We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory, the completeness of such semantics is equivalent to the Boolean prime ideal theorem. Using a result of McCarty, we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory, to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over (...) ZF, between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories. (shrink)

We discuss first order theories in which individual concepts are admitted as mathematical objects along with the things that reify them. This allows very straightforward formalizations of knowledge, belief, wanting, and necessity in ordinary first order logic without modal operators. Applications are given in philosophy and in artificial intelligence. We do not treat general concepts, and we do not present any full axiomatizations but rather show how various facts can be expressed.

In lieu of an abstract, here is a brief excerpt of the content:Martin Heidegger and OntologyEmmanuel Levinas (bio)The prestige of Martin Heidegger 1 and the influence of his thought on German philosophy marks both a new phase and one of the high points of the phenomenological movement. Caught unawares, the traditional establishment is obliged to clarify its position on this new teaching which casts a spell over youth and which, overstepping the bounds of permissibility, is already in vogue. For once, (...) Fame has picked one who deserves it and, for that matter, one who is still living. Anyone who has studied philosophy cannot, when confronted by Heidegger’s work, fail to recognize how the originality and force of his achievements, stemming from genius, are combined with an attentive, painstaking, and close working-out of the argument—with that craftsmanship of the patient artisan in which phenomenologists take such pride. In this study, it is important for us to understand, above all, the true intentions of our author, to illuminate what he thinks really needs to be said, and to surmise what is most critical for him.To get to the heart of Heidegger’s system, it seems fitting to begin with a problem that is generally familiar. We choose the problem of knowledge, a deeper understanding of which takes us to the very threshold of Heidegger’s thought. For this problem, central to modern philosophy, is one of the main obstacles of modern philosophy that Heidegger wishes to surmount. Neo-Kantianism, which takes knowledge as the philosophical problem of the first rank, is the movement against which Heidegger rebels with all his strength. We have, thus, every chance of gaining access to his thought by the main door, so to speak. Once inside the system we will try to trace its outlines, 2 reserving, for a second part of our work, the determination of Heidegger’s place in the history of ideas, especially in the phenomenological movement, as well as of his relations with the philosopher to whom he owes so much—Edmund Husserl. 31In its most general form, the problem of the theory of knowledge has a critical significance. It consists in delineating a domain where knowledge can be certain and in determining the criteria for the legitimate scope of knowledge. But this problem, as normal and as simple as it may appear, has deeper roots. That knowledge should need a criterion at all presupposes that truth is not identical to all that is known and that the course [End Page 11] of things can fail to correspond with the course of thought. “How does knowledge correspond to being?” is a more profound formulation of the problem of knowledge.But we are not going to touch anymore on the primordial phenomenon that generates the problem. The problem of correspondence between thing and thought presupposes a free activity of thought and its isolation in relation to the object. It is precisely this presupposition which renders their harmony and even their contact problematic. “How does the subject take leave of itself to attain the object?” is what the problem of knowledge, in the last analysis, boils down to. Its true source is thus the concept of “subject” as elaborated by modern philosophy. The cogito presided over the subject’s birth. The cogito was the affirmation of the privileged nature of the subject’s immanent sphere, of its unique place in existence; hence, the cogito was the specificity of the subject’s connection to the rest of reality, the sui generis nature which opens up the passage from immanence to transcendence, the passage from ideas contained in the thinking substance to 4 their “formal existence.”The concept of the subject, understood as a substance having a specific position in the entire domain of being, presents us with difficulties of two kinds. First, how do we understand this leave-taking from the self which the thinking substance brings about and which displays an entirely original aspect? Indeed, we could say that thought, in reaching out toward objects, does not actually take leave of itself, since its objects—considered as ideas and contents of thought—are, in a certain sense, already within it. In order to make sense of this paradox, Descartes... (shrink)

De Vries' mutation theory has not stood the test of time. The supposed mutations of Oenothera were in reality complex recombination phenomena, ultimately explicable in Mendelian terms, while instances of large-scale mutations were found wanting in other species. By 1915 the mutation theory had begun to lose its grip on the biological community; by de Vries' death in 1935 it was almost completely abandoned. Yet, as we have seen, during the first decade of the present century it achieved (...) an enormous popularity. As this paper has tried to suggest, one of the principal reasons for this was that de Vries' theory served as a banner around which a whole crowd of disaffected Darwinians or anti-Darwinians could rally. However, not all of those who favored de Vries did so for quite the same reasons. Underlying the multitude of views ran several common threads: a dissatisfaction with current Darwinian theory born out of misunderstanding natural selection, a general misunderstanding of the nature of species, and a prejudice against speculative, nontestable theories in biology.Supporters of de Vries were not the only opponents of Darwinism, nor was the mutation theory the only alternative to natural selection. In the early twentieth century a number of theories had been proposed to explain away the problems which Darwin had left unsolved. There was the idea of orthogenesis, championed by the American paleontologists Cope, Osborn and others; organic selection (or orthoplasy) was championed by M. M. Baldwin and C. Lloyd Morgan; there were the concepts of convergent evolution proposed by Hermann Friedmann, the theory of physiological selection by John George Romanes, and the concepts of reproductive divergence by H. M. Vernon. Virtually none of these men either accepted or were strong supporters of the de Vriesian theory, for each had his own particular ‘ism” to advocate as the major factor in evolution. The existence of a large number of such theories, each purporting to be the explanation, was characteristic of evolutionary theory at the turn of the century. It is to a large extent the emphasis on such fragmentary concepts that retarded development of the comprehensive theory of evolution which emerged in the 1920's and 1930's. For the historian, however, a study of these alternative theories is instructive in trying to understand the inherent difficulties which Dawwinian theory posed to biologists at the time. De Vries' mutation theory serves historically as a mirror to reflect the critical mood of a generation hostile to the theory of natural selection.It has often been claimed that it was impossible to understand the mechanism of natural selection until it could be placed in genetic and mathematical terms. It is certainly true that great strides have been made in population genetics and the treatment of evolutionary concepts with mathematical tools in the last forty years. But the very people who developed the genetical and mathematical approach to evolution were already convinced of the essential correctness of Darwinian theory before they started. Advances in an understanding of Mendelian heredity aided greatly in solving one important issue for evolutionists: the origin of variations. And the rigor with which selection acted could best be studied by observing changes in gene frequencies (calculated mathematically) over a number of generations. But as this paper has shown, two of the basic problems which biologists faced in evaluating Darwinian theory at the turn of the century-the nature of species, and the criteria of what constituted an acceptable explanation in biological science-could not be answered directly by mathematics. What mathematical and genetical theory did do was to help convince the skeptics of the validity of the Darwinian proposition.The change in explanatory criteria which many hailed as de Vries' most important contribution to evolutionary theory seems to have been part of a general emergence of twentieth-century biology from the domination of theorizers in the nineteenth. It also marked the emergence of America from the domination of biological, and particularly evolutionary, influence of Europeans. The change occurred in three areas: in the kinds of questions asked: testable versus non-testable; in the kind of data sought: quantitative versus qualitative; and in the kinds of theories proposed: analytical and reductive—the attempt to see complex processes in terms of simpler components-as opposed to synthetic and speculative. Although ultimately wrong in his idea, de Vries and his theories rode high on the wave of “experimentalism” which was the harbinger of a new era in evolutionary theory. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each (...) topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)

Both second-order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second-order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first-order set theory as a very high-order logic.

This book articulates and defends Fregean realism, a theory of properties based on Frege's insight that properties are not objects, but rather the satisfaction conditions of predicates. Robert Trueman argues that this approach is the key not only to dissolving a host of longstanding metaphysical puzzles, such as Bradley's Regress and the Problem of Universals, but also to understanding the relationship between states of affairs, propositions, and the truth conditions of sentences. Fregean realism, Trueman suggests, ultimately leads to (...) a version of the identity theory of truth, the theory that true propositions are identical to obtaining states of affairs. In other words, the identity theory collapses the gap between mind and world. This book will be of interest to anyone working in logic, metaphysics, the philosophy of language or the philosophy of mind. (shrink)

Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and G0, (...) in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

DEFINING OUR TERMS A “paradox" is an argumentation that appears to deduce a conclusion believed to be false from premises believed to be true. An “inconsistency proof for a theory" is an argumentation that actually deduces a negation of a theorem of the theory from premises that are all theorems of the theory. An “indirect proof of the negation of a hypothesis" is an argumentation that actually deduces a conclusion known to be false from the hypothesis alone (...) or, more commonly, from the hypothesis augmented by a set of premises known to be true. A “direct proof of a hypothesis" is an argumentation that actually deduces the hypothesis itself from premises known to be true. Since `appears', `believes' and `knows' all make elliptical reference to a participant, it is clear that `paradox', `indirect proof' and `direct proof' are all participant-relative. PARTICIPANT RELATIVITY In normal mathematical writing the participant is presumed to be “the community of mathematicians" or some more or less well-defined subcommunity and, therefore, omission of explicit reference to the participant is often warranted. However, in historical, critical, or philosophical writing focused on emerging branches of mathematics such omission often invites confusion. One and the same argumentation has been a paradox for one mathematician, an inconsistency proof for another, and an indirect proof to a third. One and the same argumentation-text can appear to one mathematician to express an indirect proof while appearing to another mathematician to express a direct proof. WHAT IS A PARADOX’S SOLUTION? Of the above four sorts of argumentation only the paradox invites “solution" or “resolution", and ordinarily this is to be accomplished either by discovering a logical fallacy in the “reasoning" of the argumentation or by discovering that the conclusion is not really false or by discovering that one of the premises is not really true. Resolution of a paradox by a participant amounts to reclassifying a formerly paradoxical argumentation either as a “fallacy", as a direct proof of its conclusion, as an indirect proof of the negation of one of its premises, as an inconsistency proof, or as something else depending on the participant's state of knowledge or belief. This illustrates why an argumentation which is a paradox to a given mathematician at a given time may well not be a paradox to the same mathematician at a later time. -/- The present article considers several set-theoretic argumentations that appeared in the period 1903-1908. The year 1903 saw the publication of B. Russell's Principles of mathematics, [Cambridge Univ. Press, Cambridge, 1903; Jbuch 34, 62]. The year 1908 saw the publication of Russell's article on type theory as well as Ernst Zermelo's two watershed articles on the axiom of choice and the foundations of set theory. The argumentations discussed concern “the largest cardinal", “the largest ordinal", the well-ordering principle, “the well-ordering of the continuum", denumerability of ordinals and denumerability of reals. The article shows that these argumentations were variously classified by various mathematicians and that the surrounding atmosphere was one of confusion and misunderstanding, partly as a result of failure to make or to heed distinctions similar to those made above. The article implies that historians have made the situation worse by not observing or not analysing the nature of the confusion. -/- RECOMMENDATION This well-written and well-documented article exemplifies the fact that clarification of history can be achieved through articulation of distinctions that had not been articulated (or were not being heeded) at the time. The article presupposes extensive knowledge of the history of mathematics, of mathematics itself (especially set theory) and of philosophy. It is therefore not to be recommended for casual reading. AFTERWORD: This review was written at the same time Corcoran was writing his signature “Argumentations and logic”[249] that covers much of the same ground in much more detail. https://www.academia.edu/14089432/Argumentations_and_Logic . (shrink)

So-called 'dynamic' semantic theories such as Kamp's discourse representation theory and Heim's file change semantics account for such phenomena as cross-sentential anaphora, donkey anaphora, and the novelty condition on indefinites, but compare unfavorably with Montague semantics in some important respects (clarity and simplicity of mathematical foundations, compositionality, handling of quantification and coordination). Preliminary efforts have been made by Muskens and by de Groote to revise and extend Montague semantics to cover dynamic phenomena. We present a new higher-order (...) class='Hi'>theory of discourse semantics which improves on their accounts by incorporating a more articulated notion of context inspired by ideas due to David Lewis and to Craige Roberts. On our account, a context consists of a common ground of mutually accepted propositions together with a set of discourse referents preordered by relative salience. Employing a richer notion of contexts enables us to extend our coverage beyond pronominal anaphora to a wider range of presuppositional phenomena, such as the factivity of certain sentential-complement verbs, resolution of anaphora associated with arbitrarily complex definite descriptions, presupposition 'holes' such as negation, and the independence condition on the antecedents of conditionals. Formally, our theory is expressed within a higher-order logic with natural number type, separation-style subtyping, and dependent coproducts parameterized by the natural numbers. The system of semantic types builds on proposals due to Thomason and to Pollard in which the type of propositions (static meanings of sentential utterances) is taken as basic and worlds are constructed from propositions (rather than the other way around as in standard Montague semantics). (shrink)

Fitch’s paradox of knowability is an apparently valid reasoning from the assumption (typical of semantic anti-realism) that every true proposition is knowable to the unacceptable conclusion that every true proposition is known. The paper develops a critical dialectic wrt one of the best motivated solutions to the paradox which have been proposed on behalf of semantic anti-realism—namely, the intuitionistic solution. The solution consists, on the one hand, in accepting the intuitionistically valid part of Fitch’s reasoning while, on the other hand, (...) exploiting the characteristic weakness of intuitionistic logic in order to preserve the consistency of such acceptance with the denial of omniscience. It is first remarked how the solution still commits one to acceptance of modal claims which are unwarranted even by the lights of standard intuitionistic semantics. A novel form of the paradox is then introduced, which focuses on infallibility rather than omniscience and derives, from semantic anti-realism and a highly plausible constraint on knowledge, that every believed proposition is not untrue. Because of the logical form of this conclusion, an analogue of the intuitionistic solution for the novel form of the paradox would require drawing the characteristic intuitionistic distinctions wrt decidable propositions, which cannot be done. Semantic anti-realism still intuitionistically entails the unacceptable conclusion that every believed (decidable) proposition is true. (shrink)

This thesis mounts an attack against accounts of perceptual justification that attempt to analyze it in terms of evidential justifiers, and has defended the view that perceptual justification should rather be analyzed in terms of non-evidential justification. What matters most to perceptual justification is not a specific sort of evidence, be it experiential evidence or factive evidence, what matters is that the perceptual process from sensory input to belief output is reliable. I argue for this conclusion in the following way. (...) Chapter 1: The Arguments from HallucinationChapter 1 presents a skeptical argument from hallucination that starts from a premise about the alleged introspective indistinguishability of perception and hallucination, and concludes that perceptual knowledge is impossible. Underlying this argument are certain assumptions about the nature of perceptual justification, and different responses to the argument portray different views of perceptual justification. Evidentialism and dogmatism both reject the premise that appearances do not provide evidence sufficient for perceptual knowledge, although they disagree about the precise nature of the appearances. Epistemological disjunctivism rejects the premise that introspectively indistinguishable experiences provide the same evidence, while upholding the idea that some sort of evidence, namely factive evidence, is necessary for perceptual justification and knowledge. Process reliabilism rejects the premise that evidence is necessary for perceptual knowledge on the grounds that even perceptual justification can be obtained without evidence, as long as the cognitive process producing the belief is reliable.Chapter 2: EvidentialismChapter 2 starts by introducing some motivations for experientialist views of perceptual justification, such as evidentialism and dogmatism, which hold that conscious experiences evidentially justify perceptual beliefs. These motivations have to do with their ability to accommodate the New Evil Demon Intuition that demon-deceived subjects have justified perceptual beliefs and the Blindsight Intuition that blindsighters do not have justified perceptual beliefs. However, experientialist views are also faced with a Sellarsian dilemma: either experience is construed as non-propositional, but then the proposed evidential relation is unclear, or experience is construed as propositional, but then it is ad hoc to hold that they can evidentially justify without being justified themselves.Chapter 2 continues with a discussion of evidentialism, which grasps the first horn of this dilemma. I argue that evidentialism cannot adequately explain why a non-propositional experience constitutes evidence for this rather than that belief, and also does not provide any account of how to determine which belief fits an evidence-set which partly consists out of non-propositional experiences. Although a reliabilist version of evidentialism can provide a better account of these matters, it suffers from counter-intuitive predictions about justification and a conception of evidence that is so liberal that it could even include blindsighters and clairvoyants as having such evidence.Chapter 3: DogmatismChapter 3 focuses on variants of dogmatism, a theory of perceptual justification that grasps the second horn of the Sellarsian dilemma by holding that a perceptual experience with the content that p is sufficient for immediate prima facie justification of the belief that p. I first argue that dogmatism should identify perceptual experiences with high-level conscious states to accommodate the problem of novice vs. expert identification and the problem of the speckled hen. I then argue that this version of dogmatism still fails to meet a certain explanatory challenge which I dub the `Distinctiveness Problem'. Dogmatism should explain what is so distinctive about perceptual experience that enables it, in contrast with desire and imagination, to evidentially justify the belief that p without, in contrast with belief, being justified itself.Phenomenalist answers to this problem fail because they either do not provide a distinctive property of perceptual experience, or else provide a property that is reducible to or possibly caused by beliefs. In both cases, dogmatism would have the result that intuitively unjustifying experiences, such as imaginations, nevertheless become capable of justifying beliefs. Moreover, phenomenalists still need some explanation of why phenomenology is epistemically relevant at all.A new variant of the problem of the speckled hen is an intuitive illustration of this point against dogmatists. If a perceptual experience that p is not reliably connected to the fact that p or the belief that p, then beliefs based on this experience simply will not be intuitively justified. One possible response to this problem incorporates reliability into the content of experience in the sense that an experience cannot have the content that p if it is not reliably connected to both the fact that p and the belief that p. However, this will lose the motivation from the New Evil Demon Scenario, and will make large concessions towards reliabilism.Chapter 4: Epistemological DisjunctivismGiven these problems for experientialist views of perceptual justification, chapter 4 focuses on a different evidential account of justification: epistemological disjunctivism. According to the Justified True Belief version of Epistemological Disjunctivism, the fact that I see that p is the factive and reflectively accessible evidence in virtue of which I know that p. I argue that JTBED faces formidable problems: first, it cannot adequately deal with the basis problem, second, it cannot accommodate the New Evil Demon Intuition, third,it faces the problem of hyper-intellectualization, and fourth, it has no good account of reflective access.Knowledge First Epistemological Disjunctivism, in contrast, holds that seeing that p just is a way of knowing that p. The fact that I see that p constitutes my justification for believing that p, even though it is not necessary for my knowledge that p. Although this view solves most of the above problems, I argue that KFED still faces a challenge of its own: its notion of justification is so strong that justification entails knowledge but is not itself entailed by it. This means that it too faces a hyper-intellectualization objection because it cannot accommodate animal justification, and that it too will struggle to accommodate the New Evil Demon Intuition.Nevertheless, there is something to the epistemological disjunctivist idea that higher-order capabilities are important to perceptual justification. Specifically, higher-order capabilities provide one with a way of accommodating the Blindsight Intuition without appealing to the notion of experiential evidence: blindsighters are epistemically worse off than normal human perceivers because they lack higher-order beliefs about how they are gaining their information. If one holds that these higher-order beliefs provide a special sort of justification for perceptual beliefs, then blindsighters lack this type of justification. Chapter 5: Process ReliabilismAfter displaying the problems of several evidential accounts of perceptual justification, chapter 5 introduces a non-evidential account: process reliabilism. The classic version of this view holds that the reliability of a belief-forming process is necessary and sufficient for the prima facie justification of a specific set of beliefs, namely, those that arise out of belief-independent processes. Two classic major problems for this view consist of counterexamples to both the necessity and sufficiency of reliability for justification. Although the Generality Problem raises some difficulties for reliabilism, they do not appear to be insurmountable.The second part of chapter 5 focuses on a variety of process reliabilism, inferentialist reliabilism, that is explicitly aimed at responding to clairvoyance cases. According to inferentialist reliabilism, one should distinguish between basic and non-basic beliefs, where basic beliefs are those that result out of the non-inferential operation of an inferentially opaque cognitive system that developed in a natural way. Reliability is only sufficient for the justification of basic beliefs, while non-basic beliefs also require conditional reliability and justification of the beliefs on which they are based.This distinction between basic and non-basic beliefs is meant to answer the challenge arising out of clairvoyance cases because these are all supposedly cases in which the resulting beliefs arenon-basic. However, I argue that the beliefs in these cases come out as non-basic primarily because of the etiological constraint on cognitive systems, a constraint that is unfortunately undermotivated. It seems that beliefs can be justified even if they are produced by a cognitive system that developed by accident, or by a system that was artificially designed to produce those beliefs. So even though the inferentialist reliabilist might have a point in distinguishing between basic and non-basic beliefs, it is still in need of a different response to clairvoyance cases. Chapter 6: Rejoinders for ReliabilismChapter 6 improves on the inferentialist reliabilist account of justification by presenting a different way of dealing with the clairvoyance cases and New Evil Demon Scenario. I start from the empirically plausible idea that we know that we are perceiving when we are perceiving, not because of the use of experiential evidence, but rather, because of an unconscious source-monitoring mechanism. I then argue that the existence of such a source-monitoring mechanism could be used to explain why clairvoyance cases actually have to do with defeat rather than absence of prima facie justification: beliefs that pop up in our head should be recognized as stemming from untrustworthy sources, and so we acquire defeating higher-order beliefs about the first-order beliefs. I also argue that this defeat should be able to occur even if the higher-order beliefs are themselves unjustified. This defeater-account of clairvoyance is preferable to several externalist alternatives. Making reliability relative to a specific world seems rather ad hoc, and further requirements on justification, such as evidence-based belief, cognitively integrated belief, or properly produced belief, all appear unnecessary and are susceptible to adapted versions of the clairvoyance case.I further argue for an alternative inferentialist reliablist response to the New Evil Demon Intuition which treats it as mistaken rather than correct. The New Evil Demon Intuition might arise because of a mistaken view of actual perceptual justification as having to do with experiential evidence. This mistaken view could be explained by the fact that experience is actually reliably connected to belief and is often cited in response to a knowledge-challenge. If this is a correct analysis of the New Evil Demon Intuition, then there is an element of question-begging involved when this scenario is still pressed against inferentialist reliabilists who do not adhere to the experientialist assumption it presupposes.With the inferentialist reliabilist account in hand,we are able to see that many of the premises of the epistemological argument from hallucination are problematic. Evidence does not appear to be required for perceptual knowledge, as perceptual justification does not even require such evidence. Moreover, if we accept a reliabilist model of introspection, then we are often able to know on the basis of introspection that we are perceiving rather than hallucinating. Once one gets clear about the alternatives, it becomes apparent that the argument thrives on internalist assumptions about justification and knowledge that one need not accept. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture on the very (...) same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

It is discerned what light can bring the recent historical reconstructions of maxwellian optics and electromagnetism unification on the following philosophical/methodological questions. I. Why should one believe that Nature is ultimately simple and that unified theories are more likely to be true? II. What does it mean to say that a theory is unified? III. Why theory unification should be an epistemic virtue? To answer the questions posed genesis and development of Maxwellian electrodynamics are elucidated. It is enunciated (...) that the Maxwellian Revolution is a far more complicated phenomenon than it may be seen in the light of Kuhnian and Lakatosian epistemological models. Correspondingly it is maintained that maxwellian electrodynamics was elaborated in the course of the old pre-maxwellian programmes’ reconciliation: the electrodynamics of Ampére-Weber, the wave theory of Young-Fresnel and Faraday’s programme. To compare the different theoretical schemes springing from the different language games James Maxwell had constructed a peculiar neutral language. Initially it had encompassed the incompressible fluid models; eventually – the vortices ones. The three programmes’ encounter engendered the construction of the hybrid theory at first with an irregular set of theoretical schemes. However, step by step, on revealing and gradual eliminating the contradictions between the programmes involved, the hybrid set is “put into order” (Maxwell’s term). A hierarchy of theoretical schemes starting from ingenious crossbreeds (the displacement current) and up to usual hybrids is set up. After the displacement current construction the interpenetration of the pre-maxwellian programmes begins that marks the commencement of theoretical schemes of optics, electricity and magnetism real unification. Maxwell’s programme surpassed that of Ampére-Weber because it did absorb the ideas of the Ampére-Weber programme, as well as the presuppositions of the programmes of Young-Fresnel and Faraday properly co-ordinating them with each other. But the opposite statement is not true. The Ampére-Weber programme did not assimilate the propositions of the Maxwellian programme. Maxwell’s victory over his rivals became possible because the gist of Maxwell’s unification strategy was formed by Kantian epistemology looked in the light of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and Sir William Hamilton. Maxwell did put forward as basic synthetic principles the ideas that radically differed from that of Ampére-Weber approach by their open, flexible and contra-ontological, genuinly epistemological, Kantian character. For Maxwell, ether was not the ultimate building block of physical reality, from which all the charges and fields should be constructed. “Action at a distance”, “incompressible fluid”, “molecular vortices”, etc. were contrived analogies for Maxwell, capable only to direct the researcher at the “right” mathematical relations. Key words: J.C. Maxwell, unification of optics and electromagnetism, I. Kant, T. Reid, W. Hamilton . (shrink)

According to many actualists, propositions, singular propositions in particular, are structurally complex, that is, roughly, (i) they have, in some sense, an internal structure that corresponds rather directly to the syntactic structure of the sentences that express them, and (ii) the metaphysical components, or constituents, of that structure are the semantic values — the meanings — of the corresponding syntactic components of those sentences. Given that reference is "direct", i.e., that the meaning of a name is its denotation, (...) an apparent consequence of this view is that any proposition expressed by a sentence containing a name that denotes a contingent being S is itself contingent — notably, the proposition [S does not exist]. Assuming that an entity must exist to have a property, necessarily, [S does not exist] must exist in order to be true. It seems to follow that, necessarily, [S does not exist] is not true and, hence, that S is not contingent after all. Past approaches to the problem — notably, those of Prior and Adams — lead to highly undesirable consequences for quantified modal logic. In this paper, several solutions to this puzzle are developed that preserve actualism, the structured view of propositions, the direct theory of reference, and the intuition that [S does not exist] is indeed possible without the adverse consequences for QML of previous solutions. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of (...) properties endorsed by Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as (...) a system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind. (shrink)

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle (...) of abstraction, axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc. (shrink)

In this paper, I have attempted to make the role of mathematical thinking clear in Husserl’s theory of sciences. Husserl believed that phenomenology could afford to provide a safe foundation for individual sciences. Hence, the first task of the project was reorganizing the system of sciences and to show the possibility of apodictic knowledge regarding the world. Husserl was inspired by the progress of mathematics at that time because mathematics is the most logical discipline and deals with abstract (...) objects. It was the most suitable model for Husserl’s project. In fact, we can find structural similarities between his project and F. Klein’s Erlangen Program; further, the procedure of the essence intuition can be explained by a mathematical induction. Mathematics is certainly a new path for understanding Husserl’s phenomenology. In order to clarify the relation between Husserl’s theory of sciences and mathematics, this study focused on the problem of classification. Lastly, another implication of Husserl’s phenomenology as a theory of sciences is that his work is still meaningful for today’s dynamic reality of sciences. (shrink)

One conclusion has already been reached, namely, the diagnosis of the problems of Rousseau's political thought. Again, this is not to say that Locke or Hobbes is correct and Rousseau incorrect, but only to observe that one cannot mix myths without getting into the deepest trouble. But there are several other observations of a more general nature that I want to make. First, the considerations introduced above are intended to point out something of the character of the language of political (...) theory. It is an error to treat such language as if it were assimilable to the verifiability model of scientific language (this was the mistake of both Hume and Weldon). The language of political theory, such as social contract theory, bears a much closer relationship to religious language than it does to at least that model of scientific discourse, for it is basically mythic in character. Speaking still in terms of the verifiability model of scientific discourse, political theories such as social contract are neither true nor false. If one means by “cognitively meaningful” the same thing as “true or false,” then one would be constrained to say that such political theories are meaningless. But the point is that although such theories cannot be shown to be either true or false in that rather simple-minded way that both Hume and Weldon require, they are in fact meaningful, and cognitively so. The essential difference is that the Hume-Weldon model requires that cognitive discourse be descriptive, while the view here advanced rejects this.It is perhaps worth noting, that the Hume-Weldon (positivistic) model has rather gone out of fashion even in the paradigmatic case of the language of science. The difficulties of the verifiability theory of meaning are legion and are well-known to philosophers of science. The problem is, however, that those difficulties and the more recent developments in the theory of scientific language arising out of those difficulties are not equally well known to all philosophers and not known at all outside philosophy. Thus a great many philosophers and practically all nonphilosophers operate with an outmoded theory of language and this theory, unfortunately, governs most of the discussion about political theory.Second, a problem related to that of verification arises out of a consideration of the relationships among mythic structures. For example, it is sometimes maintained that the American democratic system rests on a certain religious point of view: that we are a Christian nation. I am not concerned to attack or defend this position. I want only to observe that it is conceivable that myths may “stack” in certain ways, that hierarchies of mythic structures are possible, such that under one very general kind of myth (such as Marxism), certain subordinate or lower-order myths are permissible and others are not. Among these high-order myths may be certain metaphysics, such as Spinozism or Platonism, or certain religions. Consequently, in addition to such “verification” procedures as have already been suggested, there may be another, namely, the determination of the compatibility of the given myth with others of higher order. If this is so, then one must be careful not to assume that his particular political myth is compatible with just any other point of view he might hold.Finally, returning once more to Rousseau, there is the difficulty involved in getting out of one myth and into another. Religious thinkers have always known about this, for it is the problem of conversion. But philosophers have tended to think that all one had to do was present some arguments and everyone would come around to that point of view. What has been missed is the blikish element, and this undoubtedly explains why there have been virtually no philosophical conversions by argument. Scientific disputes can usually (though not always) be resolved by the introduction of evidence, but if religious points of view and political myths (like the social contract) are bliks, then they determine what will be accepted as evidence and consequently no evidence will count against them. As I have already said, I think this is too strong. There is a blikish element in myths, but in a very large, although recognizable, sense, they are “verifiable.” Even so, the fact that in myth one is presented with a picture, a drama, of which one is a part and not, as in the descriptive model of scientific language, with a set of propositions of which the knowing subject is independent, this fact, I say, makes all the difference in the world so far as switching positions is concerned. Thus, as Rousseau's case suggests, it is extremely hard to get out of one myth, one blik, one way of looking at the world, and into another. This suggests why Democrats do not often become Republicans, why Buddhists do not often become Christians, and why teen-agers complain that their parents do not understand them. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. (...) Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$ -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Historical scholarship suggests that a robust cult of the saints may have helped some European regions to resist inroads by Protestantism. Based on a neo-Durkheimian theory of rituals and social order, I propose that locally based cults of the saints that included public veneration lowered the odds that Protestantism would displace Catholicism in sixteenth-century German cities. To evaluate this proposition, I first turn to historical and theoretical reflection on the role of the cult of the saints in late medieval (...) history. I then test the hypothesis with a data set of sixteenth-century German cities. Statistical analysis provides additional support for the ritual and social order thesis because even when several important variables identified by materialist accounts of the Reformation in the social scientific literature the presence of shrines as an indicator for the cult of the saints remains large and significant. Although large-scale social change is usually assumed to have politico-economic sources, this analysis suggests that cultural factors may be of equal or greater importance. (shrink)

This is a very high quality book with a slightly misleading title. It is difficult to see how it could serve as an introduction for anyone except the mathematically mature or, at least, a student who has already been introduced to formal logic through the lower predicate calculus. Not that these topics are not covered in the book—they comprise the first 92 pages; but the discussion quickly moves into intellectual high gear with sophisticated treatments of the independence of systems of (...) propositional logic, first-order theories, isomorphism of interpretations, and completeness and decidability. With this caveat, Mendelson's book is one of the best of the "second generation" of books on mathematical logic. It is lucid and rigorous, and contains no "frills." It does, however, contain numerous exercises in which the student will have the pleasure of developing for himself some of the more important theorems in the various subjects treated. Among these are chapters on formal number theory, including Gödel's incompleteness theorem, recursive undecidability, Tarski's theorem; a chapter on axiomatic set theory with Hartog's theorem, the axiom of choice, and the axiom of restriction ; and a truly excellent chapter on effective computability which takes off from the notion of an algorithm, and then uses this as a basis upon which to develop the theories of Markov algorithms, Turing algorithms, the theories of Herbrand and Gödel, recursively enumerable sets and undecidable problems. An appendix contains a version of K. Schütte's consistency proof for formal number theory.—H. P. K. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition (...) for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

The central question of this conference is whether the media can contribute to high quality social dialogue. The prospects for resolving that question positively in the “sound and fury” depend on recovering the idea of truth. At present the news media are lurching along from one crisis to another with an empty centre. We need to articulate a believable concept of truth as communication's master principle. As the norm of healing is to medicine, justice to politics, critical thinking to education, (...) craftsmanship to engineering, and stewardship to business, so truth-telling is the news profession's occupational norm. Truth-telling is the ethical framework that fundamentally reorders the media's professional culture and enables them to enrich social dialogue rather than undermine it.Historically the mainstream press has defined itself in terms of an objectivist worldview. Centred on human rationality and armed with the scientific method, the facts in news have been said to mirror reality. The aim has been true and incontrovertible accounts of a domain separate from human consciousness. In Bertrand Russell's formula, “truth consists in some form of correspondence between belief and fact” . In the received view, truth is defined in elementary epistemological terms as accurate representation. News corresponds to context-free neutral algorithms, and ethics is equated with impartiality.The attacks on this misguided view of human knowledge had already originated in Giambattista Vico's fantasia and Wilhelm Dilthey's verstehen in the counter-Enlightenment of the 18th century. They have continued with hermeneutics, critical theory in the Frankfurt School, American pragmatism, Wittgenstein's linguistic philosophy, Gramsci, and in their own way, Lyotard's denial of master narratives and Derrida's sliding signifiers; until the anti-foundationalism of our own day indicates a crisis in correspondence views of truth. Institutional structures remain Enlightenment-driven, but in principle the tide has turned currently toward restricting objectivism to the territory of mathematics, physics, and the natural sciences. In reporting, objectivity has become increasingly controversial as the working press' professional standard, though it will remain entrenched in our ordinary practices of news production and dissemination until an alternative mission for the press is convincingly formulated.The demise of correspondence views of truth has created a predicament for the notion of truth altogether. However, instead of appealing to coherence versions or abandoning the concept, truth needs to be relocated in the moral sphere. Truth is a problem of axiology rather than epistemology. With the dominant scheme no longer tenable, truth should become the province of ethicists who can reconstruct it as the news media's contribution to social dialogue.When truth is articulated in terms of a moral framework, we can mold its richly textured meaning around the Hebrew emeth , the Greek aletheia . In Serbo-Croatian the true is justified as with plumbline in carpentry. In the powerful wheel imagery of the Buddhist tradition, truth is the immovable axle. The Truth and Reconciliation Commission in South Africa presumes that sufferings from apartheid can be healed through truthful testimony. In Ghandhi's “satyagrapha,” the power of truth through the human spirit eventually wins over force . Dietrich Bonhoeffer's Ethics contends correctly that a truthful account lays hold of the context, motives, and presuppositions involved .Telling the truth depends on the quality of discernment so that penultimates do not gain ultimacy. Truth means, in other words, to strike gold, to get at “the core, the essence, the nub, the heart of the matter” . For Henry David Thoreau – though addressing a different issue – when we are truthful, we attempt to “drive life into a corner and¼if it proves to be mean, why then to get the genuine meanness out of it and publish its meanness to the world; or if it were sublime, to know it by personal experience and be able to give a true account of the encounter” . For the former secretary general of the United Nations, Dag Hammarskjold, “the most dangerous of all moral dilemmas is when we are obliged to conceal truth in order to help the truth be victorious” . In the Talmud, the liar's punishment is that no one believes him.Augustine , professor of rhetoric at Milan and later Bishop of Hippo, illustrates a non-correspondence view of truth. His rhetorical theory is a major contribution to the philosophy of communication, contradicting the highly secular and linear view of the ancient Greeks. As with Aristotle, rhetoric entails reasoned judgement for Augustine; however, he “break[s] away from Graeco-Roman rhetoric, moving instead toward ¼rhetoric as aletheiac act” . Rhetoric for him is not knowledge-producing or opinion-producing but truth-producing . The Epistolae Doctrina Christiana scourges the value-neutral, technical language of “word merchants” without wisdom.Truth is not fundamentally a prescriptive statement. The aletheiac act in Augustine “tends to be more relational than propositional, a dialogically interpersonal, sacramentally charitable act rather than a statement¼taking into account and being motivated by [the cardinal virtues] faith, hope, and charity” . The truth for him does not merely make things clear, but motivates us to belief and action. In truthful communication for Augustine, “it is not enough to seek to move men's minds, merely for the sake of power; instead, the power to move is to be used to lead men to truth”. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

The high point of the falsification of physical theories in a standard view of the philosophy of science is the so-called crucial experiment. This experiment is a kind of manipulated empirical test, which provides the criterion for distinguishing between two rival hypotheses, where one is an acceptable theory due to passing the test, and the other turns out to be an unacceptable theory as it does not pass the test. The crucial experiment was supposed to play a significant (...) role because, in virtue of an empirical disconfirmation of one theory, the experiment was assumed to confirm the other as true. However, in 1906, in La théorie physique, son object et la structure (hereafter quoted in English translation as The Aim and Structure of Physical Theory (1906/1954)), Pierre Duhem famously argued against this view and stated that crucial experiments in physics are impossible as they are necessarily ambiguous and logically incomplete. His contention rested on the claim that, “[a] physical theory is not an explanation [of true reality in itself in virtue of some broad metaphysical ramification of physics]. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as accurately as possible a set of experimental laws” (ibid., p. 19). Furthermore, different theories could be equally suitable to represent a given group of experimental laws. And, assuming holism, no hypothesis could be tested in isolation, but merely as a part of a set of an entire scientific theory. The problem which Duhem identified in 1906 was slightly overshadowed and neglected in mainstream philosophy of science until the appearance of a challenging paper by Willard Van Orman Quine published in 1951 and entitled “Two Dogmas of Empiricism”. Quine’s paper caused a revival of interest in Duhem’s original formulation and gave a new impulse towards the problem in the form of the so-called Duhem-Quine thesis. The aim of this paper is to reconsider whether Duhem was right to argue that there are no crucial experiments in physics. In order to assess the validity of the thesis, first, this paper makes an exposition of Duhem’s arguments in their favour, and analyses the major criticisms of this position offered in the subject-literature of Adolf Grünbaum, who explicitly attacked the arguments for the thesis as inconclusive and false. Then, this paper presents possible modes of defence of the Duhem-Quine thesis and argues that the original formulation of the thesis is well qualified and plausible. Finally, this paper offers a pragmatic interpretation of the theory choice. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:HUME'S THEORY OF IMAGINATION* Historians of philosophy seem increasingly to agree with the view that David Hume is the greatest philosopher ever to have written in English. This high esteem of the Scottish empiricist, however, is a phenomenon of the last decades. As late as 1925 Charles W. Hendel could write "that Hume is no longer a living figure." And Stuart Hampshire reports that in the Oxford of (...) the Thirties the results of Humean philosophizing were regarded as "extravagances of 2 scepticism which no one could seriously accept." A cursory glance at the Hume bibliography edited by Roland Hall demonstrates that it has become standard practice in the English-speaking world to seek an understanding of Hume's arguments when analysing philosophical problems. Critical literature nevertheless continues to neglect the majority of his arguments. It is sufficient, I think, to mention Hume's remarks on space and time, his theory of emotions, his inquiries into the problem of justice, his economic writings and those on the psychology of roliqion. Dut 3 even in the frequently quoted passages of the Treatise Hume's central notion is not, in my opinion, at the heart of the analysis. Here I refer to Hume's remarks on the phenomenon of imagination (fancy). Although he mentions this curious faculty of our cognition on almost every page, there are only a few sporadic attempts at an interpretation of this "supreme Humean faculty." This may be due to the fact that until quite recently the 'problem of imagination' has hardly even belonged to the marginal topics of modern philosophy. This paper therefore attempts to demonstrate not only that Hume dealt with the phenomenon of imagination, but also that an analysis of this concept is of paramount significance for the understanding of his entire philosophy. In the first three sections three different faculties or functions of the imagination will be distinguished. Not only the artist's - 91 92. works, but also the insights of the scientist and the illusions of metaphysicians are, according to Hume, decisively based on this very cognitive faculty. Section 4 contains a brief discussion of the sceptical arguments of the Treatise, particularly of the Humean analysis of causation. Their impact on Hume's reflections on the problem of imagination is the topic of Section 5. I hope to prove that in the Enquiry Hume interprets these sceptical arguments differently than in the Treatise. While in his early work he feels himself exposed to Pyrrhonian doubts, he assumes the position of an 'academic sceptic' in the Enquiry. The concluding section eventually tries to determine which motives may have been decisive for Hume's change of opinion. I shall defend the thesis that in his late work Hume developed an 'ethics of belief on the basis of natural beliefs, in which the artistic capacity of the imagination assumes a decisive role in justifying philosophical and scientific propositions. 1. The Metaphysical Faculty of the Imagination In the Introduction to the Treatise Hume describes his endeavour as developing a propaedeutic basic science which should provide a better understanding of all products of the human mind (science, mathematics, metaphysics, ethics, aesthetics, politics). In the Abstract he calls his early work "a system of the sciences" (?7). Besides this empirical interest Hume saw the importance of justifying the choice of method, i.e. to extoll the experimental procedure as the only rational one. Metaphysical approaches are interpreted as products of a blind imagination. Beginning with the Introduction to the Treatise it has been a constant motive of Hume's philosophy to constrain metaphysics in order to make way for science. In the sections Of the effects of other relations and other habite (T106-117), Of unphilosophiaal probability 93. (T143-155), Of scepticism with regard to the senses (T187218 ), Of the antient philosophy (T219-225) Hume specifies those beliefs which, in his opinion, are based on figments of our imagination. He "had ^en that all the errors of philosophy arose from the fact that imagination determined men much more than they recognized - there could be no secure truth for him, then, until he had explored this aspect of the nature of man." He compares imagination to a ship which, having... (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency (...) about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of _possible_, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

Chapter 1: An Introduction: The ‘Semantic Sting’ Argument Describes Dworkin’s theory as concerning the conditions of legal validity. “A legal system is a system of norms. Validity is a logical property of norms in a way akin to that in which truth is a logical property of propositions. A statement about the law is true if and only if the norm it purports to describe is a valid legal norm…It follows that there must be certain conditions which render (...) certain norms, but not others, legally valid. Hence it also follows that there can be two types of disagreement over the truth or falsehood of propositions about the law. People can disagree over the question ‘What are the conditions of legal validity?’, in which case their disagreement is a theoretical one. Or they can agree on the conditions of validity, and disagree as to whether or not those conditions actually obtain in a given case or not.” Argues that the best way to interpret Dworkin’s ‘semantic-sting argument’ is as a general argument against conventionalism in legal positivism. Summarizes the outcome of Dworkin’s argument as follows: “[I]f Dworkin is right about the legal reasoning of lawyers and judges, conventionalism would be self-defeating: if lawyers and judges recognize as legally binding not only those norms which are uncontroversially identifiable under the Rule of Recognition, that is, if what they recognize as binding is not only source-based law, then conventionalism turns out to be false on its own terms. In other words, either law is not what lawyers and judges think that it is, in which case law is not a matter of conventions, or – if it is what lawyers and judges think – conventionalism is false, as they do not see the law as purely a matter of conventions.” Dworkin’s theory, then, is a new conception of jurisprudence aiming to present itself as a rival to conventionalism. The book will be concerned with re-examining positivism in light of the interpretative challenge. Chapter 2: Meaning and Interpretation “[R]oughly, interpretation can be defined as an understanding or explanation of the meaning of an object.” Interpretation in this narrow sense is distinct from a less formal and broader use of interpretation that makes it equivalent to explanation. “Hence also, only those objects which are capable of bearing some meaning qualify as objects of interpretation.” [Why shouldn’t we think of interpretation as object-identification? This seems to be closer to what Dworkin has in mind.] Thus, a consideration of the philosophical work on meaning is appropriate. First, a consideration of Davidson on interpretation. The thesis is that Davidson’s “radical interpretation” can only account for sentence meaning and not for linguistic interpretation in general. Davidson cannot account for the non-rule-governed aspects of communication. “Semantics, as opposed to interpretation, concerns those aspects of communication which are rule or convention governed…but such rules are normally unavailable as reasons or justifications for an interpretation. On the contrary, interpretation is usually required because the issue is not determined by rules or conventions.” Interpretation, rather, is parasitic with understanding the meaning of an expression and not equivalent with it. As Dummett notes, any reflection on the meanings of words assumes prior knowledge of ordinary meanings. “This leads to the conclusion that understanding or explaining the meaning of an expression and interpreting it, are two conceptually separate things. It also indicates that seman ics can only be employed, if at all, to elucidate the concept of interpretation by way of contrast: interpretation concerns those aspects of communication which are under-determined by semantic rules or conventions.” Pragmatics, the study of problems posed by discrepancies between utterer’s meaning and sentence meaning, would seem to be closer to inquiry into interpretation. Both are concerned with understanding how meaning is possible where conventional rules leave off. Contemporary pragmatics, moreover, recognizes the logically indeterminate character of communicative inferences beyond conventional meaning and that the contextual conditions that account for the possibility cannot be realized in the semantic structure of the sentence. However, different criteria of success may be at work in pragmatics than interpretation. Pragmatics is concerned with how communication is achieved and thus the criteria of success would be grasping a speaker’s intentions. Many think, however, that communication is unimportant in certain spheres of communication, e.g. artistic interpretation, and that in these cases the aim of interpretation is not to uncover the actual artist’s intentions. What kind of meaning is interpretation aimed at then? It cannot be the meaning the object has for the interpreter. “Interpretation purports to be a statement about the object interpreted, not about the subject who offers the interpretation…it is crucial to remember that not everything we say about a work of art, or any other text, amounts to an interpretation of it. Only those aspects of a text which can shed light on its meaning form part of the text’s interpretation. What the text means for someone rarely entails anything about what the text means.” [This is puzzling to some extent for it remains somewhat ambiguous what it means to talk of subject-independent meaning.] Interpretive meaning attribution is, rather, counterfactual. “In general, I will suggest that the answer to this question consists in the fact that meaning is assigned by a counterfactual statement. Given that x is the meaning attributed to, for instance a text T, and x is not the literal meaning of T, nor is it the meaning of T intended by its author, then the attribution of meaning x to T can only be understood as the contention that on the basis of certain assumptions a certain fictitious or stipulated speaker would have meant x by expressing T.” Thus, on this account, interpretation, when it does not concern an actual speaker, turns out to be a kind of counterfactual intention attribution. Twofold point: “First, that interpretation is essentially a matter of attributing intentions, that is, in the pragmatics sense of ‘meaning’, namely, meaning that such-and-such by an act or expression. At the same time, interpretations need not be based on the intentions of actual authors; the meaning of an act or expression is understandable in terms of counterfactual intentions.” [Are we really restricted to understanding the coherence and integrity of meaning in such a way? Presumably the author is a kind of device for organizing meaning, much like Dworkin’s integrity, but need it take that format?] The logic of interpretation is typically reducible to intention attribution. [What role does the counterfactual author play in structuring meaning?] The criteria of interpretive success will depend upon the object, whether it is the real author or the counterfactual author. If the former, success will depend upon the retrieval of the author’s actual intentions. “Likewise, if the author is characterized in terms of some ideal representative of a certain genre, for instance, the presumptions which are taken to determine this c aracterization would provide the criteria of success for the particular interpretation offered.” [The criteria of success would seem, then, to be the principles, ideals, etc. that we attribute to the counterfactual author. These attributed qualities would be doing all the work – is the attribution of intention to the hypothetical author, then, really necessary at all?] [Does this account presuppose a weird dichotomy between what an object is and what it means? This dichotomy is different, if present at all, in Dworkin.] Chapter 3: Dworkin’s Theory of Interpretation and the Nature of Jurisprudence Three main insights of Dworkin’s theory of interpretation: “First, that interpretation strives to present its object in its best possible light. Second, that interpretation is essentially genre-dependent. And third, that there are certain constraints that determine the limits of possible interpretations of a given object.” Why the best light? Dworkin doesn’t really address this question in depth but claims that otherwise we are left with no way of justifying and choosing an interpretation or we are left with the author’s intention model, which we should reject. [Marmor doesn’t seem clear on what precisely it means to present something in its best light – seems to conflate it with most appealing light which, I take it, is different than what Dworkin has in mind.] “Dworkin is quite right to maintain that without having some views about the values inherent in the genre to which the text is taken to belong, no interpretation can take off the ground. The values we associate with the genre partly, but crucially, determine what would make sense to say about the text, what are the kinds of meaning we could ascribe to it.” There is no necessity for an interpretation to present an object in its best light with regards to the genre it is taken to be part of. Interpretations “could simply strive to present it in a certain light, perhaps better than some, worse than others, but in a way which highlights an aspect of the meaning of the text which may be worth paying attention to for some reason or another.” [What kind of reasons would count in choosing to present it in one light rather than another?] There is simply zero grounds for Dworkin’s assertion that we have no reason to pay attention to an interpretation that does not present it in its best light. Take a psychoanalytic interpretation of Hamlet, for example – not the best light, but a legitimate one. [This seems related to the next criticism – clearly Dworkin would have to loosen his insistence on the best, though not to detriment.] Moreover, especially with regards to art, it doesn’t appear possible to make an all things considered judgment about the best interpretation given the incommensurability of different values that are taken to inhere in a given genre. [Dworkin can respond here that we may have different available interpretations that could be justified with respect to the relevant value. Moreover, a lack of common denominator does not prevent any comparison whatsoever.] Constructive interpretation concerns social practices constituted by norms – it imposes a purpose or value rendering intelligible the normative character of the practice. There is a separate aspect of constructive interpretation that maintains that elements of the practice are sensitive to the point of the practice. It is important to note that different social practices will institutionalize themselves in different ways – thus, different practices will establish varying conditions on how the practice can be altered. The position of the positivists is that evaluative judgments concerning what the law should be are insufficient for determining what the law is given the nature of law’s institutionalization.&nb p; [This would seem to reside on a certain conception of law, i.e. an interpretation about where the value of law lies.] “[T]his being one of the main points of dispute between Dworkin and his positivist opponents, Dworkin cannot at this initial state presume law to be sensitive to its value in the manner that other, non-institutional practices might be, without incurring the charge of having assumed the very point at issue.” The upshot for Dworkin is that any explanation of a social practice such as law requires the same kind of reasoning as participation in that practice. This will be called the hermeneutic thesis. We can try to reconstruct a justification for this view based on the necessity of shared background assumptions – but this will not do since Dworkin’s claim is stronger: it contends that participants’ and theorists’ must “adopt one and the same normative point of view.” [The sense of normative here is unclear. It is true, though, that shared background assumptions do not imply viewing the law as valuable in any particular way, or at least it does not imply this in any strong sense.] Dworkin makes two distinct points with the hermeneutic thesis: “First, that the explanation of a social practice, like law or the arts, is essentially interpretative, and as such, necessarily value laden. Second, that the interpretation of such social practices, which Dworkin calls ‘argumentative’, is unique in the sense that the practice itself is an evaluative enterprise, and that therefore the interpreter of such a practice must form an evaluative judgment of her own about those values which are inherent in the practice that she purports to interpret.” [As stated, the first thesis is terribly ambiguous – for Dworkin, it clear implies a commitment to constructive interpretation. For Marmor, who accepts it, it is totally unclear what commitment he intends to make. He seems to mean simply that we have criteria for success concerning what counts as a successful explanation.] Mamor denies the second thesis. “What Dworkin seems to ignore here is that there is a crucial difference between forming a view about the values which are manifest in a social practice, like law, and actually having evaluative judgments about them.” [The problem, however, it that it is a contentious matter what values are part of the law – a theorist’s account of what values are constitutive of a particular legal system must rely on the kind of arguments which participants rely on. Of course, a theorist might simply note a debate about values and attempt to remain agnostic, but this will fail to be a full theoretical account of what law is!] Dworkin’s thesis seems to depend exclusively on his account of constructive interpretation. [This seems exactly right.] Chapter 4: Coherence, Holism, and Interpretation: The Epistemic Foundation of Dworkin’s Legal Theory Begins with discussion of Rawls’ reflective equilibrium. Sums up dual criticism as follows: “[T]he assumption that intuitions are independently true and the converse one, that their truth depends on fitting a coherent scheme, both seem to yield paradoxical results.” Dworkin, in his article “The Original Position”, claims that Rawls cannot commit himself to ethical realism. “[T]he natural model presupposes some form of ethical realism, while the constructive model does not. One important difficulty arising from an attempt to apply the natural model to Rawls is, that under the natural model, any theory which dopes not account for an intuition, at least for one which is held firmly, cannot be wholly satisfactory, just as scientific theory which does not account for certain observational data it is supposed to cover, would not be satisfactory in a familiar way.” Rawls, rather, is committed to a constructivist view of morality. Moreover, the reason we value coherence epistemically in moral theory is that it itself is a value of political morality – fairness requires consistency and publicity in the application of moral standards. Marmor notes the following problem with such a view: “[I]f coherence is justified, as it is here, with reference to certain moral values, that is, a specific conception of fairness, then we face the following problem: the presupposed values of fairness must themselves be based upon intuitive convictions, in which case the question of their truth cannot be ignored. If they are taken to be true…we are driven back to the perplexities of the natural model.” Discusses, helpfully, the Fish/Dworkin debate. The upshot is that Dworkin’s interpretive model is best off if it commits itself to a Quineian holism. However, this undermines his distinction between internal and external skepticism since holism must deny that moral judgments constitute a closed system. [This analysis seems exactly right.] Endorses Simmonds view that Dworkin’s theory does not meet the requirements of complexity he identifies as necessary to avoid vicious circularity in interpretive judgments of fit and identity. Coherence is doing the work through and through and therefore the interpretive theory of law is circular. [Will require further investigation, though it is unclear how coherence is doing any work in the pre-interpretive stage. At this stage, the judgments, if interpretive, seem to be derived from different sources. In any case, using coherence as a value seems to be highly suspect.] Why doesn’t Dworkin just reject Fish’s assumptions: because of his jurisprudence. “If legal texts can have a meaning that is not entirely dependent on a process of interpretation, then it is at least sometimes the case that the law can simply be understood, and applied, without the mediation of interpretation. And if that is the case, then the argument from interpretation against legal positivism collapses. It is no longer the case that every conclusion about what the law is, depends on evaluative considerations about what it ought to be.” Chapter 5: Semantics, Realism, and Natural Law Assessment of Michael Moore’s legal realism and its implications. [The final criticisms of Moore are not decisive.] Chapter 6: Constructive Identification and Razian Authority A consideration of Dworkin’s denial that the communication model of interpretation is appropriate for legal interpretation. Marmor argues that intentions do play a crucial role in the identification of legal norms in a way that is incompatible with Dworkin’s “coherence thesis”. Marmor takes Dworkin to be committed to the constructive identification thesis: that the identification of something as part of the legal, artistic, etc. genre can be done sufficiently by evaluative considerations. “Here, one must maintain that evaluative considerations are sufficient to determine whether something is a legal norm or a work of art”. [Fit, however, constrains possible evaluative considerations – thus evaluative considerations are not sufficient in themselves.] “Now the crucial point here is this: if you maintain the possibility of constructive identification in art, you must assume that works of art can be identified as such on the basis of certain features they happen to possess, features which contain no reference to any particular intention to create a work of art…unless we take intentions into account, how can we discriminate between the concept of an aesthetic artifact and the con ept of a work of art?” [It is unclear, and I think false, that constructive interpretation of anything requires identifying something irrespective of an artificers intentions. Those intentions may be relevant, dependent upon the genre under which we are trying to classify the object. Nothing, seemingly, about constructive interpretation in general excludes this possibility.] Marmor’s point is that it turns out to be impossible to identify art as art without reference to an artist’s intentions – especially given the state of contemporary art. [It is notable that Marmor’s response to counterexample implicitly depends upon a fully developed theory of aesthetics. His cursory comments on the matter of identifying art seem insufficient to say the least.] The rest of the chapter considers the incompatibility of Raz’s account of authority with constructive interpretation. The incompatibilities mentioned do not move beyond what Raz identifies in his article “Authority, Law and Morality”, but a defense of this notion of authority is defended in several [inconclusive] respects. [Raz and Marmor want to separate identification from content. This is not clearly a defensible separation.] Chapter 7: No Easy Cases? Chapter defends notion that there is a distinction between easy and hard cases in the positivist sense. Takes a Hartian approach and ultimately defends the distinction by associating it Wittgenstein’s conception of rule following. The point is that one need to look the purpose of rule in order to understand what the rule requires. Chapter 8: Legislative Intent and the Authority of Law Examines the following doctrine concerning the role of legislative intent in adjudication: “[F]irst, it would hold that laws, at least in certain cases, are enacted with relatively specific intentions, and that this is a matter of fact which is discernible through an ordinary fact-finding procedure. Second, that in certain cases the presence of such a fact, namely, that the law was enacted with a certain intention, provides judges with a reason to decide the legal dispute in accordance with the relevant legislative intent.” [It is important that for Marmor, interpretive strategies are only appropriate in hard cases. However, we should wonder what grounds positivism can offer for deciding a hard case in any manner which is not the morally best manner according to the judgment of the judge. There are no legal grounds, by definition, in hard cases and so what justificatory strategy would endorse any judicial decision except for the morally best one?] We can ascribe intentions to a legislative body, when this is possible at all, by taking the shared intentions of the majority, rather than some group intention, to be the significant intentions. Moreover, we should expect a good deal of consensus about intentions for otherwise it is hard to explain how legislative bodies are able to produce so much legislation. Discussion of kinds of intentions concludes as follows: “I have distinguished between three main types of intention that are potentially relevant from the legal point of view. Apart from the intentions that are manifest in the language of the law itself, legislators typically have further intentions in enacting a given law, and sometimes they would have certain intentions bearing on its proper application. I have also suggested that some of these further intentions may be essentially non-avowable, in which case they are rendered initially irrelevant. Finally, I have pointed out that considerations of consistency require that the legislator’s application intentions be taken into account only if, and to the extent that, they are in accord with his further intentions.” The manner in which intentionalism is to be justif ed relies on the distinction, Marmor maintains, between expert and mere collective action authority. “The point I wanted to make is strictly conditional: if, and only if, a certain law is justified on the basis of the expertise branch of the normal justification thesis, would it make sense to defer to the legislature’s intentions in the interpretation of the law, that is, to the extent that there is, in fact,; such an intention and it can clarify something that needs clarification.” Insofar as a legislature can be considered an expert on the matter requiring adjudication and interpretation should the body’s intention be taken into account. Chapter 9: Constitutional Interpretation Begins by laying out general necessary features of a constitution. Then Marmor moves on to consider general questions of constitutional legitimacy. First, he considers what grounds might be offered for the legitimacy of a constitution at all. He notes that the central question is what permits a single generation to bind future generations. He then examines four arguments to avoid this problem. The first is based on the moral legitimacy of the constitution; Marmor rejects this solution for the [truly dubious] reason that the constitution would make no practical difference because it would not supply reasons in addition to those provided by morality [but, presumably, the point is that these legally valid rules would be institutionally enshrined]. The second is based on the moral authority of the framers and Marmor rightly rejects this on the basis of the idealization of the framers it requires and less rightly on the basis that there cannot be moral authorities. Third is the “argument from interpretation” which claims that “as long as the particular content of the constitution is determined by its interpretation, and the authoritative interpretation at any time correctly instantiates the values which ought to be upheld in the community, the constitution would be morally legitimate”. The fourth is Raz’s which argues that a constitution is valid so long as its constraints are morally permissible because conventions of this type might aid the continuity of the legal system, i.e. it is important to have a convention of this type and the constitution is a kind of this type. Marmor appears to endorse these two approaches and concludes: “The conditions for the legitimacy of a constitution must comprise the following conditions. First, the values and principles enshrined in it must be morally permissible…Second, when certain choices are made in particular cases, they would be legitimate if they are either morally underdetermined, or else, morally correct…It follows from this that both arguments must assume that at least in those areas in which the constitution would make a moral difference, it can be interpreted to make the difference that it should, that is, according to the true moral principles that should apply to the particular case...the moral legitimacy of constitutions very much dependent on the practices of their interpretation. In other words, a great deal of the burden of moral legitimacy is shifted by these arguments to the application of the constitution, thus assuming that the constitution is legitimate only if the courts are likely to apply the constitution in a morally desirable way.” [This implies that the only significant value constitutions have is in providing a conventional practice where some such practice is required.] Moves on to argue that courts typically rely, and rightly so, on moral considerations when deciding constitutional issues because typically, in cases of the kind that reach the supreme or constitutional court, there is no law on the case. Much of this relies on his earlier discussion about the nature of interpretation in the law – that legal interpretation is always a matter of cha ging the law. (shrink)

The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received (...) adequate justification: the assumption that a proposition invariant under the appropriate group is therefore meaningful, and the assumption that representations should be unique up to a transformation of the appropriate group. A general theory of representational meaningfulness is offered, based on a semantic and syntactic analysis of representational propositions. Two neglected features of representational propositions are formalized and made use of: (a) that such propositions are induced by more general propositions defined for other structures than the one being represented, and (b) that the true purpose of representation is the application of the theory of the representing system to the represented system. On the basis of these developments, justifications are offered for the two problematic assumptions made by the existing theories. (shrink)

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, (...) and strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

“Every step and every movement of the multitude, even in what are termed enlightened ages, are made with equal blindness to the future; and nations stumble upon establishments, which are indeed the result of human action, but not the execution of any human design.”—_Adam Ferguson_ During the Scottish Enlightenment, David Hume, Adam Smith, Adam Ferguson, and other lesser thinkers described a theory of spontaneously generated social order. Ronald Hamowy discusses their contributions to this significant area of social theory, (...) noting that the revolutionary aspect of these philosophers’ thoughts was “the proposition that social phenomena of a high degree of intricacy are not the product of intentional design.” Hamowy shows that Hume is best known for his application of the theory to justice, developing it most thoroughly in the _Treatise of Human Nature. _Adam Smith’s first treatment of it is in the _Theory of Moral Sentiment, _published before the significant _Wealth of Nations. _There Smith uses it to explain the development of general rules that compose the moral fabric of societies. All of these thinkers used their notion of spontaneously generated social order to explain systems of moral rules and social systems. In addition Adam Ferguson and David Hume applied the theory to language, as well as to economic development. (shrink)

A reasoned argument or tarka is essential for a wholesome vāda that aims at establishing the truth. A strong tarka constitutes of a number of elements including an anumāna based on a valid hetu. Several scholars, such as Dharmakīrti, Vasubandhu and Dignāga, have worked on theories for the establishment of a valid hetu to distinguish it from an invalid one. This paper aims to interpret Dignāga’s hetu-cakra, called the wheel of grounds, from a modern philosophical perspective by deconstructing it into (...) a simple probabilistic mathematical model. The objective is to understand how and why a vāda based on a probabilistically weaker hetu can degrade into a Jalpa or vitaṇḍā. To do so, the paper maps the concept of ‘Bounded Rationality’ onto the hetu-cakra. Bounded Rationality, an idea coined by the management thinker Herbert Simon, is often employed in understanding decision-making processes of rational agents. In the context of this paper, the concept would state that the prativādin and ālocaka (debater) may not hold unbounded information to back their pratijñā (proposition). The paper argues that within the probabilistically deconstructed hetu-cakra model, most people argue in the ‘Zone of Bounded Rationality’, and thus, the probability of a debate degrading into Jalpa or vitaṇḍā is high. -/- . (shrink)

Colonius suggests that, in using standard set theory as the language in which to express our computational-level theory of human memory, we would need to violate the axiom of foundation in order to express meaningful memory bindings in which a context is identical to an item in the list. We circumvent Colonius's objection by allowing that a list item may serve as a label for a context without being identical to that context. This debate serves to highlight (...) the value of specifying memory operations in set theoretic notation, as it would have been difficult if not impossible to formulate such an objection at the algorithmic level. (shrink)

This is a concise book that introduces students to the basics of logical thinking and important mathematical structures that are critical for a solid understanding of logical formalisms themselves as well as for building the necessary background to tackle other fields that are based on these logical principles. Despite its compact and small size, it includes many solved problems and quite a few end-of-section exercises that will help readers consolidate their understanding of the material. This textbook is essential reading (...) for anyone interested in the logical foundations of Informatics, Computer Science, Data Science, Artificial Intelligence, and other related areas. Written with undergraduate students in these disciplines in mind, this book can very well serve the needs of interested and curious readers who wish to get a grasp of the logical principles upon which these fields are built. This book does not require readers to possess math skills beyond those learned in high school. (shrink)

How do we explain the truth of true propositions? Truthmaker theory is the branch of metaphysics that explores the relationships between what is true and what exists. It plays an important role in contemporary debates about the nature of metaphysics and metaphysical enquiry. -/- In this book Jonathan Tallant argues, controversially, that we should reject truthmaker theory. In its place he argues for an 'explanationist' approach. Drawing on a deflationary theory of truth he shows that it (...) allows us to explain the truth of true propositions and respond to recent arguments that purport to show otherwise. He augments this with a distinction between internally and externally quantified claims: externally quantified claims are claims that quantify over elements of our ontology that play an indispensable explanatory role; internally quantified claims do not. He deploys this union of deflationism and a distinction between kinds of quantification to pursue metaphysical inquiry, sketching the implications for a number of first-order debates, including those in the philosophy of time, modality and mathematics, and also shows how this explanationist model can be used to solve the key problems that afflicted truthmaker theory. -/- Truth and the World is an important contribution to debates about truth and truthmaker theory as well as metametaphysics, the metaphysics of time and the metaphysics of mathematics, and is essential reading for students and scholars engaged in the study of these topics. (shrink)

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive (...) notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

"The terrain of the self is vast," notes renowned psychiatrist Arnold Ludwig, "parts known, parts impenetrable, and parts unexplored." How do we construct a sense of ourselves? How can a self reflect upon itself or deceive itself? Is all personal identity plagiarized? Is a "true" or "authentic" self even possible? Is it possible to really "know" someone else or ourselves for that matter? To answer these and many other intriguing questions, Ludwig takes a unique approach, examining the art of biography (...) for the insights it can give us into the construction of the self. In The Biography of the Self, he takes readers on an intriguing tour of the biographer's art, revealing how much this can tell us about ourselves. Drawing on in-depth interviews with twenty-one of our most esteemed biographers--writers such as David McCullough (the biographer of Truman and Theodore Roosevelt), Wallace Stegner (John Wesley Powell), Gloria Steinem (Marilyn Monroe), Leon Edel (Henry James), Peter Gay (Freud), Diane Middlebrook (Anne Sexton), and many others--and interweaving fascinating observations of his own practice, Ludwig takes us through the labyrinthine hall of mirrors we term the self and shows us how malleable, elusive, and paradoxical it can be. In chapters such as "The 'Real' Marilyn," "Psychoanalyzing Freud," "How Did Hitler Live With Himself?" and "What Madness Reveals," we sit in as biographers talk not only about their work, but about their subjects (Allan Bullock on Hitler and Stalin, for instance, or Arnold Rampersad on Langston Hughes) and how their subjects saw themselves. Ludwig describes how biographers must impose a narrative structure on their subjects' lives to create order out of a mass of often contradictory views, baffling behavior, and inconsistent self-representations, much in the same way that psychotherapists try to foster self-awareness and understanding in their patients. In his concluding chapter, Ludwig introduces a new concept--biographical freedom--which brilliantly reconciles free will and determinism. We can, he asserts, become biographers of ourselves. Like the biographer, we are constrained to consider all the available facts of our lives--the personal experiences, cultural forces, and predetermined scripts that shape us--but we remain free to interpret, emphasize, and fashion these givens into a cohesive and meaningful narrative of our own choosing. This thought-provoking volume offers not only a wide-ranging and informative commentary on the biographer's art, but also a highly original theory of the self. Readers interested in biography and in the lives of others will come away with a new sense of what it means to be a "person" and, in particular, who they are. (shrink)

It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.

From 1929 through 1944, Wittgenstein endeavors to clarify mathematical meaningfulness by showing how (algorithmically decidable) mathematical propositions, which lack contingent "sense," have mathematical sense in contrast to all infinitistic "mathematical" expressions. In the middle period (1929-34), Wittgenstein adopts strong formalism and argues that mathematical calculi are formal inventions in which meaningfulness and "truth" are entirely intrasystemic and epistemological affairs. In his later period (1937-44), Wittgenstein resolves the conflict between his intermediate strong formalism and his (...) criticism of set theory by requiring that a mathematical calculus (vs. a "sign-game") must have an extrasystemic, real world application, thereby returning to the weak formalism of the Tractatus. (shrink)

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency (...) about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

In papers published between 1930 and 1935. Zermelo outlines a foundational program, with infinitary logic at its heart, that is intended to (1) secure axiomatic set theory as a foundation for arithmetic and analysis and (2) show that all mathematical propositions are decidable. Zermelo's theory of systems of infinitely long propositions may be termed "Cantorian" in that a logical distinction between open and closed domains plays a signal role. Well-foundedness and strong inaccessibility are used to (...) systematically integrate highly transfinite concepts of demonstrability and existence. Zermelo incompleteness is then the analogue of the Problem of Proper Classes, and the resolution of these two anomalies is similarly analogous. (shrink)

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures (...) that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)