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)

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)

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)

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)

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)

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)

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)

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)

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)

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.

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.

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)

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.

If the reliability of a source of testimony is open to question, it seems epistemically illegitimate to verify the source’s reliability by appealing to that source’s own testimony. Is this because it is illegitimate to trust a questionable source’s testimony on any matter whatsoever? Or is there a distinctive problem with appealing to the source’s testimony on the matter of that source’s own reliability? After distinguishing between two kinds of epistemically illegitimate circularity—bootstrapping and self-verification—I argue for a qualified version of (...) the claim that there is nothing especially illegitimate about using a questionable source to evaluate its own reliability. Instead, it is illegitimate to appeal to a questionable source’s testimony on any matter whatsoever, with the matter of the source’s own reliability serving only as a special case. (shrink)

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)

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)

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)

A natural view of testimony holds that a source's statements provide one with evidence about what the source believes, which in turn provides one with evidence about what is true. But some theorists have gone further and developed a broadly analogous view of memory. According to this view, which this essay calls the “diary model,” one's memory ordinarily serves as a means for one's present self to gain evidence about one's past judgments, and in turn about the truth. This essay (...) rejects the diary model's analogy between memory and testimony from one's former self, arguing first that memory and a diary differ with respect to their psychological roles, and second that this psychological difference underwrites important downstream epistemic differences. The resulting view stands opposed to prominent discussions of memory and testimony, which either, like the diary model, treat memory by analogy to what we naively wish to say about testimony, or which instead attempt to extend to testimony the epistemically preservative role of memory. (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)

This paper critically examines currently influential transparency accounts of our knowledge of our own beliefs that say that self-ascriptions of belief typically are arrived at by “looking outward” onto the world. For example, one version of the transparency account says that one self-ascribes beliefs via an inference from a premise to the conclusion that one believes that premise. This rule of inference reliably yields accurate self-ascriptions because you cannot infer a conclusion from a premise without believing the premise, and so (...) you cannot infer from a premise that you believe the premise unless you do believe it. I argue that this procedure cannot be a source of justification, however, because one can be justified in inferring from p that q only if p amounts to strong evidence that q is true. This is incompatible with the transparency account because p often is not very strong evidence that you believe that p. For example, unless you are a weather expert, the fact that it will rain is not very strong evidence that you believe it will rain. After showing how this intuitive problem can be made precise, I conclude with a broader lesson about the nature of inferential justification: that beliefs, when justified, must be underwritten by beliefs, when justified, must be underwritten by evidential relationships between the facts or propositions which those beliefs represent. (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)

Self-verifying judgments like _I exist_ seem rational, and self-defeating ones like _It will rain, but I don’t believe it will rain_ seem irrational_._ But one’s evidence might support a self-defeating judgment, and fail to support a self-verifying one. This paper explains how it can be rational to defy one’s evidence if judgment is construed as a mental performance or act, akin to inner assertion. The explanation comes at significant cost, however. Instead of causing or constituting beliefs, judgments turn out to (...) be mere epiphenomena, and self-verification and self-defeat lack the broader philosophical import often claimed for them. (shrink)

This chapter defines "epistemology," introduces the key epistemological questions, and briefly outlines how the field has evolved over time. It serves as the introduction to the edited collection, Introduction to Philosophy: Epistemology (a volume in the Introduction to Philosophy open textbook series edited by Christina Hendricks).

Is self-knowledge a requirement of rationality, like consistency, or means-ends coherence? Many claim so, citing the evident impropriety of asserting, and the alleged irrationality of believing, Moore-paradoxical propositions of the form < p, but I don't believe that p>. If there were nothing irrational about failing to know one's own beliefs, they claim, then there would be nothing irrational about Moore-paradoxical assertions or beliefs. This article considers a few ways the data surrounding Moore's paradox might be marshaled to support rational (...) requirements to know one's beliefs, and finds that none succeed. (shrink)

I argue that, unlike your brain, you are not composed of other things: you are simple. My argument centers on what I take to be an uncontroversial datum: for any pair of conscious beings, it is impossible for the pair itself to be conscious. Consider, for instance, the pair comprising you and me. You might pinch your arm and feel a pain. I might simultaneously pinch my arm and feel a qualitatively identical pain. But the pair we form would not (...) feel a thing.1 Pairs of people themselves are incapable of experience. Call this The Datum. What explains The Datum? I think the following exhaust the reasonable options. (1) Pairs of people lack a sufficient number of immediate parts. (2) Pairs of people lack immediate parts capable of standing in the right sorts of relations to each other and their environment. (3) Pairs of people lack immediate parts of the right nature. (4) Pairs of people are not structures (they are unstructured collections of their two immediate parts). (5) Some combination of (1) – (4). Finally, (6) pairs of people are not simple. (shrink)

An internalist slogan says that justification depends on internal factors. But which factors are those? This paper examines some common motivations favoring internalism over externalism, and says they are compatible with including dispositional and even past mental states in the internal.

According to a traditional Cartesian epistemology of perception, perception does not provide one with direct knowledge of the external world. Instead, your immediate perceptual evidence is limited to facts about your own visual experience, from which conclusions about the external world must be inferred. Cartesianism faces well-known skeptical challenges. But this chapter argues that any anti-Cartesian view strong enough to avoid these challenges must license a way of updating one’s beliefs in response to anticipated experiences that seems diachronically irrational. To (...) avoid this result, the anti-Cartesian must either license an unacceptable epistemic chauvinism, or else claim that merely reflecting on one’s experiences defeats perceptual justification. This leaves us with a puzzle: Although Cartesianism faces problems, avoiding them brings a new set of problems. (shrink)

A common view among nontheists combines the de jure objection that theism is epistemically unacceptable with agnosticism about the de facto objection that theism is false. Following Plantinga, we can call this a “proper” de jure objection—a de jure objection that does not depend on any de facto objection. In his Warranted Christian Belief, Plantinga has produced a general argument against all proper de jure objections. Here I first show that this argument is logically fallacious (it makes subtle probabilistic fallacies (...) disguised by scope ambiguities), and proceed to lay the groundwork for the construction of actual proper de jure objections. (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)

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.

이 책자는 형식 논리의 일종인 양상논리에 입문하고 싶으신 분들을 위한 짧은 교재입니다. “양상논리 맛보기” 라는 말마따나, 이 책자는 양상논리에 관심은 있지만 아직 본격적으로 공부를 시작하진 않은 분들께서 ‘맛보기’를 하기에 적합한 안내 책자입니다. 아무쪼록 이 책자가 양상논리를 공부해나가시는데 유용한 첫 발판이 될 수 있기를 바랍니다. / 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)

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)

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)

According to the traditional analysis of propositional knowledge (which derives from Plato's account in the Meno and Theaetetus), knowledge is justified true belief. This chapter develops the traditional analysis, introduces the famous Gettier and lottery problems, and provides an overview of prospective solutions. In closing, I briefly comment on the value of conceptual analysis, note how it has shaped the field, and assess the state of post-Gettier epistemology.

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)

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on (...) the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)

On an optimistic version of realist moral epistemology, a significant range of ordinary moral beliefs, construed in realist terms, constitute knowledge—or at least some weaker positive epistemic status, such as epistemic justification. The “debunking challenge” to this view grants prima facie justification but claims that it is “debunked” (i.e., defeated), yielding the final verdict that moral beliefs are ultima facie unjustified. Notable candidate “debunkers” (i.e., defeaters) include the so-called “evolutionary debunking arguments,” the “Benacerraf-Field Challenge,” and persistent moral disagreement among epistemic (...) peers. Such defeaters are best treated as higher-order evidence—viz., evidence contesting the merits of the first-order evidence on which moral beliefs are based. This chapter first develops a theory of higher-order defeat in general, which it then applies to debunking in particular. The result: the challenge fails entirely on epistemic grounds—regardless of whether or not its empirical and metaphysical presuppositions are correct. An advantage of this purely epistemic defense over alternative strategies is that the former extends even to laypersons who themselves lack the expertise necessary to formulate an adequate response. However, this leaves open the prospects for non-epistemological interpretations of debunking (e.g., moral or ontological). The chapter therefore concludes with brief suggestions in that direction. (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.

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.

Any intermediate propositional logic can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon $ - (...) and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic. (shrink)

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various ways can (...) be approximated by finite-valued logics. It is shown that the minimal m-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics. (shrink)

A concise open-access textbook intended for an undergraduate audience, which brings together essential selections from Gandhi on nonviolence with supplementary materials, including: a preface; boxes providing examples, historical notes, extended explanations, and related philosophical work; overviews of post-Gandhian developments in nonviolence; diagrams, tables, and photos; discussion questions; reading and viewing suggestions; and a glossary.

Introduction to Philosophy: Epistemology engages first-time philosophy readers on a guided tour through the core concepts, questions, methods, arguments, and theories of epistemology—the branch of philosophy devoted to the study of knowledge. After a brief overview of the field, the book progresses systematically while placing central ideas and thinkers in historical and contemporary context. The chapters cover the analysis of knowledge, the nature of epistemic justification, rationalism vs. empiricism, skepticism, the value of knowledge, the ethics of belief, Bayesian epistemology, social (...) epistemology, and feminist epistemologies. Along the way, instructors and students will encounter a wealth of additional resources and tools: chapter learning outcomes, key terms, images of philosophers and related art, useful diagrams and tables, boxes containing excerpts and other supplementary material, questions for reflection, suggestions for further reading, and a glossary. For an undergraduate survey epistemology course, Introduction to Philosophy: Epistemology is ideal when used as a main text paired with primary sources and scholarly articles. For an introductory philosophy course, select book chapters are best used in combination with chapters from other books in the Introduction to Philosophy open textbook series (edited by Christina Hendricks). (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)

Higher-order evidence is, roughly, evidence of evidence. The idea is that evidence comes in levels. At the first, or lowest, evidential level is evidence of the familiar type—evidence concerning some proposition that is not itself about evidence. At a higher evidential level the evidence concerns some proposition about the evidence at a lower level. Only in relatively recent years has this less familiar type of evidence been explicitly identified as a subject of epistemological focus, and the work on it remains (...) relegated to a small circle of authors and a short stack of published articles—far disproportionate to the attention it deserves. It deserves to occupy center stage for several reasons. First, higher-order evidence frequently arises in a strikingly diverse range of epistemic contexts, including testimony, disagreement, empirical observation, introspection, and memory, among others. Second, in many of the contexts in which it arises, such evidence plays a crucial epistemic role. Third, the precise role it plays is complex, gives rise to a number of interesting epistemological puzzles, and for these reasons remains controversial and is not yet fully understood. As such, higher-order evidence merits systematic investigation. This thesis undertakes such an investigation. It aims to produce a thorough account of higher-order evidence—what it is, how it works, and its epistemic consequences. Chapter 1 serves as a general introduction to the topic and an overview of the existing literature, but primarily aims to further elucidate the concept of higher-order evidence and build a theoretical framework for later chapters. Chapter 2 develops an account of what I call “higher-order support”: the bearing higher-order evidence has, not on corresponding “lower-order evidence” (roughly, the evidence the higher-order evidence is about), but on corresponding “object-level propositions” (roughly, the propositions the higher-order evidence alleges the lower-order evidence to be about). Chapter 3 develops an account of “levels interaction”: the effect on overall support when the different evidential levels combine. Chapter 4 identifies important consequences of the theoretical results of the previous two chapters and applies the theory to four select cases of current epistemological controversy—testimony, memory, the closure of inquiry, and disagreement. (shrink)

Decision theory has had a long-standing history in the behavioural and social sciences as a tool for constructing good approximations of human behaviour. Yet as artificially intelligent systems (AIs) grow in intellectual capacity and eventually outpace humans, decision theory becomes evermore important as a model of AI behaviour. What sort of decision procedure might an AI employ? In this work, I propose that policy-based causal decision theory (PCDT), which places a primacy on the decision-relevance of predictors and simulations of agent (...) behaviour, may be such a procedure. I compare this account to the recently-developed functional decision theory (FDT), which is motivated by similar concerns. I also address potentially counterintuitive features of PCDT, such as its refusal to condition on observations made at certain times. (shrink)

On the heels of Franzén's fine technical exposition of Gödel's incompleteness theorems and related topics (Franzén 2004) comes this survey of the incompleteness theorems aimed at a general audience. Gödel's Theorem: An Incomplete Guide to its Use and Abuse is an extended and self-contained exposition of the incompleteness theorems and a discussion of what informal consequences can, and in particular cannot, be drawn from them.