In this paper, I will argue that there is a version of possibilism—inspired by the modal analogue of Kit Fine’s fragmentalism—that can be combined with a weakening of actualism. The reasons for analysing this view, which I call Modal Fragmentalism, are twofold. Firstly, it can enrich our understanding of the actualism/possibilism divide, by showing that, at least in principle, the adoption of possibilia does not correspond to an outright rejection of the actualist intuitions. Secondly, and more specifically, it can enrich (...) our understanding of concretism, by proving that, at least in principle, the idea that objects have properties in an absolute manner is compatible with transworld identity. (shrink)

Belief storage is often modeled as having the structure of a single, unified web. This model of belief storage is attractive and widely assumed because it appears to provide an explanation of the flexibility of cognition and the complicated dynamics of belief revision. However, when one scrutinizes human cognition, one finds strong evidence against a unified web of belief and for a fragmented model of belief storage. Using the best available evidence from cognitive science, we develop this fragmented model into (...) a nascent theory of the cognitive architecture of belief storage. (shrink)

The thesis of mental fragmentation has recently attracted increased attention as a way of explaining facts about mind and language. This volume provides an accessible introduction and essays on foundations and applications of fragmentation.

Fragments presents a set of brilliantly intriguing interviews with Jean Baudrillard whose work today occupies center stage in the analysis of consumerism, terrorism, and contemporary culture. In these frank discussions with François L'Yvonnet, Baudrillard reveals for the first time in detail the thinkers who have been the dominant influences on his work during his career. Instead of examining his work as a project of intellectual accumulation, he challenges all the major interpretations of his work by suggesting he has always adopted (...) an anti-system, anti-totality strategy. Even globalization is accompanied in his view by a Western culture which itself is no longer a well-founded confident universalism. Perhaps most interestingly, Baudrillard discusses his life's work in relationship to his contemporaries -- Bataille , Barthes, Lyotard and Deleuze - and explores his position as an outsider in the field of French philosophy. Baudrillard in these interviews is in sparkling form, and in his ownwords he presents not only a lively introduction to this great thinker but also gives readers a window into a brilliant mind. (shrink)

To reconcile the standard possible worlds model of knowledge with the intuition that ordinary agents fall far short of logical omniscience, a Stalnakerian strategy appeals to two components. The first is the idea that mathematical and logical knowledge is at bottom metalinguistic knowledge. The second is the idea that non-ideal minds are often fragmented. In this paper, we investigate this Stalnakerian reconciliation strategy and argue, ultimately, that it fails. We are not the first to complain about the Stalnakerian strategy. But (...) in contrast to existing complaints, we want to cause trouble for the strategy directly on its home turf. That is, we will advance our objection while granting both the plausibility of the fragmentation component—save for an extreme version of it—and that of the metalinguistic component. Once our central objection to the Stalnakerian strategy is in place, we will show how it negatively affects Adam Elga and Augustín Rayo’s recent attempt to apply the Stalnakerian strategy in the context of Bayesian decision theory. (shrink)

In order to predict and explain behavior, one cannot specify the mental state of an agent merely by saying what information she possesses. Instead one must specify what information is available to an agent relative to various purposes. Specifying mental states in this way allows us to accommodate cases of imperfect recall, cognitive accomplishments involved in logical deduction, the mental states of confused or fragmented subjects, and the difference between propositional knowledge and know-how .

Une centaine de fragments inédits attribués au plus célèbre représentant de l'Ecole cynique sommeillait depuis près de 2 500 ans sous le doux soleil d'Orient. La voix de Diogène de Sinope, le philosophe-chien, l'homme au tonneau qui embrasse les statues l'hiver, retraverse les âges. Au terme d'une véritable chasse au trésor remontant jusqu'aux penseurs arabes du Xe siècle, Adeline Baldacchino propose pour la première fois en langue française ces nouveaux fragments. On y retrouve avec délice le Diogène insolent dont les (...) paraboles sont si riches d'enseignements. Maître de l'invective, il témoigne de son amour de la vie, de son intransigeance et de sa résistance au pouvoir. Une occasion unique de redécouvrir l'esprit cynique, sa force critique, sa passion pour la liberté et son invraisemblable modernité. (shrink)

It would be good to have a Bayesian decision theory that assesses our decisions and thinking according to everyday standards of rationality — standards that do not require logical omniscience (Garber 1983, Hacking 1967). To that end we develop a “fragmented” decision theory in which a single state of mind is represented by a family of credence functions, each associated with a distinct choice condition (Lewis 1982, Stalnaker 1984). The theory imposes a local coherence assumption guaranteeing that as an agent's (...) attention shifts, successive batches of "obvious" logical information become available to her. A rule of expected utility maximization can then be applied to the decision of what to attend to next during a train of thought. On the resulting theory, rationality requires ordinary agents to be logically competent and to often engage in trains of thought that increase the unification of their states of mind. But rationality does not require ordinary agents to be logically omniscient. (shrink)

The growing interest in fragmentalism is one of the most exciting trends in philosophy of time and is gradually reshaping the contemporary debate. Providing an extensive interpretation of this view, Samuele Iaquinto and Giuliano Torrengo articulate a novel theory of the passage of time and argue that it is the most effective in vindicating the inherent dynamism of reality. Iaquinto and Torrengo offer the first full-range application of fragmentalism to a number of metaphysical topics, including the open future, causation, the (...) A-theoretic interpretation of special relativity and time travel. The resulting picture, they argue, conveys the potential of a radically new understanding of time. (shrink)

Posthumous Fragments: Spring–Autumn 1881 [Excerpts].

In this paper, we articulate a version of non-standard A-theory—which we call Flow Fragmentalism—in relation to its take on the issue of supervenience of truth on being. According to the Truth Supervenes on Being (TSB) Principle, the truth of past- and future-tensed propositions supervenes, respectively, on past and future facts. Since the standard presentist denies the existence of past and future entities and facts concerning them that do not obtain in the present, she seems to lack the resources to accept (...) both past and future-tensed truths and the TSB Principle. Contrariwise, positions in philosophy of time that accept an eternalist ontology (e.g., B-theory, moving spotlight, and Fine’s and Lipman’s versions of fragmentalism) allow for a “direct” supervenience base for past- and future-tensed truths. We argue that Flow Fragmentalism constitutes a middle ground, which retains most of the advantages of both views, and allows us to articulate a novel account of the passage of time. (shrink)

Greek text, english translation and commentary on the surviving fragments of Empedocles (fragments as known in 1981, does not include more recent finds).

Fragmentalism allows incompatible facts to constitute reality in an absolute manner, provided that they fail to obtain together. In recent years, the view has been extensively discussed, with a focus on its formalisation in model-theoretic terms. This paper focuses on three formalisations: Lipman’s approach, the subvaluationist interpretation, and a novel view that has been so far overlooked. The aim of the paper is to explore the application of these formalisations to the alethic modal case. This logical exploration will allow us (...) to study (i) cases of metaphysical incompatibility between modal facts and (ii) cases of modal dialetheias. In turn, this will enrich our understanding of the role of impossibility in the fragmentalist framework. (shrink)

Brenda Almond throws down a timely challenge to liberal consensus about personal relationships. She maintains that the traditional family is fragmenting in Western societies, causing serious social problems. She urges that we reconsider our attitudes to sex and reproduction in order to strengthen our most important social institution, the family.

This paper explores taste fragmentalism, a novel approach to matters of taste and faultless disagreement. The view is inspired by Kit Fine’s fragmentalism about time, according to which the temporal dimension can be constituted—in an absolute manner—by states that are pairwise incompatible, provided that they do not obtain together. In the present paper, we will apply this metaphysical framework to taste states. In our proposal, two incompatible taste states (such as the state of rhubarb’s being tasty and the state of (...) rhubarb’s being distasteful) can both constitute reality in an absolute manner, although no agent can have joint access to both states. We will then develop a formalised version of our view by means of an exact truthmaker semantics for taste assertions. Within this framework—we argue—our linguistic and inferential practices concerning cases of faultless disagreement are elegantly vindicated, thus suggesting that taste fragmentalism is worth of further consideration. (shrink)

Dialectic of Enlightenment is undoubtedly the most influential publication of the Frankfurt School of Critical Theory. Written during the Second World War and circulated privately, it appeared in a printed edition in Amsterdam in 1947. "What we had set out to do," the authors write in the Preface, "was nothing less than to explain why humanity, instead of entering a truly human state, is sinking into a new kind of barbarism." Yet the work goes far beyond a mere critique of (...) contemporary events. Historically remote developments, indeed, the birth of Western history and of subjectivity itself out of the struggle against natural forces, as represented in myths, are connected in a wide arch to the most threatening experiences of the present. The book consists in five chapters, at first glance unconnected, together with a number of shorter notes. The various analyses concern such phenomena as the detachment of science from practical life, formalized morality, the manipulative nature of entertainment culture, and a paranoid behavioral structure, expressed in aggressive anti-Semitism, that marks the limits of enlightenment. The authors perceive a common element in these phenomena, the tendency toward self-destruction of the guiding criteria inherent in enlightenment thought from the beginning. Using historical analyses to elucidate the present, they show, against the background of a prehistory of subjectivity, why the National Socialist terror was not an aberration of modern history but was rooted deeply in the fundamental characteristics of Western civilization. Adorno and Horkheimer see the self-destruction of Western reason as grounded in a historical and fateful dialectic between the domination of external nature and society. They trace enlightenment, which split these spheres apart, back to its mythical roots. Enlightenment and myth, therefore, are not irreconcilable opposites, but dialectically mediated qualities of both real and intellectual life. "Myth is already enlightenment, and enlightenment reverts to mythology." This paradox is the fundamental thesis of the book. This new translation, based on the text in the complete edition of the works of Max Horkheimer, contains textual variants, commentary upon them, and an editorial discussion of the position of this work in the development of Critical Theory. (shrink)

Bayesian confirmation theory is our best formal framework for describing inductive reasoning. The problem of old evidence is a particularly difficult one for confirmation theory, because it suggests that this framework fails to account for central and important cases of inductive reasoning and scientific inference. I show that we can appeal to the fragmentation of doxastic states to solve this problem for confirmation theory. This fragmentation solution is independently well-motivated because of the success of fragmentation in solving (...) other problems. I also argue that the fragmentation solution is preferable to other solutions to the problem of old evidence. These other solutions are already committed to something like fragmentation, but suffer from difficulties due to their additional commitments. If these arguments are successful, Bayesian confirmation theory is saved from the problem of old evidence, and the argument for fragmentation is bolstered by its ability to solve yet another problem. (shrink)

From Descartes to Popper, philosophers have criticized and tried to improve the strategies of reasoning invoked in science and in everyday life. In recent years leading cognitive psychologists have painted a detailed, controversial, and highly critical portrait of common sense reasoning. Stephen Stich begins with a spirited defense of this work and a critique of those writers who argue that widespread irrationality is a biological or conceptual impossibility.Stich then explores the nature of rationality and irrationality: What is it that distinguishes (...) good reasoning from bad? He rejects the most widely accepted approaches to this question approaches which unpack rationality by appeal to truth, to reflective equilibrium or conceptual analysis. The alternative he defends grows out of the pragmatic tradition in which reasoning is viewed as a cognitive tool. Stich's version of pragmatism leads to a radical epistemic relativism and he argues that the widespread abhorrence of relativism is ill founded. (shrink)

Edited with New Translation by Richard McKirahan With a New Preface by Malcolm Schofield This book is a revised and expanded version of A.H. Coxon's full critical edition of the extant remains of Parmenides of Elea—the fifth-century B.C. philosopher by many considered "one of the greatest and most astonishing thinkers of all times." Coxon's presentation of the complete ancient evidence for Parmenides and his comprehensive examination of the fragments, unsurpassed to this day, have proven invaluable to our understanding of the (...) Eleatic since the book's first publication in 1986. This edition, edited by Richard McKirahan and with a new preface by Malcolm Schofield, is released on the 100th anniversary of Coxon's birth. This new edition for the first time includes English translations of the testimonia and of any Ancient Greek throughout the book, as well as an English/Greek glossary by Richard McKirahan, and revisions by the late author himself. The text consists of Coxon's collations of the relevant folios of manuscripts of Sextus Empiricus, Proclus and Simplicius and includes all extant fragments, a commentary, the testimonia, a complete list of sources, linguistic parallels from both earlier and later authors, and the fullest critical apparatus that has appeared since Diels’ _Poetarum Philosophorum Fragmenta _. The collection of testimonia includes the philosophical discussions of Parmenides by Plato, Aristotle and the Neoplatonists, most of which had been omitted by Diels. The introduction discusses the history of the text, the language and form of the poem, Parmenides’ use and understanding of the verb ‘to be’, his place in the history of earlier and later philosophy and the biographical tradition. In the commentary Coxon deals in detail with both the language and the subject matter of the poem and pays full attention to Parmenides’ account of the physical world. The appendix relates later Eleatic arguments to those of Parmenides. (shrink)

Πόλεμος πάντων μὲν πατήρ ἐστι War is the father of all. New : Publication of my book : Histoire du libéralisme in Editions Ellipses, on Fnac or Amazon.1) HERACLITUS : 139 Fragments.a) Heraclitus (PDF) Original Greek text : Diels; English translation : John Burnet (1912), French translation of the English translation (1919), in PDFb) Heraclitus (unicode) : Parallel version or Interlinear version (Work in Progress) Original Greek text : Diels; English translation : John Burnet (1912), French translation of the English (...) translation (1919)c) Heraclitus : Original greek text : original pages of Hermann Diels edition (1912)2) The Encyclopædia Britannica Eleventh Edition (1910–1911) : HERACLITUS (Ἡράκλειτος; c. 540-475 B.C.), Greek philosopher, was born at Ephesus of distinguished parentage. Of his early life and education we know nothing; from the contempt with which he spoke of all his fellow-philosophers and of his fellow-citizens as a whole we may gather that he regarded himself as self-taught and a pioneer of wisdom. So intensely aristocratic (hence his nickname ὀχλολοίδορος, "he who rails at the people") was his temperament that he declined to exercise the regal-hieratic office of βασιλεύς which was hereditary in his family, and presented it to his brother. It is probable, however, that he did occasionally intervene in the affairs of the city at the period when the rule of Persia had given place to autonomy; it is said that he compelled the usurper Melancomas to abdicate. From the lonely life he led, and still more from the extreme profundity of his philosophy and his contempt for mankind in general, he was called the "Dark Philosopher" (ὁ σκοτεινός), or the "Weeping Philosopher," in contrast to Democritus, the "Laughing Philosopher." κακοὶ μάρτυρες Heraclitus is in a real sense the founder of metaphysics. Starting from the physical standpoint of the Ionian physicists, he accepted their general idea of the unity of nature, but entirely denied their theory of being. The fundamental uniform fact in nature is constant change (πάντα χωρεῖ καὶ οὐδὲν μένει); everything both is and is not at the same time. He thus arrives at the principle of Relativity; harmony and unity consist in diversity and multiplicity. The senses are "bad witnesses" (κακοὶ μάρτυρες); only the wise man can obtain knowledge. To appreciate the significance of the doctrines of Heraclitus, it must be borne in mind that to Greek philosophy the sharp distinction between subject and object which pervades modern thought was foreign, a consideration which suggests the conclusion that, while it is a great mistake to reckon Heraclitus with the materialistic cosmologists of the Ionic schools, it is, on the other hand, going too far to treat his theory, with Hegel and Lassalle, as one of pure Panlogism. Accordingly, when he denies the reality of Being, and declares Becoming, or eternal flux and change, to be the sole actuality, Heraclitus must be understood to enunciate not only the unreality of the abstract notion of being, except as the correlative of that of not-being, but also the physical doctrine that all phenomena are in a state of continuous transition from non-existence to existence, and vice versa, without either distinguishing these propositions or qualifying them by any reference to the relation of thought to experience. " Every thing is and is not'"; all things are, and nothing remains. So far he is in general agreement with Anaximander (q.v.), but he differs from him in the solution of the problem, disliking, as a poet and a mystic, the primary matter which satisfied the patient researcher, and demanding a more vivid and picturesque element. Naturally he selects fire, according to him the most complete embodiment of the process of Becoming, as the principle of empirical existence, out of which all things, including even the soul, grow by way of a quasi condensation, and into which all things must in course of time be again resolved. But this primordial fire is in itself that divine rational process, the harmony of which constitutes the law of the universe (see LOGOS). Real knowledge consists in comprehending this all-pervading harmony as embodied in the manifold of perception, and the senses are "bad-witnesses," because they apprehend phenomena, not as its manifestation, but as "stiff and dead." In like manner real virtue consists in the subordination of the individual to the laws of this harmony as the universal reason wherein alone true freedom is to be found." The law of things is a law of Reason Universal (λόγος), but most men live as though they had a wisdom of their own." Ethics here stands to sociology in a close relation, similar, in many respects, to that which we find in Hegel and in Comte. For Heraclitus the soul approaches most nearly to perfection when it is most akin to the fiery vapour out of which it was originally created, and as this is most so in death, " while we live our souls are dead in us, but when we die our souls are restored to life." The doctrine of immortality comes prominently forward in his ethics, but whether this must not be reckoned with the figurative accommodation to the popular theology of Greece which pervades his ethical teaching, is very doubtful. The school of disciples founded by Heraclitus flourished for long after his death, the chief exponent of his teaching being Cratylus. A good deal of the information in regard to his doctrines has been gathered from the later Greek philosophy, which was deeply influenced by it. BIBLIOGRAPHY.—The only authentic extant work of Heraclitus is the περὶ φύσεως. The best edition (containing also the probably spurious Ἐπιστολαί) is that of I. Bywater, Heracliti Ephesii reliquiae (Oxford, 1877); of the epistles alone by A. Westermann (Leipzig, 1857). See also in A. H. Ritter and L. Preller's Historia philosophiae Graecae (8th ed. by E. Wellmann, 1898); F. W. A. Mullach, Fragm. philos. Graec. (Paris, 1860); A. Fairbanks, The First Philosophers of Greece (1898); H. Diels, Heraklit von Ephesus (2nd ed., 1909), Greek and German. English translation of By water's edition with introduction by G. T. W. Patrick (Baltimore, 1889). For criticism see, in addition to the histories of philosophy, F. Lassalle, Die Philosophie Herakleitos' des Dunklen (Berlin, 1858; 2nd ed., 1892), which, however, is too strongly dominated by modern Hegelianism; Paul Schuster, Heraklit von Ephesus (Leipzig, 1873); J. Bernays, Die heraklitischen Briefe (Berlin, 1869); T. Gomperz, Zu Heraclits Lehre und den Uberresten seines Werkes (Vienna, 1887), and in his Greek Thinkers (English translation, L. Magnus, vol. i. 1901); J. Burnet, Early Greek Philosophy (1892); A. Patin, Heraklits Einheitslehre (Leipzig, 1886); E. Pfleiderer, Die Philosophie des Heraklit us von Ephesus im Lichte der Mysterienidee (Berlin, 1886); G. T. Schäfer, Die Philosophie des Heraklit von Ephesus und die moderne Heraklilforschung (Leipzig, 1902); Wolfgang Schultz, Studien zur antiken Kultur, i.; Pythagoras und Heraklit (Leipzig, 1905); O. Spengler, Heraklit. Eine Studie ilber den energetischen Grundgedanken seiner Philosophie (Halle, 1904); A. Brieger, "Die Grundzüge der heraklitischen Physik " in Hermes, xxxix. (1904) 182-223, and "Heraklit der Dunkle" in Neue Jahrb. f. das klass. Altertum (1904), p. 687. For his place in the development of early philosophy see also articles IONIAN SCHOOL OF PHILOSOPHY and LOGOS. Ancient authorities: Diog. Laert. ix.; Sext. Empiric, Adv. mathem. vii. 126, 127, 133; Plato, Cratylus, 402 A and Theaetetus, 152 E; Plutarch, Isis and Osiris, 45, 48; Arist. Nic. Eth. vii. 3, 4; Clement of Alexandria, Stromata, v. 599, 603 (ed. Paris). (J. M. M.) Copyright. (shrink)

In PHILOSOPHICAL FRAGMENTS, Søren Kierkegaard (writing under the pseudonym Johannes Climacus), seeks to explain the nature of Christianity in such as way as to bring out its demands on the individual, and to emphasize its incompatibility with the theology based on the work of Hegel that was becoming progressively more influential in Denmark. If one were to read only two or three of Kierkegaard's works, this is unquestionably one of the ones to read. One cannot understand Kierkegaard's thought without reading (...) this book, and along with its sequel represents the heart of what he was trying to achieve in what he called his "Authorship." Through PHILOSOPHICAL FRAGMENTS, Kierkegaard purports to present the logic of Christianity. (shrink)

Life in Fragments is a continuation of the themes and motifs explored in Zygmunt Bauman's acclaimed study, Postmodern Ethics (Blackwell, 1993).

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined (...) in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the (...) extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. (shrink)

One of the dominating themes in the first part is the negative treatment that Marx’s concept of labor has received by late critical theorists, particularly Habermas. While supportive of the rejection of Marx’s economic functionalism entailed by Habermas’s adoption of communicative action as the basic category of critical theory, Honneth worries about the indifference towards the normative potential of labor that he sees in most twentieth-century social theory. Honneth agrees with critics of reductionism that labor is neither the only form (...) of social reproduction, nor a primary source of emancipatory consciousness, as the fragmentation and depolitization of a working class that has suffered the effects of Taylorism clearly manifests. But Honneth is still convinced that the experience of recognition is to a large extent associated with work, and that Habermas tends to understand interaction in terms of instrumental action, making the concept of interaction inadequate to capture the sort of normative claims expressed by workers who do not see themselves sufficiently recognized in the labor they do. Although Honneth acknowledges that he has not settled how to navigate between these competing interests, the clarity of the call for a more complex concept of labor justifies the inclusion of these papers in the volume. (shrink)

Kris McDaniel argues that there are different ways in which things exist. For instance, past things don't exist in the same way as present things. Numbers don't exist in the same way as physical objects; nor do holes, which are real, but less real than what they are in. McDaniel's theory of being illuminates a wide range of metaphysical topics.

We strengthen a result of Harrington and Shelah by showing that, unless ω1 is an inaccessible cardinal in L, a relatively weak fragment of Martin's axiom implies that there exists a δ13 set of reals without the property of Baire.

I criticize the idea that theories of ‘fragmented’ or ‘compartmentalized’ belief (as found in, e.g., Lewis 1982, Egan 2008) can help to account for the puzzling phenomena they are often taken to account for. After introducing fragmentationalism and a paradigm case that purportedly motivates it, I criticize the view primarily on the grounds that the models and explanations it offers are at best trivial—as witnessed by examples of over-generation—and should be seen as merely re-describing in figurative terms the phenomena it (...) is designed to account for. I also point out that fragments, as used in these theories, are not likely to be psychologically real in any robust sense and so cannot be appealed to on such grounds. (shrink)

This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic |$FL_{ew}$| (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. |$FL_{ew}$|-algebras) that includes both Heyting and Nelson algebras (...) and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics. (shrink)

The publication of FRAGMENTS OF A POETICS OF FIRE is a milestone in Bachelard studies that will influence the way we think about his themes & method for a long time to come. Dissatisfied with his earlier attempt to come to terms with the element of fire in "The Psychoanalysis of Fire" (1937), Bachelard returned to this theme in the book he was working on at the time of his death in 1962. Because of delays in &, eventually, the abandonment (...) of a projected edition of his complete works, these FRAGMENTS OF A POETICS OF FIRE remained unpublished & their very existence unknown to all but a handful of Bachelard's readers. The author's daughter, Suzanne Bachelard, edited them for separate publication over a quarter-century later in 1988. For the first time we have an insight into the way Bachelard constructed his remarkable books. Miss Bachelard's introduction & extensive notes are an indispensable guide to the workings of his mind as "he shapes a meandering series of observations on the phoenix, Prometheus, & Empedocles into a coherent & engaging structure that respects the fluidity & openness of a living image - the powerful image of fire.". (shrink)

We identify complete fragments of the simple theory of types with infinity and Quine’s new foundations set theory. We show that TSTI decides every sentence ϕ in the language of type theory that is in one of the following forms: ϕ=∀x1r1⋯∀xkrk∃y1s1⋯∃ylslθ where the superscripts denote the types of the variables, s1>⋯>sl, and θ is quantifier-free, ϕ=∀x1r1⋯∀xkrk∃y1s⋯∃ylsθ where the superscripts denote the types of the variables and θ is quantifier-free. This shows that NF decides every stratified sentence ϕ in the language (...) of set theory that is in one of the following forms: ϕ=∀x1⋯∀xk∃y1⋯∃ylθ where θ is quantifier-free and ϕ admits a stratification that assigns distinct values to all of the variables y1,…,yl, ϕ=∀x1⋯∀xk∃y1⋯∃ylθ where θ is quantifier-free and ϕ admits a stratification that assigns the same value to all of the variables y1,…,yl. (shrink)

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)