Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and (...) add Zermelo's choice function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

The year 1897 saw the publication of the first of the modern logical paradoxes. It was published by Cesare Burali-Forti, the Italian mathematician whose name it has come to bear. Burali-Forti's own formulation of the paradox was not altogether satisfactory, as he had confused well-ordered sets as defined by Cantor with what he himself called “perfectly ordered sets”. However, he soon realized his mistake, and published a note admitting the error and making the correction. He (...) concluded the note with the observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the “perfectly ordered sets” for which it had first been obtained. We shall reproduce his results in their corrected form. (shrink)

The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We (...) contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali-Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic. (shrink)

After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( (...) 1930 ) to secure ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair. (shrink)

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like (...) Russell’s paradox, is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership (...) distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. (shrink)

Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a (...) predicative logic. Such a retreat has implications for other neo-Fregean theories, including the cardinal arithmetic on which Hale has focused. The discussion will touch on a topic which has been at the centre of Hale’s more recent work—namely, the interpretation of plural and higher-order quantifiers. (shrink)

A key idea in both Frege's development of arithmetic in theGrundlagen[7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a “type reducing” correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains betweenconceptandnumber, in Zermelo's (through the axiom of choice), betweensetandmember. In this paper, a formulation is given and a detailed investigation undertaken of a system ℱ of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of (...) type reducing correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within ℱ, any correspondencecbetween second-level entities (here calledconcepts) and first-level ones (here calledobjects) induces a well-ordering relationW(c) in a canonical manner. We shall see that, whencis the “Fregean” correspondence between concepts and cardinal numbers,W(c) is (the well-ordering of) the ordinalω+ 1, and whencis a “Zermelian” choice function on concepts,W(c) is a well-ordering of the universal concept embracing all objects.In ℱ an important role is played by the notion ofextensionof a concept. To each conceptXwe assume there is assigned an objecte(X) in such a way that, for any conceptsX, Ysatisfying a certain predicateE, we havee(X) =e(Y) iff the same objects fall underXandY. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

In lieu of an abstract, here is a brief excerpt of the content:Objective Reality of Ideas in Descartes, Caterus, and Su irez NORMAN j. WELLS IT HAS LONG BEEN ACKNOWLEDGEDthat Francisco Sufirez's distinction between a formal and an objective concept exercised some influence upon Descartes's teaching on 'idea'.' It would appear, however, that not enough attention has been given to that distinction of Sufirez (and especially to another to be mentioned shordy) to aid in dispelling what I take to be (...) a widespread confusion and misunderstanding with regard to 'idea' in Descartes, especially with respect to 'idea' taken 'objectively'. Much, if not all, of the confusion in question has to do with Descartes's admittedly equivocal use of the term 'idea'. For he had allowed that 'idea' could be taken 'materially' (materialiter), on the one hand, for the knowing activity of the intellect. On the other hand, it could be taken 'objectively' (objective) for the thing (res) represented by that knowing activity.' At this point, the contemporary reader can only be brought up short, See E. Gilson, Index scolastico-cart~sien (New York: Burt Franklin Reprint, 1912), 48-49; Discours de la m~thode: Texte et commentazre (Paris: J. Vrin, 1947), 321; G. Rodis-Lewisin her edition of Meditationes de Prima PhRosoph.m (Paris: J. Vrin, 1953), 41 n. 1; L'oeuure de Descartes (Paris: J. Vrin, 1971), 53~ n. 8; J. Collins, Modern European Philosophy (Milwaukee: Bruce, 1954), 165;T.J. Cronin, S.J., Objecttve Being m Descartes and SuArez (Rome: Gregorian University Press, 1966); "Objective Reality of Ideas in Human Thought: Descartes and Su~rez," Wisdom in Depth: Essays in Honor of Henri Renard, SJ., eds. V. F. Daues, S.J., M. R. Holloway, S.J., L. Sweeney, S.J. (Milwaukee : Bruce, 1966), 68-79; L. Gilen, S.J., "Uber die Beziehungen Descartes zur zeitgenossischen Scholastik," Scholastik 32 (1957): 41-66; R. Dalbiez, "Les sources scolastiques de la th~orie cart6sienne de l'&re objectif," Revue d'histoire de la philosophic 3 (1929): 464-72;J. Owens, C.S.S.R., in The Cambridge HtstoU of Later Medu,val Philosophy, eds. N. Kretzmann, A. Kenny, J. Pinborg (Cambridge University Press, 1982), 459; G. Nuchelmans,Judgment and Propositwnfrom Descartesto Kant (Amsterdam: North Holland Publishing Co., 1983), 36-54. " Oeuvres de Descartes, ed. C. Adam et P. Tannery, 11 vols. (Paris: J. Vrin, 1897-19o9; hereafter cited according to volume, page and line), Praefatio ad Lectorem, 7: 8. 19-251 "Sed respondeo hic subesse aequivocationem in voce ideae: sumi enim potest vel materialiter, pro operatione intellectus, quo sensu me perfectior dici nequit, vel objective, pro re per istam operationem repraesentata, quae res, etsi non supponatur extra intellectum existere, potest tamen me esse perfectior ratione suae essentiae." It is clear that two realities are to be causally accounted for. The [33] 34 JOURNAL OF THE HISTORY OF PHILOSOPHY 28"1 JANUARY 1990 conditioned as we are by an appreciation of 'objectivity' altogether extramental! It makes for a considerable wrench of perspective to do justice to Descartes's acknowledged doctrine of intramental objectivity and an intramental 'thing represented' (resrepraesentata). 'Idea', then, in traditional Cartesian fashion, designates the act of representing as well as the 'thing represented '. S Yet, for all this, and despite repeated expressions of the same doctrine, there is a persistent tendency in the history of Cartesianism and its scholarship to interpret 'idea' taken 'objectively' as in some way representative. Contrary to what I take to be Descartes's constant position, such a rendering unwittingly transfers that function which is proper to the cognitive activity, mind is the efficient cause of its cognitive activity. With respect to ideas taken 'objectively', such as the ideas of God and the triangle on the levels of metaphysical and mathematical intelligibility, an efficient cause, extrinsic to the mind, is certainly at issue. For here we are in the presence of what is independent of the mind, however intramental it may be. On the triangle, see Medit. 5; 7: 64. ]4s7 : "...est tamen profecto determinata quaedam ejus natura, sire essentia, sive forma, immutabUis et aeterna, quae a me non efficta est, nec a mente mea dependet." For the idea of God, see Medit. 5; 7: 68. l o-12: "Nam sane multis modis inteUigo illam [ideam... (shrink)

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only use (...) arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

The book reconstructs the history of Western ethics. The approach chosen focuses the endless dialectic of moral codes, or different kinds of ethos, moral doctrines that are preached in order to bring about a reform of existing ethos, and ethical theories that have taken shape in the context of controversies about the ethos and moral doctrines as means of justifying or reforming moral doctrines. Such dialectic is what is meant here by the phrase ‘moral traditions’, taken as a name for (...) threads of moral discourse, made in turn of other interwoven threads, including transmission of shared codes, appeals to reform of prevailing custom, rational argument about the justification of some precept on the basis of some shared general teaching or principle, and rational argument about the ultimate basis for principles and justification of authoritative teaching. That is, the approach adopted to the reading of ethical texts depends on a firm belief in the fact that philosophers hardly created any ethical doctrine out of nothing. The main point this book tries to highlight is how different philosophical theories emerged and followed each other as a result of attempts at accounting for what was going on in the world of moral traditions. Changes were propelled by controversies between different schools, and highly abstract arguments were the unintended effects of moves made by controversialists forced to transform (and occasionally to turn upside down) their own doctrines in order to face the challenge posed by other arguments. This is the reason why the book examines not only texts that already enjoy pride of place in the history of philosophy (Aristotle, Kant, Hegel), but also other texts usually treated in the histories of religions (the Bible, the Talmud, the Quran), and others considered to be much less philosophical (Plutarchus, Pufendorf). -/- 1. Plato and a response to ethical scepticism. Two different traditions of morality in VI-V century Greece are described. The birth of philosophical questioning of traditional morality and temporal and spatial variation of custom is described within the context of the v century crisis, the demise of traditional aristocratic and tyrannical rule and the birth of democracy. Two conflicting answers to the challenge are reconstructed, namely conventionalist or immoralist theories formulated by the Sophists and the eudemonist and intellectualist Socratic theory. Plato’s own reformulation of Socrates answer to the Sophists is reconstructed. His psychological views, his classification of the four cardinal virtues and his political theory are described as parts of a unitary system, culminating in an extremely realist moral ontology identifying the idea of the good with the essence of the (moral and extra-moral) world itself. -/- 2. Aristotle and the invention of practical philosophy. Aristotle’s invention of practical philosophy as a field separated from first philosophy is shown to be an implication of his break with Plato moral ultra-realism. Aristotle’s agenda in his moral works is arguably dependent on a polemical intention, namely dismantling Socratic intellectualism. The semi-inductive or virtuously circular method of practical philosophy is illustrated, starting with the received opinions of the better and wiser individuals and trying dialectically to sift what is left of mistake and inconsistence in such opinions, finally trying to correct mistakes and make the overall practical science more consistent. The chapter illustrates then the relationship of individual ethics, or ‘monastics’, with the art and the science of the pater familias, or ‘economics’, and the science of the ruler and citizen, or politics. The nature of virtues, or better, excellences of character, is discussed, highlighting the basic role of hexis, or ‘disposition’. Prudence, or better practical wisdom, is the focus of the chapter. Its relationship with bouleusis or deliberation is examined, and its autonomous status vis-à-vis theoretical knowledge is stressed. -/- 3. Diogenes and philosophy as a way of life. The chapter provides an overview of Hellenistic ethics, which almost amounts to Hellenistic schools of philosophy, in so far as ‘philosophy’ became in these centuries primarily the name for a way of life. The typical character of the Cynical movement is highlighted, that of a school of life, not a school aimed at providing any kind of intellectual training was to be provided. -/- 4. Epicurus and ethics as self-care. The peculiar character of the Epicurean school is described, a combination of a science of well-being aiming, more than at pleasure as in the popular view, at reduction of useless suffering, of unnecessary needs, and at a balanced selection of pleasures of the best and most durable kind. -/- 5. Epictetus and ethics as therapy of the passions. The various phases of Stoicism are described, and the shifting place given to ethics in the Stoic system of idea, culminating in the paradoxical view of ethics, its impossibility in principle notwithstanding, as the only truly significant and necessary part of philosophy. Cicero is treated, showing how his own synthesis of various Hellenistic trends is as a truly philosophical enterprise, deserving serious consideration after one or two centuries when he was confined to the role of literate. Epictetus is chosen as the best example of what the Stoic tradition could yield, an art of living based on sophisticated introspection, in turn aimed at making a kind of cognitive therapy possible, dismantling obnoxious passions at their root by systematically correcting false representations. -/- 6. Philo and the reconciliation of Torah and Platonism. A reconstruction of basic ideas from a few books of the Hebrew Bible is provided, starting with the Prophetic tradition and the focus on God’s mercy as the source of motivation and standard for human behaviour. Then a comparative analysis is undertaken of a parallel tradition, namely the three codifications of the Torā (Law or, better, Instruction), highlighting how a core of moral ideas may be recognized as a basis and preamble of codification of civil law, cultural practice, and regulation of ritual purity. The importance of Leviticus is stressed as the turning point when emphasis mercy, typical of the Prophetic tradition, is combined with the legal tradition, yielding the change in sensibility of the Second Temple time. Philo of Alexandria is described as one of the three leading figures – together with Rabbi Hillel and Rabbi Yeshua – at the apex the emergence of the mentioned new sensibility, gradually including mercy as an essential part of justice and establishing the starting point of both the Rabbinic and the Christian tradition. This consists precisely in the precept of one’s neighbour’s love or of the golden rule, two eventually identical precepts whose meaning is arguably more sober and sensible than the long-lasting Christian tradition deriving from John and Augustine has made us believe and no novelty vis-à-vis so-called Ancient-Testament teachings. -/- 7. Augustine and Christianity as Neo-Platonism. The first section examines the moral doctrines of so-called 'ethical Treaties' from the Talmud, a group of treaties, among which the best known is Pirqé Avot, that were left out of the six "orders" of the canon as they did not fit in any of the six groups of issues ritual or legal on which the division was based. According to Maimonides, their peculiar theme is provided by the Deòt, 'opinions', i.e., mental dispositions, that is, the translation of the Greek term hexeis and Arab akhlak (in turn providing in this language the name for ethics as such). The three topics I reconstruct are: i) the notion of Torah: The Torah is understood as the world order itself, or as the ‘Wisdom’ that existed even before creation and was ‘the tool by which the world was built’; however, the Torah is an earthly and human entity, as it was "received" by humans, and from that very moment belongs to them; ii) the relationship between love of God and love of neighbour; the treaties require us to study and practice the Torah ‘for its own sake’, that is, require us to act out of love, not out of fear or hope of reward; iii) the idea of sanctification of daily life: having disappeared with the destruction of the Temple the possibility of any conflict between liturgical service and everyday life, the latter is assumed to be in itself divine service: to give food to the poor has the same value as sacrifices in the Temple, and as an implication, the insistence become recurrent on the goodness of created things in themselves along with a polemic against ascetic currents. The conclusions drawn are: i) the moral teachings of the Talmud and those of Yeshua are, rather than similar, virtually identical; one may safely say that the precept of love and the golden rule are central ones for all Talmudic rabbis, that mercy plays an indispensable role alongside with justice, and the latter is not a different thing from one’s neighbour’s love; ii) a peculiarity of Talmud rabbis facing Yeshua is the idea of study as worship, and knowledge as a source of justice; but this is an idea of Judaism after the Temple's destruction that cannot be attributed to the Pharisees of Yeshua’s time; iii) the relation of study and practice in the Talmud parallels that between faith and deeds in Paul's epistles, that is, respectively faith or learning are a necessary and sufficient condition to be recognized as righteous , but deeds are the inevitable effect of either faith or learning. The sayings ascribed to Yeshua are examined first, yielding the conclusion that a close equivalent may be found for every saying in Talmudic literature and yet the whole is ascribed to one rabbi, with rather consistent stress on God’s mercy and unconditional forgiving as the mark of true imitation of God. Thus, Yeshua’s teaching is pure Judaism. The third section describes briefly the galaxy of Gnostic currents and Manichaeism, trying to sketch the profile of moral teachings resulting from an encounter of Asian spiritual traditions, Hellenistic lore and sparks of teachings from apocalyptic Jewish currents. The last section summarizes the turbulent history of several encounters between Christian currents and Hellenistic philosophical schools. The first one was with late Cynicism. Recent, rather controversial, literature discovered the jargon and a few topics from the Stoic-Cynical popular philosophy in a few books from the New Testament itself. This, far from proving that Rabbi Yeshua had been influenced by cynic preachers, is a proof of the necessity felt by Christian preachers to translate the original ‘Christian’ teaching into a Greek lexicon deeply impregnated with cynic terminology. The second was with Platonism, yielding the mild and temperate moral teaching of Clemens of Alexandria, teaching the sanctity of nature and the human body, the joy of moderate fruition of ‘natural’ kinds of pleasures, and the beauty of the marital life – in short, the opposite of the standard picture of Medieval Christianity. Ambrose of Milan brings about a different kind of synthesis, namely with Middle Platonism, where Stoic themes prevail. The most shocking case is Augustine, where an early Manichean education is overcome in a former phase by a synthesis of Plotinian Neo-Platonism and Christian preaching, yielding a sustained polemic with the Manicheans and rather optimistic views on life and Creation, the body and sexuality, and Hebrew-Judaic tradition not far from Clemens of Alexandria. In a later phase, occasioned by controversy on the opposite front, with such Christian currents as the Pelagians and Donatists, Augustine comes back to heavily anti-Judaic and world-denying ascetic attitudes where is earlier Manichean upbringing seems to emerge again. The tragedy of medieval Christianity will be the later Augustine’s overwhelming influence. A final section is dedicated to the monastic tradition where a curious mixture of world-denying asceticism with an astonishingly penetrating ‘science’ of introspection emerges. -/- 8. Al Farabi and the reconciliation of Islam and Platonism. The Qurān and the qadit, that is, collections of sayings ascribed to the prophet Muhammad contain a wealth of precepts and catalogues of virtues mirroring moral contents from the Jewish and the Christian traditions, among them the basic notion of imitation of the Deity, where mercy is assumed to be its basic trait. An important tradition within Islam, namely Sufism, stressed to the utmost degree the role of mercy, turning Islam into a mystic doctrine centred on retreat from the world, abandonment to God’s will, and a peaceful and fraternal attitude to our fellow-beings. A secular literary tradition originating in the Sassanid Empire, the literature of advice to the Prince had for a time a widespread influence in the newly constituted Arabic-Islamic commonwealth. In a later phase, the legacy of Hellenic philosophy made its way into the intellectual elite of the Islamic world. The first important legacy was Platonic, and the Republic became the main text for Islamic Platonism. Al-Farabi wrote an enjoyable remake of the Platonic Republic, arguing that the citizen’s virtues were the basis on which the political building needs to rest. The secular lawgiver is enlightened by the light of reason in his everyday practice of governance, but room is made for the Prophet as the voice of revealed truth that confirms and completes the rational truth that philosophy may afford. In a second phase also Aristotelian and Stoic influence were assimilated. Miskawayh’s treatise was the masterwork at the time Arabic philosophy reached its Zenith. It is a treatment of the soul’s diseases and their remedies, combining the Aristotelian doctrine of the golden mean with the Stoic doctrine of the passions and elements of Galenic medicine. Towards the eleventh-twelve centuries a war raged among theologians and philosophers, finally won by the former with disappearance of philosophy as such. The newly established mainstream, yet, was no kind of intolerant fanaticism. It drew from the work of mystic theologian Al-Ghazālī, the best heir of Sufism, teaching a tolerant and peaceful attitude to our fellow-beings and a passive attitude to destiny as an expression of the Divine will. -/- 9. Moshe ben Maimon and the reconciliation of Torah and Aristotelian ethics. The encounter between the Talmudic tradition and Hellenic philosophy had taken place a first time in Alexandria at the time of Philo but the two traditions had parted way again. In fact, the kind of Platonic Judaism founded by Philo survived only within Christianity, in the fourth Gospel and then in writings by Clemens of Alexandria. The Talmudic literature had absorbed just less striking traits from the Hellenic Philosophy, namely an idea of ethics as care of the self and a role for the education of character as propaedeutic to theoretical knowledge. In the Arabic-Islamic world, a second round started when Jewish authors writing in Arabic undertook the task to prove the full compatibility of the tradition deriving from Torah and Platonic philosophy. The culmination of this attempt is provided by Moshe ben Maimon who tried to use Aristotelian ethics as a language into which the teaching from the Pirké Avot could be translated. -/- 10. Thomas Aquinas and the reconciliation of Christian morality and Aristotelian ethics. In the first section the fresh start is described of philosophical ethics in Latin Europe at the turn of the Millennium. In a first phase, Anselm and Peter Abelard built on a Platonic and Augustinian legacy. In this phase. remarkable innovations are introduced, including Abelard’s claim of the obliging character of erring consciousness. In a second phase, the rediscovery of Nicomachean Ethics thanks to Latin translation of Arabic versions gives birth to a new wave of ethical studies, recovering the very idea of Aristotelian practical philosophy, with the potential implication of full legitimization of natural morality, i.e. ethics without Revelation. Aquinas’s ethics is a theological ethics out of which it would be vain to try to extract a self-contained philosophical ethics. His treatment of topics in philosophical ethics, yet, does not boil down to repetition of Aristotelian arguments but is rather a creative reshaping of such arguments. For example, he introduces also in the practical philosophy a first principle parallel to the principle of non-contradiction; and he also carries out a synthesis of Aristotelianism, Stoicism, and neo-Platonism. Even though it is essentially moral theology, Aquinas’s doctrine - unlike Augustine – grants full citizenship to "natural" morality, firstly by rejecting the claim that the corruption of human nature due to the original sin is so radical as to leave "pure nature" incapable of moral goodness. The doctrine is presented in a more sophisticated formulation in a few of the Quaestiones, such as De Malo and De Veritate, in the Summa contra Gentes and in the commentaries to Aristotle than in the famous Summa Theologica, but the latter work includes the only or the largest exposition of some decisive part of the theory. Thus, the Summa Theologica should be read for what it is more than criticized for not being what it was not meant to be. It was not meant to be the brilliant synthesis of all that Reason he had been able to produce with what Revelation had added about which the Neo-Thomists used to dream, but rather a manual for the training of preachers and confessors, where theoretical claims are not too demanding and a few serious tensions are left. Besides a jump between the Prima Secundae and the Secunda Secundae, being the former an essay in virtue theory and the latter a handbook for confessors, the most serious tension is perhaps the one between the ethics of right reason presented in most of Ia-IIae and the eudemonistic ethics developed in quaestiones 1-5 of the same part; the alternative ethical theory which also may be found in Augustine, the Stoic view of a cosmic reason eventually coincident with the moral law, was believed by Anselm (followed by John Duns Scotus and William of Ockham) to be incompatible with eudemonism. It is questionable whether Thomas could reduce the tension proving that it is just an apparent tension, in so far as the right reason and bliss derived from knowledge of God tend to coincide, but this is just a conjecture. Thomas’s ethics is a virtue ethic, not a law-based one, and moral judgment focuses on the virtues, particularly charity, not on the commandments and even less on absolute prohibitions; Thomas, however, would not have considered a drastic alternative between law and the virtues such as the one which has been advanced in late twentieth-century philosophy to be justified. Nonetheless, when it discusses ‘special’ virtues, it ceases to be an ethics of virtue and becomes a disappointing and often contradictory discussion of legal and illegal acts. Such a discussion takes most of the time ‘reasonable’ middle positions on controversial issues but not the alternative approach that Aristotelianism would have made possible; even when some occasional Aristotelian claim shows up, such as money’s barrenness as a reason against usury, this seems to be made by an author who apparently ignores the Aristotelian Thomas of the Prima Secundae. It is an ethic of human autonomy which recognizes the binding character of the individual conscience and, potentially, even a duty to disobey unjust laws. It is true that what Thomas writes in his discussion of death penalty and persecution of heretics is simply disgusting, and yet we should blame Thomas the man, not Thomist ethical theory. Finally, Thomas’s ethics is not ethical ‘absolutism’, as both traditionalist Catholic theologians and secularist enemies of dogmatic morality tend to believe. It is instead an ethic of prudential judgment. Exceptions to this approach – or better results of logical fallacies – are such claims as the absolute character of negative precepts and of the existence of "intrinsically evil acts", claims that simply cease to make sense in the light of Thomas’s own distinction between human act and natural act, carrying consequences Thomas did not live long enough to draw or was not consistent enough to dare to draw. -/- 11. Francisco de Vitoria and casuistry. The first topic illustrated is the discussion about the notion of pura natura, namely the human condition after the original sin and before divine revelation. The core notion was already there in Aquinas but was later developed by Caietanus (Thomas de Vio) with a number of interesting implications: firstly a view of human nature as such much more positive than Augustine’s and his followers’ view, including both Calvinists and Jansenists; secondly, the implication that the philosopher’s morality, as opposed to the Christian’s morality, is a quite respectable kind of morality; thirdly, that theocracy and prosecution of the unfaithful lacked justification, with more far-fetching consequences than Aquinas himself had dared to draw. The second topic is casuistry, a new name for a comparatively older way of thinking, which was already there in late Stoicism and Cicero as well as in the Talmudic literature. This is an approach aimed at yielding probable enough solutions for doubtful cases even in absence of completely safe staring points. The genre developed from medieval reference books for confessors and became by the sixteenth-century a sophisticated tool-box for dealing with various fields of applied ethics. Francisco Vitoria, the main authority of Baroque Scholasticism, was a controversial figure, among the proponents of the new discipline of casuistry and a consistent proponent of more separation between the religious and the civil authority, stricter limits to the legitimacy of war, innate rights of non-Christian nations in the New World based on the notion of pura natura, providing a standard of ‘natural’ goodness, previous to revelation, on which the Indian nations were judged to live in a naturally good condition, provided with the institutions of family, justice, and religion had accordingly a right to full respect by Christian sovereigns. Bartolomé de Las Casas, arguing along similar lines, was the leader of a historical battle in defence of the rights of the Indios. -/- 12. Michel de Montaigne and the art of living. A short description of one of the Renaissance Phyrronism, one of the classical philosophies revived in the fifteenth and sixteenth century. Montaigne combines the sceptical epistemology with an understanding of ethics – indeed of philosophy as a whole – in terms of an art of living inspired by two basic ideas, sagesse, that is, practical wisdom as the only kind of rationality available after theology, science, and tradition have declared bankrupt, and an aesthetic ideal of self-transformation through the art of writing and introspection. -/- 13. Pierre Nicole and neo-Augustinianism. The chapter illustrates first the vicissitudes of Augustinianism newly born after the late medieval triumph of Thomist Aristotelianism in the alternative Protestant and Catholic versions processed by the Calvinists and the Jansenists. Then it illustrates the doctrines of Pierre Nicole, Blaise Pascal’s best disciple, on the impossibility of introspection, on the ubiquity of self-deception, on the incurably evil character of the passion of love, and on the two moralities, the one of the man of the world, the morality of honesty which is indispensable for granting peace and order but devoid of any true value for eternal salvation, since the same external behaviour may be prompted by opposite motivations, and the morality of charity, the only true one but also useless to society. The third topic is Pierre Malebranche’s view of self-deception as a ubiquitous phenomenon accounting for human action and responsibility and his reformulated theory of self-love, no less ubiquitous than for Nicole and yet not as incurably evil, given the distinction between morally indifferent and even necessary amour de soi and vicious amour proper, a distinction on which the whole of Rousseau’s moral and political theory rests. -/- 14. Samuel von Pufendorf and the new science of morality. The chapter is dedicated to the discovery of the idea of a ‘new moral science’. Hugo Grotius is discussed first highlighting the real character of his project, much closer to the Thomist idea of a natural law accessible to non-corrupted human reason when human beings are living in a state of pura natura. Then Hobbes’s combination of scepticism, voluntarism, and conventionalism is described and both the continuity with Grotius’s new science, in the search for non-revealed rational morality, and the break with him, in the adoption of voluntarism and refusal of an intellectualist view of natural law, are illustrated. Pufendorf’s work is illustrated as a synthesis of the two previous attempts and the – up to Schneewind underestimated – paradigmatic example of the new science of morality. -/- 15. Richard Cumberland and consequentialist voluntarism. The chapter gives an overview of eighteenth-century Anglican ethics, noticing how the Cambridge tradition gave special weight to natural theology as opposed to positive or revealed theology – and how two Cambridge fellows, John Gay and Thomas Brown, elaborated on Cumberland’s (and Malebranche’s, as well as Leibniz’s) strategy for finding a third way between intellectualist view and voluntarist view of the laws of nature. The result of their elaboration was a kind of a rational-choice account of the origins of natural laws, where a law-giver God chooses among a number possible sets of laws on a maximizing criterion, and God’s maximandum is happiness for his creatures. The chapter notices also how such a solution aimed at solving at once the problem of evil and that of the foundation of moral obligation by proving how God’s choice was justified as far as it was the one minimising the amount of suffering in the world. 16. Richard Price and intuitionism. The chapter describes first the doctrines of the Cambridge Platonists, an example of hyper-rationalist reaction to Calvinism. Secondly it describes the sophisticated and universally ignored – from Sidgwick to Anscombe –version of what was later labelled ‘ethical intuitionism’ – showing how it escapes familiar objections and misrepresentations of intuitionism – from Mill to Rawls – in grounding its argument on transcendental arguments while carefully avoiding recourse to introspection and psychological evidence, which has been taken as a too easy target by critics of intuitionism. Thirdly, the chapter discusses Whewell’s ‘philosophy of morality’, as opposed to ‘systematic morality’, not unlike Kant’s distinction between a pure and an empirical moral philosophy. Whewell worked out a systematization of traditional normative ethics as a first step before its rational justification; he believed that the point in the philosophy of morality is justifying a few rational truths about the structure of morality such as to rule hedonism, eudemonism, and consequentialism; yet a system of positive morality cannot be derived solely from such rational truths but requires consideration of the ongoing dialectics between idea and fact in historically given moralities. Whewell’s intuitionism turns out closer to Kantian ethics than commentators have made us believe until now, and quite different from what Sidgwick meant by intuitionism. -/- 17. Adam Smith and the morality of role-switch. The chapter describes first, Hutcheson’s attempt at basing the ‘new science’ of natural law on different bedrock than Pufendorf and the English Platonists, namely a moral sense, a faculty whose existence is assumed to be proved on an empirical basis. The second step is a reconstruction of Hume’s rejoinder in terms of a new science of man including morality on an ‘experimental’ basis, that is, a ‘Newtonian’ hypothetical-deductive approach, distinguished from Hutcheson’s allegedly uncritical descriptive approach to human nature. The third step is a reconstruction of Adam Smith theory of morality understood as emerging from a spontaneous interplay of exchanges of situations (the most basic meaning of the word ‘sympathy’ as construed by this author). -/- 18. Jeremy Bentham and utilitarianism. The bulk of the chapter is dedicated to the doctrines of Jeremy Bentham. The Enlightenment spirit that suggested the idea of a new morality, free from religion, is illustrated. The notion of utility is illustrated as well as the subsequent formulations of the principle of utility. The idea of felicific calculus is discussed, showing how its inner difficulties prompted several reformulations of the principle of utility in order to avoid undesirable implications of proposed formulations. The role of the thesis of spontaneous convergence between interest and duty is discussed, showing how it left numberless questions open, and the distinction between the virtues of prudence, justice and beneficence is described as a way out of the deadlocks of classical utilitarianism. After Bentham, Mill’s reformed utilitarianism is reconstructed, showing how it is a kind of mixed system – as closer to common sense as it gives up Bentham’s claims to consistency and simplicity – resulting as unintended consequences from the controversies in which Mill was too keen to engage. The hidden agenda of the controversy between Utilitarianism and Intuitionism, going beyond the image of the battle between Prejudice and Reason, is described, showing how both competing philosophical outlooks turn out to be more research programs than self-contained doctrinal bodies, and such programs appear to be implemented, and indeed radically transformed while in progress thanks to their enemies no less than to their supporters. -/- 19. Immanuel Kant, practical reason and judgment. The chapter argues that Kant took from Moses Mendelssohn the idea of a distinction between geometry of morals and a practical ethic. He was drastically misunderstood by his followers precisely on this point. He had learned from the sceptics and the Jansenists the lesson that men are prompted to act by deceptive ends, and he was aware that human actions are also empirical phenomena, where laws like the laws of Nature may be detected. His practical ethics made room for judgment as a holistic procedure for assessing the saliency of relevant moral qualities in one given situation; this procedure yields the same results as the geometry of morals does for abstract cases but does so immediately and without balancing conflicting duties with each other, since what makes for the salient quality of a situation is perceived from the very beginning. Kant's practical philosophy is richer than the received image, making room for an ‘empirical moral philosophy’ or moral anthropology including treatment of commerce, needs, value and price, happiness and well-being; the overall social theory and philosophy of history is less different from Adam Smith's than the received image makes believe; the paradox of happiness is central to Kant's philosophy; a distinction between happiness and well-being is clearly drawn; the distinction plays a basic role in establishing a link between the economic and the moral life. -/- 20. Georg Wilhelm Friedrich Hegel and the critique of abstract universalism. The chapter describes first the Romantic movement and the implications some of its concerns, rescuing passions, community, tradition, the individual as the bearer of a unique destiny, and the longing for organic unity between the individual, mankind and nature. Hegel’s contribution is discussed then, highlighting how, on the one hand, he shared a number of these concerns and on the other he had more rationalist leanings. The notion of morality is the pivotal point of the reconstruction, highlighting how Hegel construes this notion as a key to his own diagnosis of the malaise of modernity – the separation of individual and Gemeinschaft – and how his attack on Kant turns around this very idea. -/- 21. Friedrich Nietzsche against Christianity and the Enlightenment. The chapter illustrates first the idea of deconstruction of the back-world of values. Nietzsche claims to be legitimate heir of the French moralists, Leibniz, Kant and Hegel, in so far as he allegedly carries out to its deepest implications their discovery of what lies behind traditional naïve belief in the existence of an objective realm of values just waiting for description by the philosopher. The two exemplars form which genealogy draws inspiration are the classical philologist’s historical reconstructions of lost meanings and the chemist’s decomposition of elements. Then the genealogy of the notions of good and evil carried out in the first dissertation of Genealogy of morals is illustrated with its paradoxical conclusion that will to power is in fact the only ‘genuine’ kind of goodness. The third point illustrated is the dialectic of ascetism and self-realization with its ambiguous outcome. The suggested interpretation of such outcome is that Nietzsche’s normative ethics is a kind of virtue ethics taking an aesthetic ideal as a normative standard -/- 22. George Edward Moore and ideal utilitarianism. The chapter discusses the ideas of common sense and common-sense morality in Sidgwick. I argue that, far from aiming at overcoming common-sense morality, Sidgwick aimed purposely at grounding a consist code of morality by methods allegedly taken from the natural sciences to reach, also in ethics, the same kind of “mature” knowledge as in the natural sciences. His whole polemics with intuitionism was vitiated by the a priori assumption that the widespread ethos of the educated part of humankind, not the theories of the intuitionist philosophers, was what was worth considering as the expression of intuitionist ethics. In spite of a naïve positivist starting-point, Sidgwick was encouraged by his own approach in exploring the fruitfulness of coherentist methods for normative ethics. Thus, Sidgwick left an ambivalent legacy to twentieth-century ethics: the dogmatic idea of a “new” morality of a consequentialist kind, and the fruitful idea that we can argue rationally in normative ethics albeit without shared foundations. Then it reconstructs the background of ideas, concerns and intentions out of which Moore’s early essays, the preliminary version, and then the final version of Principia Ethica originated. I stress the role of religious concerns, as well as that of the Idealist legacy. I argue that PE is more a patchwork of rather diverging contributions than a unitary work, not to say the paradigm of a new school in Ethics. I add a comparison with Rashdall’s almost contemporary ethical work, suggesting that the latter defends the same general claims in a different way, one that manages to pave decisive objections in a more plausible way. I end by suggesting that the emergence of Analytic Ethics was a more ambiguous phenomenon than the received view would make us believe, and that the wheat (or some other gluten-free grain) of this tradition, that is, what logic can do for philosophy, should be separated from the chaff, that is, the confused and mutually incompatible legacies of Utilitarianism and Idealism. -/- 23. Edmund Husserl and the a-priori of action. The chapter illustrates first the idea of phenomenology and the Husserl’s project of a phenomenological ethic as illustrated in his 1908-1914 lectures. The key idea is dismissing psychology and trying to ground a new science of the a priori of action, within which a more restricted field of inquiry will be the science of right actions. Then the chapter illustrates the criticism of modern moral philosophy carried out in the 1920 lectures, where the main target is naturalism, understood in the Kantian meaning of primacy of common sense. The third point illustrate is Adolph Reinach’s theory of social acts as a key the grounding of norms, a view that basically sketches the very ideas ‘discovered’ later by Clarence I. Lewis, John Searle, Karl-Otto Apel and Jürgen Habermas. A final section is dedicated to Nicolai Hartman, who always refused to define himself a phenomenologist and yet developed a more articulated and detailed theory of ‘values’ – with surprising affinities with George E. Moore - than philosophers such as Max Scheler who claimed to Husserl’s legitimate heirs. -/- 24. Bertrand Russell and non-cognitivism. The chapter reconstructs first the discussion after Moore. The naturalistic-fallacy argument was widely accepted but twisted to prove instead that the intuitive character of the definition of ‘good’, its non-cognitive meaning, in a first phase identified with ‘emotive’ meaning. Alfred Julius Ayer is indicated as a typical proponent of such non-cognitivist meta-ethics. More detailed discussion is dedicated to Bertrand Russell’s ethics, a more nuanced and sophisticated doctrine, arguing that non-cognitivism does not condemn morality to arbitrariness and that the project of a rational normative ethics is still possible, heading finally to the justification of a kind of non-hedonist utilitarianism. Stevenson’s theory, another moderate version of emotivism is discussed at some length, showing how the author comes close to the discovery of the role of a pragmatic dimension of language as a basis for ethical argument. Last of all, the discussion from the Forties about Hume’s law is described, mentioning Karl Popper’s argument and Richard Hare’s early non-cognitivist but non-emotivist doctrine named prescriptivism. -/- 25. Elizabeth Anscombe and the revival of virtue ethics. The chapter discusses the three theses defended by Anscombe in 'Modern Moral Philosophy'. I argue that: a) her answer to the question "why should I be moral?" requires a solution of the problem of theodicy, and ignores any attempts to save the moral point of view without recourse to divine retribution; b) her notion of divine law is an odd one more neo-Augustinian than Biblical or Scholastic; c) her image of Kantian ethics and intuitionism is the impoverished image manufactured by consequentialist opponents for polemical purposes and that she seems strangely accept it; d) the difficulty of identifying the "relevant descriptions" of acts is not an argument in favour of an ethics of virtue and against consequentialism or Kantian ethics, and indeed the role of judgment in the latter is a response to the difficulties raised by the case of judgment concerning future action. A short look at further developments in the neo-naturalist current is given trough a reconstruction of Philippa Foot’s and Peter Geach’s critiques to the naturalist-fallacy argument and Alasdair MacIntyre’s grand reconstruction of the origins and allegedly unavoidable failure of the Enlightenment project of an autonomous ethic. -/- 26. Richard Hare and neo-Utilitarianism. The chapter addresses the issue of the complex process of self-transformation Utilitarianism underwent after Sidgwick’s and Moore’s fatal criticism and the unexpected Phoenix-like process of rebirth of a doctrine definitely refuted. A glimpse at this uproarious process is given through two examples. The first is Roy Harrod Wittgensteinian transformation of Utilitarianism in pure normative ethics depurated from hedonism as well as from whatsoever theory of the good. This results in preference utilitarianism combined with a ‘Kantian’ version of rule utilitarianism. The second is Richard Hare’s two-level preference utilitarianism where act utilitarianism plays the function of eventual rational justification of moral judgments and rule-utilitarianism that of action-guiding practical device. -/- 27. Hans-Georg Gadamer and rehabilitation of practical philosophy. The post-war rediscovery of ethics by many German thinkers and its culmination in the Sixties in the movement named ‘Rehabilitation of practical philosophy’ is described. Among the actors of such rehabilitation there were a few of Heidegger’s most brilliant disciples, and Hans-Georg Gadamer is chosen as a paradigmatic example. His reading of Aristotle’s lesson I reconstructed, starting with Heidegger’s lesson but then subtly subverting its outcome thanks to the recovery of the central role of the notion of phronesis. -/- 28. Karl-Otto Apel and the revival of Kantian ethics. Parallel to the neo-Aristotelian trend, there was in the Rehabilitation of practical philosophy a Kantian current. This started, instead than the rehabilitation of prudence, with the discovery of the pragmatic dimension of language carried out by Charles Peirce and the Oxford linguistic philosophy. On this basis, Karl-Otto Apel singled out as the decisive proponent of the linguistic and Kantian turn in German-speaking ethics, worked out the performative-contradiction argument while claiming that this was able to provide a new rational and universal basis for normative ethics. An examination of his argument is some detail is offered, followed by a more cursory reconstruction of Jürgen Habermas’s elaboration on Apel’s theory. -/- 29. John Rawls and public ethics as applied ethics. Rawls’s distinction between a “political” and a “metaphysical” approach to one central part of ethical theory, namely the theory of justice, is interpreted as a formulation of the same basic idea at the root of both the principles approach and neo-casuistry, both discussed in the following chapter, namely that it is possible to reach an agreement concerning positive moral judgments even though the discussion is still open – and in Rawls’ view never will be close – on the basic criteria for judgment. -/- 30. Beauchamp and Childress and bioethics as applied ethics. The chapter presents the revolution of applied ethics while stressing its methodological novelty, exemplified primarily by Beauchamp and Childress principles approach and then by Jonsen and Toulmin’s new casuistry. -/- . (shrink)

In a recent paper, Lyngzeidetson [1990] has claimed that a type of parallel computer called the ‘Connection Machine’ instantiates architectural principles which will ‘revolutionize which "functions" of the human mind can and cannot be modelled by (non-human) computational automata.’ In particular, he claims that the Connection Machine architecture shows the anti-mechanist argument from Gödel's theorem to be false for at least one kind of parallel computer. In the first part of this paper, I argue that Lyngzeidetson's claims are not supported (...) by his arguments; in the second part I consider some other aspects of parallel computation which may be of theoretical significance in cognitive science. (shrink)

This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's Incompleteness (...) Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. (shrink)

Theorems of hyperarithmetic analysis occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed iterations of the Turing jump but below ATR $_{0}$. There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They (...) seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity.This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis. When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic. (shrink)

Context: Radical constructivism claims that we have no final truth criteria for establishing one ontology over another. This leaves us with the question of how we can come to know anything in a viable manner. According to von Glasersfeld, radical constructivism is a theory of knowledge rather than a philosophy of the world in itself because we do not have access to a human-independent world. He considers knowledge as the ordering of experience to cope with situations in a satisfactory (...) way. Problem: Von Foerster and Krippendorff show that the central goal of a constructivist theory of knowing must be to find a way of putting the knower into a known that is constructed so as to keep the knower, as well as the knowing process, viable in practice. Method: The conceptual and philosophical analysis of present theories and their necessary prerequisites suggests that such foundation for viable knowing can be built on the analysis of what the ontological prerequisites are for establishing viable observing, cognition, communication and observer-communicators, and communication media and vehicles. Results: The moment an observer chooses to accept his/her own embodied conscious presence in this world as well as language, he/she must accept other humans as partly independently existing conversation partners; if knowledge and knowing has to make sense, he/she must also accept as prerequisites for our observation and conversation a pre-linguistic reality from which our bodies come and which our conversation is often about. Furthermore, we can no longer claim that there is a reality that we do not know anything about: From being here in conversation, we know that the world can produce more or less stable embodied consciousnesses that can exchange and construct conceptual meanings through embodied conversations and actions that last over time and exist in space-time and mind, and are correlated to our embodied practices. We can also see that our communication works through signs for all living systems as well as in human language, understood as a structured and progressively developed system of communication. The prerequisite for this social semiotic production of meaning is the fourfold “semiotic star of cybersemiotics,” which includes at least four different worlds: our bodies, the combination of society, culture and language, our consciousness, and also an outer nature. Implications: The semiotic star in cybersemiotics claims that the internal subjective, the intersubjective linguistic, our living bodies, and nature are irreducible and equally necessary as epistemological prerequisites for knowing. The viable reality of any of them cannot be denied without self-refuting paradoxes. There is an obvious connectedness between the four worlds, which Peirce called “synechism.” It also points to Peirce’s conclusion that logic and rationality are part of the process of semiosis, and that meaning in the form of semiosis is a fundamental aspect of reality, not just a construction in our heads. Erratum: The paper erroneously refers to “pleroma.” The correct term is “plemora.” . (shrink)

The goal of this dissertation is two-fold. It is, first, to reconstruct Kant's argument against self-murder, and, second, to analyze the function of the principle of self-preservation of reason with regard to the prohibition of self-murder. I argue that self-murder is contrary to the principle of self-preservation of reason and violates the trustee-relationship between the homo phaenomenon and the homo noumenon. The analysis shows that moral self-preservation in Kant is a rational principle which serves to secure the possibility (...) of moral faith and self-perfection. ;In the first part of the dissertation, I provide a comprehensive analysis of Kant's argument against self-murder by examining all of the relevant statements in his works, the Nachlass, as well as the lectures recorded by Herder, Powalski, Collins, and Vigilantius. Since self-murder violates a perfect duty to oneself, key-topics of the analysis are: the dual status of the duty of self-preservation as both inner duty of right and duty of virtue, the right and end of humanity, as well as Kant's model of self-possession in terms of proprietas. The analysis makes clear that self-murder destroys the moral self and must be distinguished from sacrificing one's life in order to preserve one's moral integrity. ;In the second part of the dissertation, I interpret the duty of self-preservation and the prohibition of self-murder in the context of Kant's doctrine of the faith of reason and the principle of self-preservation of reason. I first provide an overview of the debate on self-preservation and its implications for an understanding of modern rationality and summarize the different meanings of -preservation in Kant. I then argue that the "principle of self-preservation of reason" consists in what Kant calls "the maxim of healthy reason" and that it grounds the faith and the unity of reason. Against the background of this analysis, I interpret the trustee-relationship between the homo phaenomenon and the homo noumenon as a model of moral self-obligation. Entrusting the homo phaenomenon to the homo noumenon, I contend, means to entrust life to practical reason and to subject oneself to the moral law. (shrink)

Penrose proved that a computational or formalizable theory of the brainís cognitive functioning is impossible, but suggested that a physical non-computational and non-formalizable one may be viable. Arguments as to why Penroseís program is unrealizable are presented. The main argument is that a non-formalizable theory should be verbal. However, verbal paradoxes based on Cantorís diagonal processes show the impossibility of a consistent verbal theory of the brain comprising its arithmetical cognition. It is suggested that comprehensive theories of the human brain (...) and of physical experience are Kantian rather than Platonic ideas. This suggestion is likewise based on arguments related to diagonal processes. (shrink)

One of the most basic functions of human language is to convey who did what to whom. In the world's languages, the order of these three constituents (subject [S], verb [V], and object [O]) is uneven, with SOV and SVO being most common. Recent experiments using experimentally elicited pantomime provide a possible explanation of the prevalence of SOV, but extant explanations for the prevalence of SVO could benefit from further empirical support. Here, we test whether SVO might emerge because (a) (...) SOV is not well suited for describing reversible events (a woman pushing a boy) and (b) pressures to be efficient and mention subjects before objects conspire to rule out many other alternatives. We tested this by asking participants to describe reversible and non‐reversible events in pantomime, and we instructed some participants to be consistent in the form of their gestures and to teach them to the experimenter. These manipulations led to the emergence of SVO in speakers of both English (SVO) and Turkish (SOV). (shrink)

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making (...) this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Inspiration and Institution in John of Rupescissa's Liber Ostensor XI1Graziana S. CiolaIntroductionThe present study proposes a philosophical analysis of John of Rupescissa's Liber Ostensor [=LO], Treatise XI. John of Rupescissa (OFM, 1310 ca. – 1366)2 is a particularly interesting, eclectic and somewhat extraordinary [End Page 7] author writing around the second third of the 14th century in the wake of the Spiritual Franciscan movement in the South of France.3 (...) In LO XI, Rupescissa theorizes and systematically articulates a complex relationship between inspired knowledge and the normative portrait of what he believes the Franciscan Order should ideally be; he does so against the background of his eschatological concerns. Per Rupescissa's style of exposition and argumentation, these different elements — i.e., inspired epistemology, eschatology, and ecclesiology — are tightly intertwined on a theoretical and foundational level that goes beyond that of mere rhetorics.These themes are both recurrent and pivotal throughout Rupescissa's extremely prolific intellectual output, which counts numerous theological, ecclesiological, eschatological and alchemical treatises.4 LO, composed in [End Page 8] 1356 during Rupescissa's own "Avignon Captivity," is in many ways a sum of Rupescissa's prophetical and eschatological thought within the framing of the author's self-apologetic aims and of his ecclesiological views, in particular concerning the future of the Franciscan Order.5 [End Page 9]Treatise XI is undoubtedly the most theoretical book in LO; with its 489 paragraphs, it is also the longest. While this extremely dense text broaches on a great variety of topics, it is first and foremost about and for Spiritual Franciscans. Overall, LO XI is both a great portrayal of the Franciscan order as it is and as it should be, as well as a handbook addressed to the elect among the Spirituals, in order to help them face the eschaton and the tribulations to come. Per Rupescissa's eschatological views, such trials would affect the Franciscan Order the hardest and for the longest — hence the need for a detailed guide. At its core, then, LO XI is a deeply ecclesiological treatise and as such it is complementary to the larger systematization established in the Sexdequiloquium.6 It is in this framework that LO XI offers a complex theory of prophecy and illuminative knowledge as they should be embodied in the practices of a true Franciscan.7Given its highly abstract nature and cardinal function within one the most complex works of Rupescissa's maturity, LO XI, combining relevant insights on Rupescissa's ecclesiological, epistemological and prophetical positions, constitutes the ideal case study in order to assess Rupescissa's most refined and articulated account on inspired knowledge, particularly insofar as true and actual prophecy is concerned. This inspired knowledge, according to Rupescissa, is an indispensable tool for the elect to be able to victoriously face the imminent end of times. In the account presented in LO XI, such knowledge is possible exclusively through a form of institutional embedding: divine inspiration, culminating in true prophecy, is in service of the Franciscan Order's soteriological role within the Church as a whole, and can only be achieved by those who are faithful to the spirit of Francis's Rule.8 [End Page 10]Here, I analyze Rupescissa's theory of evangelical poverty and its role as a necessary condition for inspired knowledge, following the internal articulation of LO XI.Overall, in Rupescissa's view, Franciscan poverty and inspiration are two inseparable sides of the same coin. On the one hand, inspired knowledge is only achievable through a rigorous practice and thorough understanding of Franciscan poverty. On the other hand, the true institutional role of the Franciscan Order and its soteriological aims can be fulfilled exclusively by the true Spiritual Franciscans, whose moral, spiritual and epistemic perfection is divinely inspired. For Rupescissa, not only is poverty the source of all virtues both for the Franciscan Order and for the rest of the Church, but it also constitutes the ideal moral and social model, tout court. But most importantly, evangelical poverty has a genuinely epistemic role; this role is essential for Rupescissa's illuminative epistemology and for his account of prophetical exegesis — or exegetical prophecy. In turn, the... (shrink)

This study of Western philosophic systems, their types, history, relations, and projected future in the next half century, stems from Robert S. Brumbaugh’s forty-year fascination with the paradox of the many consistent overarching systems of ideas that are nevertheless mutually exclusive. Brumbaugh argues that when we isolate these systems’s patterns and look at them more abstractly, they consistently fall into four main types, and the interaction of these four types of explanation and order is a dominant theme in the history (...) of Western philosophy. In Brumbaugh’s view these four philosophic systems are not, as some critical historians and thinkers have claimed, so different that they are mutually unintelligible, forcing us to make a choice among them that is entirely arbitrary. But neither are they, as a majority of past thinkers and historians have hoped, simply parts of some single "right" or "orthodox" scheme. Their mutual understanding requires a method of transformation that interprets one to another without destroying their diversity. The history of Western philosophy from the fifth century A.D. to the present shows a pattern of alternating revolutions in systematic method and direction of explanation. Brumbaugh feels that the pattern is continuing in a change toward a revised Platonism, just beginning with the twenty-first century. He anticipates that it will be a Platonism of a new texture, one that has matured and learned a great deal in the course of the adventures of its ideas through space and time. (shrink)

This work is largely the author’s Habilitationsschrift presented at Erlangen. There are three major parts: the first describes ancient and modern forms of scientific reason, the development of which he calls a first and a second Enlightenment; the second explores more deeply certain elements in modern science such as the laws of motion, experimentation, forms of calculation, gravity and force; the third studies rational methodology, including issues of epistemology and artificial languages. The author accepts Kant’s and Nietzsche’s criticism of earlier (...) forms of metaphysics and sees signs of a coming third Enlightenment which will involve criticism of historical and linguistic reason. In parts one and two, considerable space is given to historical and social circumstances as well as to scientific development. Galileo, Newton, Descartes, Leibniz and Kant are among the major figures treated, but many detailed issues are also examined, such as scientific concerns in the late middle ages and renaissance. A forty page bibliography indicates the extent of the author’s range.—R. S. (shrink)

Substance use disorders (SUDs) constitute a growing global health crisis, yet many limitations and challenges exist in SUD treatment research, including the lack of objective brain-based markers for tracking treatment outcomes. Electroencephalography (EEG) is a neurophysiological technique for measuring brain activity, and although much is known about EEG activity in acute and chronic substance use, knowledge regarding EEG in relation to abstinence and treatment outcomes is sparse. We performed a scoping review of longitudinal and pre-post treatment EEG studies that explored (...) putative changes in brain function associated with abstinence and/or treatment in individuals with SUD. Following PRISMA guidelines, we identified studies published between January 2000 and March 2022 from online databases. Search keywords included EEG, addictive substances (e.g., alcohol, cocaine, methamphetamine), and treatment related terms (e.g., abstinence, relapse). Selected studies used EEG at least at one time point as a predictor of abstinence or other treatment-related outcomes; or examined pre- vs. post-SUD intervention (brain stimulation, pharmacological, behavioral) EEG effects. Studies were also rated on the risk of bias and quality using validated instruments. Forty-four studies met the inclusion criteria. More consistent findings included lower oddball P3 and higher resting beta at baseline predicting negative outcomes, and abstinence-mediated longitudinal decrease in cue-elicited P3 amplitude and resting beta power. Other findings included abstinence or treatment-related changes in late positive potential (LPP) and N2 amplitudes, as well as in delta and theta power. Existing studies were heterogeneous and limited in terms of specific substances of interest, brief times for follow-ups, and inconsistent or sparse results. Encouragingly, in this limited but maturing literature, many studies demonstrated partial associations of EEG markers with abstinence, treatment outcomes, or pre-post treatment-effects. Studies were generally of good quality in terms of risk of bias. More EEG studies are warranted to better understand abstinence- or treatment-mediated neural changes or to predict SUD treatment outcomes. Future research can benefit from prospective large-sample cohorts and the use of standardized methods such as task batteries. EEG markers elucidating the temporal dynamics of changes in brain function related to abstinence and/or treatment may enable evidence-based planning for more effective and targeted treatments, potentially pre-empting relapse or minimizing negative lifespan effects of SUD. (shrink)

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional. The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required properties inEL+ (...) QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide ; but inELit is impossible to prove this. (shrink)

Packing optimization problems have a wide spectrum of real-word applications. One of the applications of the problems is problem of placement of containers with spent nuclear fuel on the storage platform. The solution of the problem can be reduced to the solution of the problem of finding the optimal placement of a given set of congruent circles into a multiconnected domain taking into account technological restrictions. A mathematical model of the prob-lem is constructed and its peculiarities are considered. Our approach (...) is based on the mathematical modelling of rela-tions between geometric objects by means of phi-function technique. That allowed us to reduce the problem solving to nonlinear programming. Today, an important scientific problem is the problem of creating conditions for safe storage of spent nuclear fuel. In the process of creating any dry spent nuclear fuel storage, the following main stages can be identified: site selection, storage design, construction, operation and decommissioning. A full check for compliance of the repository and its elements with these standards usually begins at the design stage. At the stage of site selection, the inspection for compliance with safety standards is carried out only in terms of the impact of the repository as a whole on the environment. This approach cannot be considered fully appropriate, because, taking into account, for example, all the climatic features of the future storage site, it is possible to adjust the thermal storage regimes of spent nuclear fuel. Similarly, it can be considered necessary to analyze and select the shape of the storage site in order to accommo-date the maximum possible number of spent fuel containers. Such a choice, obviously, should be made taking into ac-count the norms of nuclear, radiation and thermal safety, as well as in compliance with technological limitations. The problem of finding the optimal placement of containers taking into account the given technological limitations can be formulated in the form of the problem of optimization of geometric design. Therefore, the purpose of the study is to build a mathematical model of the problem and study its characteristics to develop effective methods of solution. The proposed approach is based on mathematical modeling of relations between geometric objects using the method of phi-functions. This allowed to reduce the solution of the problem to the problem of nonlinear programming. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Science in a Democratic Society by Philip KitcherHenry S. RichardsonReview: Philip Kitcher, Science in a Democratic Society, Prometheus Books, 2011In examining the place of science in a democratic society, Philip Kitcher is ultimately asking what standards scientific activity is answerable to. Here, as in Science, Truth, and Democracy (Oxford University Press, 2001), he rejects two extreme possibilities: first, the suggestion that science is autonomous, in the sense that (...) it is answerable only to its own standards; and, second, the suggestion that it is answerable to “vulgar democracy”—that is, to unconstrained majority rule by untutored citizens. In this book, as in the earlier book, he deploys the idea of “well-ordered science,” which he seeks to characterize in part by leaning on the idea of a hypothetical, ideal conversation. Crucially informing this newer book is a much fuller account of nonvulgar democracy. Underlying that account is a much broader picture of “the ethical project”—the title idea of another important Kitcher book from 2011 (Harvard University Press).In order to motivate the friends of science to turn to these broader ethical ideas, Kitcher persuasively rehearses the case against value neutrality in science. Much current distrust of science and scientists, he argues, derives from scientists falsely pretending—whether deludedly or cynically—that science can proceed in a value-neutral way. It cannot do so in resolving competitions among theories, let alone in setting the research agenda or weighing in on practical problems of public concern. By wearing a false mask of value neutrality, scientists succeed only in convincing the public that they are ideologically driven. This, in turn, gives an opening for dissenters motivated by religion or by their own ideologies to develop supposedly scientific counter-claims of their own, such as those that assert creationism or deny the reality of climate change. Better, then, for science and scientists to be frank about its lack of value neutrality. [End Page E-106]What values ought to inform science? To answer this question, Kitcher turns to his conception of the ethical project, of which Chapter 2 gives a summary. As he conceives of ethics, it arose pre-historically as a set of mechanisms helping us to deal with deficiencies in altruistic motivation. Although inspired by evolutionary theory, Kitcher spurns the sociobiologist’s identification of the good with the adaptive. Instead, while giving an elaborate reconstruction of how our ethical practices may actually have arisen, he gives pride of place to the idea of conversation. On his reconstruction, the “egalitarian plateau” touted by Will Kymlicka (Contemporary Political Philosophy [Oxford University Press, 1990], p. 5) and others as the great achievement of the Enlightenment was merely a delayed philosophical echo of the ethical project’s initial feature, one that would have been found in the ethical discussions of human hunter-gatherer bands of forty-thousand years ago. Being more focused on how life is lived than on philosophy, Kitcher, fairly enough, is unwilling to see the Enlightenment as an egalitarian age. Rather, most recent incarnations of the ethical project have been distorted by hierarchies and inequalities. Emphasizing the word, Kitcher writes, “I propose... that we undo the distortion,” and model the ethical project, going forward, as involving an egalitarian conversation. This frankness disarms any worry about the move from ‘was’ to ‘ought’ that might have arisen in the minds of those already puzzled by the current popularity of the cave-man diet; and in any case, since Kitcher’s account of the ethical project is defended at length in his contemporaneous book of that title, doubts about it would be better addressed in that context. Rhetorically, since the ideal of treating persons fundamentally as equals is even more popular than eating meat, he is on strong ground here.The book develops this core egalitarian proposal in three distinct ways, thereby gaining a rich evaluative basis for considering the place of science in society. The first is an account of the purposes to which science is answerable. Kitcher dismisses as unrealistically high-minded and ultimately elitist a conception of scientists as driven by a disinterested quest for truth and a simple love of knowledge for its own sake. Rather, his... (shrink)

In this paper we are concerned with definably, with or without parameters, complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o-minimal if and only if all their definable subsets are finite disjoint unions of definably connected subsets. We pay attention to how simply a definable gap in an expansion is so. (...) Next we prove that over parametrically definably complete expansions of ordered fields, all one-to-one definable continuous functions are monotone and open. Moreover, in both parameter and parameter-free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine-Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets. (shrink)

This book, as its title indicates, is put forth as a criticism of our age. The author, who is especially known for his work in the tradition of Husserl and Heidegger, and who has written a book on Aristotle, has often mentioned elements of his own philosophical position in his many essays and books; this volume presents the complete view, of which the others gave only hints. Boehm defines "our age" as determined by science, a science which stems from the (...) Greek understanding of knowledge and contains theological elements from antiquity and the middle ages. Knowledge is taken to be an attempt to overcome the limits of the human condition, and Boehm says that this attempt in turn is based on a desire to escape dying. He examines Hesiod, the Bible, and Lao-Tse to find extra-philosophical statements of the interpretation of death as something that somehow might not be. After these analyses of classical and non-philosophical versions of this theme, Boehm has a chapter to show how the issue is developed in various modern ways: Pascal evades the issue of death by escaping into divertissement, and modern political and economic life becomes a process or a game of its own, cut loose from the ends it was contrived to serve; Fichte and Marx are used here, and so are the satires of Parkinson and Boorstin’s analysis of "image" as replacing real need. Boehm emphasizes the reversal of means and ends in contemporary life and sees it as the consequence of the ancient attempt to replace the human condition by knowledge; an inhuman existence results from the desire to avoid being human. In a final chapter, Boehm studies Descartes, Spinoza, Locke, Kant, and others as thinkers who further develop the theme of man as "subject," in the sense of a being that subjects itself to necessity as its only way of surviving; this conception of subjectivity is a condition for the modern subjugation of human needs to the control of the "laws" of economics, production, science, bureaucracy, and the other manifestations of sheer rationality. Human needs are reduced to the needs of rationalistic order and thinking. In conclusion, Boehm observes that philosophy can only criticize the image of the age and its foundations; the return to being human must be accomplished by other means. Boehm’s book is very interesting and puts flesh and blood on many themes we have heard from Heidegger in more abstract form. The main weakness in the book is a failure to analyse virtue, phronesis, and action, which must be seen as a counterpoise even for the understanding of Aristotle’s notion of theory. His criticisms of rationalism, and of gnostic versions of religion which attempt to solve human problems purely by knowledge or by thinking, are accurate, although it is questionable whether he gives a satisfactory picture of Christianity and Judaism in the few passages he cites and interprets; for example, once again the dimension of performance in these religions is not included. Some of his historical remarks, like his interpretation of the events of the thirteenth century, are bold, and one is inclined to read them along with his own remarks on the writing of history. The short description of a conversation with a man of science is pointed and well done. The book is lively and vivid, especially in narrating the reversals of ends and means in our present world.—R.S. (shrink)

Population axiology concerns how to evaluate populations in regard to their goodness, that is, how to order populations by the relations “is better than” and “is as good as”. This field has been riddled with impossibility results which seem to show that our considered beliefs are inconsistent in cases where the number of people and their welfare varies.1 All of these results have one thing in common, however. They all involve an adequacy condition that rules out Derek Parfit’s Repugnant Conclusion: (...) The Repugnant Conclusion: For any perfectly equal population with very high positive welfare, there is a population with very low positive welfare which is better, other things being equal.2 1 The informal Mere Addition Paradox in Parfit (1984), pp. 419ff is the locus classicus. For an informal proof of a similar result with stronger assumptions, see Ng (1989), p. 240. A formal proof with slightly stronger assumptions than Ng’s can be found in Blackorby and Donaldson (1991). For theorems with much weaker assumptions, see my (1999), (2000b), and especially (2000a), (2001), and (2009). 2 See Parfit (1984), p. 388. My formulation is more general than Parfit’s apart from that he doesn’t demand that the people with very high welfare are equally well off. Expressions such as “a population with very high positive welfare”, “a population with very low positive welfare”, etc., are elliptical for the more cumbersome phrases “a population consisting only of lives with.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Beyond Ratzinger's Republic:Communio's Postliberal TurnSam Zeno Conedera S.J. and Vincent L. Strand S.J.Is the political future of the West a postliberal one? For the past decade, numerous prominent thinkers in America and Europe have been debating this question. Matters that not long ago were merely of historical interest, such as Pope Gelasius I's understanding of the relation between sacral authority and royal power, Thomas Aquinas's thought on monarchy and (...) Robert Bellarmine's theory of the pope's indirect authority in temporal matters are now increasingly studied for their potential answers to contemporary political questions. This development, which would have been nearly unimaginable just a decade earlier, unsurprisingly has been led by Catholics. We say "unsurprisingly" in view of the long pre-modern tradition of Catholic political thought, as well as the Church's long struggle against liberalism during the nineteenth and early twentieth centuries. Some of the current postliberal writers explicitly connect themselves to both of these aspects by using the term "integralist" to describe their position.1 [End Page 889]This "new integralism" represents a radical departure from the mainstream of Catholic political discourse of the last half century, which has largely sought rapprochement with Western liberalism. A group of writers, led by John Courtney Murray, George Weigel, Michael Novak, and Robert George, have held that the American constitutional order is rooted in the natural law tradition, and therefore is basically friendly to Catholic political thought. The Church's role in American life, they say, is upholding the natural law as a public moral philosophy, while embracing the non-establishment clause of the First Amendment as a matter of principle, in accordance with the teachings of the Second Vatican Council. This "Whig Thomism," which for decades represented the predominant outlook of conservative American Catholics, must now face the challenge of a different sort of Thomism represented by the integralists.2While the engagement between Whig Thomists and integralists has received most of the attention from scholars and journalists alike, another postliberal option has emerged in recent years. Its source is a group of Catholic thinkers associated with the Communio theological movement. Their political vision, which has no single name or label, has a complex and subtle relationship with the thought of earlier Communio writers, most of whom shared the Whig Thomist desire for reconciliation with Western liberalism. [End Page 890] The appearance of a Communio postliberalism therefore represents a significant development within this intellectual tradition. The most important representative of the earlier Communio perspective on political questions is Joseph Ratzinger / Pope Benedict XVI, who has recently been lauded as the "most important Catholic thinker of the twentieth century" on matters at the intersection of theology and politics.3In an article titled "Ratzinger's Republic" that appeared in these pages in 2020, we identified a significant tension in his thought about the Church's contribution to political life.4 Exploring Ratzinger's writings on natural law, we demonstrated that two currents are present. In accord with his fundamental theological commitments about nature and grace and about faith and reason, Ratzinger often questioned the effectiveness of arguments from natural law. He claimed that, since reason is historically conditioned, and therefore in the current order marked by sin, reason needs grace and faith to function properly. Yet, on other occasions, Ratzinger strongly endorsed arguments from natural law, highlighting its role as the foundation of a just democratic political order. We traced this ambivalence about natural law back to a dilemma at the heart of Ratzinger's political thought. On the one hand, in accord with his thinking about the relations of nature and grace and faith and reason, Ratzinger argued that the state needs the Church to supply moral values that the state cannot furnish on its own. On the other hand, he believed it belongs to the very nature of the Church to be separate from the state, and he endorsed nonconfessional, religiously pluralistic states.5 In light of these two commitments, Ratzinger argued that the contribution Christians are to make to the state is to supply moral values that were discerned under the influence of grace and divine revelation but... (shrink)

Background Several studies suggest that deviations from proper authorship practices are commonplace in medicine. The aim of this study was to explore experiences of and attitudes towards the handling of authorship in PhD theses at medical faculties in Denmark, Norway, and Sweden. Methods Those who defended their PhD thesis at a medical faculty in Scandinavia during the second half of 2020 were offered, by e-mail, to participate in an online survey. Survey questions dealt with experiences of violations of the first (...) three of the ICMJE authorship criteria and misuse of authorship order in the thesis articles, as well as respondents’ attitudes to these matters. Both questions with fixed response alternatives and questions with free-text responses were used. Quantitative data were analysed statistically using the Table functions in SPSS 25 and Chi-2 tests. Free-text responses were analysed qualitatively using manifest content analysis. Results 287 valid questionnaires were returned (response rate: 34.1%). Almost half (46.0%) of the respondents reported that the ICMJE authorship criteria were not fully respected in at least one of the papers in their thesis, while a vast majority (96.7%) found it important that authorship is handled according to the ICMJE authorship criteria. 24.4% reported inadequate handling of authorship order in at least one paper. The qualitative results provide a wide spectrum of examples of how the ICMJE authorship criteria are circumvented. Conclusion Despite increasing educational efforts to reduce deviations from good research practice at Scandinavian universities, the handling of authorship in medical papers remains problematic. (shrink)

A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...) relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to be (...) “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

As persons, beings with a capacity for autonomy, we face a certain practical task in living out our lives. At any given period we find ourselves with many desires or preferences, yet we have limited resources, and so we cannot satisfy them all. Our limited resources include insufficient economic means, of course; few of us have either the funds or the material provisions to obtain or pursue all that we might like. More significantly, though, we are limited to a single (...) life and one of finite duration. We also age, and pursuits that were possible at earlier points within a life may become impossible at later stages; we thus encounter not only an ultimate time limit but episodic limits as well. Because we must live our lives with limited resources—material and temporal—we are pressed to choose among and to order our preferences. Without some selection and ordering, few if any of them would be satisfied, and we would be unable to live lives that are recognizably good at all. Moreover, we would be unable to function well as the autonomous beings that we are. Our practical task then is to form a coherent, stable, and attractive ordering of aims—to develop a conception of our good. (shrink)

Church's Undecidability Theorem is one of the meta-theoretical results of the mid-third decade of the last century, which along with other limiting theorems such as those of Gödel and Tarski have generated endless reflections and analyzes, both within the framework of the formal sciences, that is, mathematics, logic and theoretical computation, as well as outside them, especially the philosophy of mathematics, philosophy of logic and philosophy of mind. We propose, as a general purpose of this article, to formulate Church's (...) Undecidability Theorem and present the main ideas of its demonstration. In order to carry out the first objective, we need to introduce and explain the notions of recursive function and numbering used by Gödel, which will allow to formally and rigorously enunciate Church's Theorem. After we enunciate Church's Theorem of Unspeakability in a formal and rigorous manner, we will present the main ideas of the proof of Church's Undecidability Theorem for First Order Logic, which uses Robinson's axiomatic system for arithmetic and four facts about himself: In Robinson's system for arithmetic recursive functions are representable Robinson's system is undecidable, The number of axioms proper to the Robinson system is finite and The logical calculation of the Robinson system is equal to the calculation of the first-order logic. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to be (...) “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Daoism, Practice, and Politics:From Nourishing Life to Ecological PraxisEric S. Nelson (bio)I. Daoism's Multiple ModelsManhua Li, Yumi Suzuki, and Lisa Indraccola have offered evocative insights, questions, and alternatives in their contributions concerning the arguments of Daoism and Environmental Philosophy: Nourishing Life (Nelson 2021). The present brief response and sketch of the book will not address every point in their essays, but I will strive to reply, directly and indirectly, (...) to their primary concerns.First, it should be noted that I take a pluralistic conception of 'Daoisms' as my point of departure. This is necessary since Daoist studies have been hampered by overly ahistorical and one-sided definitions that prioritize one philosophical or religious moment (such as skepticism or mysticism) in historically complex and varied transmissions that encompass a multiplicity of texts and perspectives. Due to the fact of the plurality of incompatible kinds of Daoism in Chinese antiquity and in subsequent history, my text considers a range of early Daoist and Daoist-related perspectives in the context of their potential consequences for environmental philosophy, ethics, and politics. Consequently, my approach is primarily philosophical without limiting any text to being exclusively philosophical, religious, or rigidly committed to one perspective, which would be untrue to the sources themselves.Second, examining distinctive tendencies (modifying Graham 1989 and Liu 1994) in the Daodejing 道德經, the Zhuangzi 莊子, and related texts enables us to articulate distinctive and conflicting ecological, ethical, and political paradigms. In place of constructing a mythologizing ideological construction of 'Chinese culture' and Daoism as intrinsically 'Green', a familiar essentializing tendency that has provoked skepticism regarding the very possibility of Daoist environmental philosophy, my book provides a different 'critical Daoist' strategy. This alternative entails that Daoist exemplars can be deployed as therapeutic models in the present, while at [End Page 792] the same time confronting their ideological codifications and uses. Early Daoist sources have roamed on long intercultural journeys. The Daodejing and the Zhuangzi have provided elements for European thinking about civilization, technology, and nature from Martin Buber and Martin Heidegger (Nelson 2017) to Peter Sloterdijk's 'Euro-Taoism' (Sloterdijk 1989) to contemporary progressive Green movements (Joshi 2022) and even ecofascism (Lubarda 2020). Similarly, in the current Chinese context, Green Daoism has both critical and authoritarian ideological functions in visions of 'ecological civilization' (shengtai wenming 生態文明) (Schönfeld and Chen 2019) and fears of a 'green leviathan'. Accordingly, it is not accidental that my anarchistic and participatory democratic interpretation of one key thread, which I call 'ziranist' and clarify below, in early Daoist and Wei-Jin mysterious-learning (xuanxue 玄學) discourses, has led to the most intense discussions and controversies about this book. These include the contributions by Li, Suzuki, and Indraccola that propose questions and alternative possibilities concerning agency and responsiveness, authority and self-determination, freedom and fate, and adaptation and intervention, as well as purposive techniques and anti-purposive uselessness.Third, this hermeneutical strategy points toward the tertiary aspect of our preliminary definition of Daoism. It does not primarily name a philosophical theory or a religious worldview. It concerns shifting, overlapping, and diverging ensembles of discourse and practice. This is not the vague ahistorical idea of 'a way of life' but rather refers to how philosophy is enacted through a specific configuration of—to adopt early Greek expressions—logos as dis-course (e.g., argumentation, hermeneutics, logic, rhetoric), ethos (e.g., comportment, orientation, a way of dwelling), and praxis (practices of the self, with others, and with things and nature). Thus, the Laozi 老子 and Zhuangzi collections can be analyzed textually and conceptually as having multiple threads (drawing on a variety of authors and sources) that form constellations in the received texts with better and worse historical, practical, and conceptual readings. My interpretive strategy in this discussion, which does not exclude other approaches, is that these texts express forms of practical instead of merely speculative philosophy. They are suggestive of an ethos of relational freedom, irreducible to rules and techniques, which dismantles hypostatized experiential and linguistic fixations, including moralistic fixations, in order to freely encounter the myriad things and enact and practice the dao in one's own singular way.'Critical Daoism', therefore, serves as a pluralistic, diagnostic, and practical designation... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Evil in Modern Thought: An Alternative History of PhilosophyPaul S. MiklowitzSusan Neiman. Evil in Modern Thought: An Alternative History of Philosophy. Princeton: Princeton University Press, 2002. Pp. xii + 358. Cloth, $29.95.Contemporary philosophy in America tends to regard epistemological questions as the most fundamental of the discipline, but Susan Neiman's Evil in Modern Thought sets itself against this assumption in an attempt to sketch "an alternative history of (...) philosophy." "It's possible to begin to worry about the difference between appearance and reality because... a dream is so vivid that you want to grasp one of its objects for a moment or two of sleepy half-consciousness," Neiman writes, but then "you wake up in your bed, slap your face if you have to... " (5). The problem of evil, in stark contrast to such skeptical speculations, is far more urgent and trenchant; it is "the place where philosophy begins, and threatens to stop" (ibid.). Moreover, the problem of evil, far from being merely a theological quandary, actually forms a link between metaphysics and epistemology by starkly posing questions about the intelligibility of the world: Why are things like that? wants the answer Because it's best that way. For the early modern period, these questions centered on the "theodicy" debates focused by the catastrophic Lisbon earthquake of 1755, while for us the holocaust threatens to explode all our hopes and expectations for the possibility of a world that is as it should be. Thus, Lisbon and Auschwitz frame for Neiman the historical topoi that ground discussion of the relation between natural and moral evils, the evolving philosophical understanding of which defines "who we have become in the three centuries that separate us from the early Enlightenment" (10).The problem of theodicy is introduced through the figure of Alfonso X, king of Castille in 1252, and "the first Enlightenment hero" (14), according to Neiman. Having studied astronomy under the guidance of learned Jews from Toledo, Alfonso scandalously remarked "If I had been of God's counsel at the Creation, many things would have been ordered better" (15). Serving as a paradigm of blasphemy for half a millennium, Alfonso's boast also expresses the inevitable dismay that must accompany the delighted wonder at natural order exploited by the argument from design: while Creation is surely miraculous in its manifestation of seemingly intelligent structure, it is just as surely astonishing that so much of that structure seems so wantonly destructive. Can this really be the best of all possible worlds if innocent lives are cut short by cruel diseases or natural disasters while vicious villains prosper into comfortable old age?! Alfonso's remark serves Neiman as a leitmotif around which she orders central figures in the canon of western philosophy into two camps: those who struggled to reconcile natural evil with God's goodness or with reason (Leibniz, Pope, Rousseau, Kant, Hegel and Marx), and those who skeptically condemned reason by appealing to "brute appearances" (Bayle, Voltaire, Hume, Sade and Schopenhauer). Because the very categories of natural and moral evil tend to break down as a result of the development of psychology as a "science of nature" (to the extent that human motives are regarded as natural processes subject to empirical explanation, they cease to manifest a unique ontology), Nietzsche and Freud fall outside the philosophical canon thus construed—although Neiman's discussion of these two thinkers is perhaps the most compelling of the book.No new interpretive ground is broken in the resulting exegesis of western philosophical history, but Neiman's treatment of each of these figures is intelligent and skillful. In discussing Kant, for example, a deep paradox at the center of his moral thought is brilliantly harnessed to the book's argument. Whereas theodicies from Leibniz onward had focused on attempts to show that Providence ultimately guaranteed a connection between virtue and happiness despite obvious evidence to the contrary, Kant took exactly the opposite tack. As Neiman puts it, "Solving the problem of evil is not only impossible but immoral. For knowing the connections between moral and natural evils would undermine the possibility of morality" (69). If we could be sure that good deeds... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Evidence of Augustinian 'Ressourcement' in the Franciscan Summa Halensis:The Cases of Contra Faustum and De spiritu et litteraMichael S. HahnAmong the thornier issues surrounding the Parisian Franciscan collaborative compilation Summa Halensis1 is the matter of its sources, consideration of which most often involves discernment of its contributing authors and their engagement with near-contemporary texts and trends in twelfth- and thirteenth-century scholastic theology.2 Hiding in plain sight, and thus easily (...) overlooked in this array of detailed concerns, is the privileged place afforded to Augustine of Hippo—and more precisely, to historically underexamined and underutilized of his works—within certain sections of the Franciscan Summa, a dynamic that seems to indicate a deliberate Augustinian ressourcement alongside more standard scholastic uses of stock textual auctoritates.3 [End Page 59]In order to establish the plausibility and general contours of such ressourcement, the present study examines the deployment of Augustine's Contra Faustum and De spiritu et littera in the Franciscan Summa's treatise on law, outlining both the originality and significance of this usage. The study falls into three main parts. In the first part, I offer a brief introduction to these two works, noting their import vis-à-vis Augustine's broader corpus. In the second, I provide a rough sketch of their respective reception histories, limning the standards of use that obtain in twelfth- and thirteenth-century scholastic writings prior to the Franciscan Summa. And in the third part, I detail the employment of these two works in the Summa itself, attending to where and how they are used, and how this usage variously coheres with and moves beyond the preceding theological tradition. Finally, by way of conclusion, I gesture at the immediate impact of this usage, taking as my primary example the later Summa theologiae of Thomas Aquinas. As I aim to show, although these important Augustinian works were not entirely unknown in the twelfth and early thirteenth centuries, there is little evidence of their first-hand utilization in scholastic theology prior to that of the Summa Halensis. Moreover, the Franciscan Summa's ample and markedly astute use of these works can be seen not only to undergird the Augustinian cast of its teaching on the distinction and relation of law and grace, but also to initiate and direct a broader return to these sources among the next generation of mendicant scholastics, Franciscan and Dominican alike.I. Augustine's Contra Faustum and De spiritu et litteraThe present inquiry concerns two very different polemic texts of Augustine that originate from the first decade and a half of the fifth century: the Contra Faustum, composed from around 400 to 403, one of the latest—and by far the longest and most biblically focused—of Augustine's anti-Manichaean works;4 and De spiritu et littera, composed in 412, [End Page 60] among the first of his works against the Pelagians.5 The length and sprawl of the former work is noted by Augustine in his Retractationes, where he presents this as a function of the scattered array of attacks by the Manichaean Faustus of Mileve:Against Faustus the Manichaean, who blasphemed the law and the prophets and their God and the incarnation of Christ, and who asserted that the scriptures of the New Testament by which he is refuted were falsified, I wrote a massive work (grande opus), giving my replies to his words that are set forth. There are thirty-three debates (disputationes), but why may I not call them books (libros) as well? For even if some among them are brief, they are nonetheless books.6Although Augustine touches on many other points in this "massive work," his main task centers on defending and elucidating the divinely ordered harmony of the Old and New Testaments in proclaiming Christ.7 The procedure he employs is chiefly exegetical, involving assertion of the prophetic nature of the Old Testament and illustration of this via figural interpretation of particular passages. What is most distinctive about the Contra Faustum is this specific approach to the Old Testament and its signature emphasis on the prophetic status of those from whom Christ takes his flesh, who especially through the giving... (shrink)

In "Abstracts from Logical Form I/II," it was stated in the abstract that it remained necessary to put the pilot experiments into a "comprehensive theory." It is suggested here that the comprehensive theory is nothing other than classical logic modestly extended to include higher-order predicates, functions, and epistemic predicates, as well as a quantitative quantifier to deal with cases other than "all" (taken literally) or "some" in the sense of at least one. It is further suggested that up to (...) a point the theory should generalize across languages, and that logical form is very powerful. (shrink)

This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Verlen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to (...) Kruskal, the ‘tree theorem’, as well as a ‘finite miniaturization’ of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called ‘natural independence phenomena’, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γo and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence. (shrink)

Government intervention often gives rise to contests and the government can influence their outcome by choosing their type. We consider a contest with two interest groups: one that is governed by a central planner and one that is not. Rent dissipation is compared under two well-known contest success functions: the generalized logit and the all-pay auction. We also consider the case in which the government can limit the size of the non-governed interest group in order to determine the scope (...) of rent dissipation, with the goal of either increasing the rent obtained by the government or reducing the wasted resources invested in the contest. (shrink)

This is the first English translation of all of Kant's writings on moral and political philosophy collected in a single volume. No other collection competes with the comprehensiveness of this one. As well as Kant's most famous moral and political writings, the Groundwork to the Metaphysics of Morals, the Critique of Practical Reason, the Metaphysics of Morals, and Toward Perpetual Peace, the volume includes shorter essays and reviews, some of which have never been translated before. The volume (...) has been furnished with a substantial editorial apparatus including translator's introductions and explanatory notes to each text by Mary Gregor, and a general introduction to Kant's moral and political philosophy by Allen Wood. There is also an English-German and German-English glossary of key terms. (shrink)

In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind. This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundations of set theory to be shaken, he showed no apparent concern over the paradox and (...) its implications whatever. Against this account, I will argue here that in fact Cantor never saw any paradox at all, but that his conception of set at that time, and already as far back as 1883, was one in which the paradoxes cannot arise. (shrink)

The expectation is fulfilled, but in an unexpected way. 'The first studies toward this book were addressed to topics in the field of ethics' ; but our author, like Wagner composing 'Der Ring des Nibelungen', found himself becoming preoccupied with prolegomena. To these the present volume is wholly devoted. In order to establish its fundamental thesis that valuation is a form of empirical knowledge, two preparatory discussions are called for. An analysis of empirical knowledge in general is one of these; (...) but there is an inquiry properly prior even to this, an examination of the topic of meaning. The articulation of the book is accordingly tripartite; and Book I Meaning and Analytic Truth, Book II Empirical Knowledge, and Book III Valuation are of roughly equal length. (shrink)

Theoretically, a positive environment (PE) includes (a) tangible and intangible resources that satisfy human needs, (b) enablers of healthy, pro-social, and pro-environmental behaviors that guarantee socio-environmental quality and wellbeing, and (c) environmental challenges that must be faced and solved. One of the most salient challenges is the global COVID-19 pandemic. This study sought to investigate whether PEs can stimulate responsible actions (i.e., self-care and precautionary behaviors against COVID-19), while maintaining personal wellbeing. Nine hundred and forty-nine Mexicans participated in an online (...) survey encompassing five primary factors: resources, enablers, challenges, responsible health behaviors, and wellbeing. The first three factors examine “resources” such as physical infrastructure as well as family and social support, “enablers” which include information about protective health practices and perceived legitimacy of authorities in handling the pandemic, and “challenges” encompassing threat perception and social pressure to not engage in precautionary measures. Participants also self-reported hedonic wellbeing as well as self-care and precautionary behaviors, which formed the “responsible (health) behavior” factor. Structural equations model (n= 714 after list-wise deletion) showed that “resources,” “challenges,” and “enablers” form a second-order factor, “positive environments,” and this factor strongly covaries with “responsible behavior” and “wellbeing.” These results suggest that PEs are not only buffers against the negative impact of the COVID-19 pandemic but can also stimulate effective responses against a threat while maintaining individual wellbeing. These results can be used to inform the development and maintenance of PE frameworks aimed at minimizing the spread of COVID-19 and encouraging mental and physical health. (shrink)

Two thoughts about the concept of number are incompatible: that any zero or more things have a number, and that any zero or more things have a number only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any things have a number is Frege's; the thought that things have a number only (...) if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)