reviewed or academic/university press between 2011–2020; (2) “substantially” devoted to one of Aristotle’s (...) ethical or political works (e.g., Pol, EN, EE, MM, 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.). I encourage scholars to alert me about published works that I have omitted or listed incorrectly. (shrink)

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)

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 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)

I have sought to keep a running tabulation of all books, edited collections, translations, and journal articles which are primarily devoted to Aristotle’s ethical and political writings (including their historical reception but excluding neo-Aristotelian virtue ethics). Criteria for inclusion in this bibliography are: (1) published after January 1, 2021 (including pre-publication articles assigned a DOI); (2) devoted to one of Aristotle’s ethical or political works (e.g., Pol, EN, EE, MM, 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.). I encourage scholars to alert me about published works that I have omitted or listed incorrectly. (shrink)

Aeschylus’ Persians dramatically represents the Athenian victory at Salamis from the perspective of the Persian royal court at Susa. Although the play is in some sense a patriotic celebration of the Athenian victory and its democracy, nonetheless in both form and function it is a tragedy that generates sympathy for the suffering of its main character, Xerxes. Although scholars have argued whether the play is primarily patriotic or tragic, I argue that the play purposively provides both patriotic and tragic elements (...) in such a fashion as to invite its audience to reflect on the ramifications of Persia’s failed empire for Athens’ own nascent Delian League, which even in 472 was showing imperialistic tendencies. I argue that Aeschylus’ political theory in Persians is aporetic in the sense that its careful balance between patriotism and anti-imperialism generates an impasse which members of the play’s audience are thus given the occasion to puzzle over. (shrink)

Topical bibliography of Aristotle's Nicomachean Ethics, organized by books/subjects within the Ethics. Includes editions and lexica for the study of Aristotle's Eudemian Ethics and Magna Moralia.

In Winter 2017, the first author piloted a course in formal logic in which we aimed to (a) improve student engagement and mastery of the content, and (b) reduce maths anxiety and its negative effects on student outcomes, by adopting student oriented teaching including peer instruction and classroom flipping techniques. The course implemented a partially flipped approach, and incorporated group-work and peer learning elements, while retaining some of the traditional lecture format. By doing this, a wide variety of student learning (...) preferences could be provided for. (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)

The relationship between nature and normativity in Aristotle’s practical philosophy is problematic. On the one hand, Aristotle insists that ethical virtue arises through the habitual repetition of ethically good actions, and thus no one is good or virtuous by nature. Phusikê aretê or “natural virtue” is more like cleverness (demotes) than prudence (phronêsis) and it can result in wrong actions. Yet on the other hand, at times Aristotle appears to use nature to justify normative claims. Thus the problem with Aristotle’s (...) use of the notion of nature in his practical philosophy is that in some instances what is natural seems to be ethically neutral, but in other instances it seems to set the standard for human perfection. Modern Aristotle exegetes have sometimes argued that Aristotle must, consciously or not, be using the term “nature” in at least two different senses here. To use the language of Julia Annas, when Aristotle talks about “natural virtue” in the sense of the talents or abilities which one possesses from birth, he has in mind the notion of “mere nature,” but when he discusses the perfections which a virtuous individual enjoys by his or her nature, Aristotle has in mind the notion of nature as an “ethical ideal.” I would like to suggest an alternative solution to the problem of reconciling Aristotle’s different senses of nature in his Ethics. In Nicomachean Ethics V.7, Aristotle claims that political justice {to dikaion politikon) possesses a “natural” {phusikon) part and a “conventional” {nomikori) part. Aristotle claims that both nature and convention admit of variation, and his language suggests that the two are ultimately parts which need to be interwoven or combined. Scholars who have struggled with Aristotle’s apparently disparate senses of the idea of nature have assumed that nature is an ethical ideal which can be separated from and serve as a guide for that which is merely conventional. I argue that when Aristotle invokes what is “natural” as a norm, he does so under the assumption that the natural component of a norm is ultimately separable from its conventional part only in abstraction. Just like the syllable “BA” is not reducible to the letters “b” and “a,” but instead is a whole greater than its two parts, political justice is a composite unity of nature and convention which, in unifying the two parts, transforms them. (shrink)

Scholars of race in antiquity commonly claim that Aristotle holds proto-racist views about βάρβαροι or non-Greeks. But a careful examination of Aristotle’s remarks in his Politics about slavery, non-Greek political institutions, and Greek and non-Greek natural qualities calls into question such claims. No doubt, Aristotle held views at odds with modern liberalism, such as defenses of gender subordination and the exploitation of slave and non-slave labor. But claims that Aristotle holds proto-racist views are regularly but erroneously asserted without careful consideration (...) of relevant textual evidence. My article argues that Aristotle neither categorically distinguishes Greeks and non-Greeks nor does he endorse the claim that Greeks are categorically superior to non-Greeks. Indeed, Aristotle regularly draws upon non-Greek political institutions in his own formulation of the best constitution and he praises the non-Greek constitution of Carthage as superior to that of Greek constitutions such as Sparta and Crete. (shrink)

Many of the philosophical treatises which Cicero wrote in the last years of his life quote, discuss, and debate the various doctrines and philosophical systems of the Hellenistic schools of philosophy. Although Cicero sometimes represents himself as only translating or reproducing Greek ideas for a Latin or Roman audience, his actual philosophical practice is much more subtle and varies from treatise to treatise. In some treatises, such as On the Nature of the Gods or Tusculan Disputations, Cicero incorporates the views (...) of the Hellenistic schools into a philosophical presentation oriented by the methodology of the New Academy, which sought to call into question the dogmas of those schools. In other treatises, such as On Duty, Cicero philosophizes in light of Hellenistic views, but he does so critically and discriminately, arriving at a novel philosophical position which is responsive to the intellectual predicaments occasioned by the collapse of the Roman Republic. (shrink)