There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

Natural language conditionals seem to be subject to three logical requirements: they invalidate Antecedent Strengthening, they validate so-called Simplification of Disjunctive Antecedents, and they allow for the replacement of logically equivalent clauses in antecedent position. Unfortunately, these requirements are jointly inconsistent. Conservative solutions to the puzzle drop Simplification, treating it as a pragmatic inference. I show that pragmatic accounts of Simplification fail, and develop a truthmaker semantics for conditionals that captures all the relevant data. Differently from existing truthmaker semantics, my (...) semantics extends, rather than replaces, standard possible worlds semantics. The main innovation is the notion of a truthmaker in play: this notion is cognitive, rather than metaphysical, and can be defined in a purely syntactic way. (shrink)

The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze and (...) the one given by Martin-Löf for the intuitionistic type theory with a type of all types.The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast domain for foundational research. (shrink)

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

Some philosophers argue that false speech and false belief are impossible. In the Sophist, Plato addresses this 'falsehood paradox', which purports to prove that one can neither say nor believe falsehoods. In this book Paolo Crivelli closely examines the whole dialogue and shows how Plato's brilliant solution to the paradox is radically different from those put forward by modern philosophers. He surveys and critically discusses the vast range of literature which has developed around the Sophist over the past fifty (...) years, and provides original solutions to several problems that are so far unsolved. His book will be important for all who are interested in the Sophist and in ancient ontology and philosophy of language more generally. (shrink)

Quine is one of the most influential of contemporary philosophers, whose work has ranged broadly across a great number of topics and issues in a career spanning some fifty years. In this collection a group of distinguished philosophers offer a sustained critical evaluation of the full range of Quine's writings. Amongst the topics addressed are interpretation, epistemology, ontology, modality, and mathematical truth. This collection will certainly influence all future discussion of Quine. The contributors include: George Boolos, H-N. Castaneda, Donald Davidson, (...) Umberto Eco, Dagfinn Follesdal, James Higginbotham, Charles Parsons, Hilary Putnam, Barry Stroud, and Bas van Fraassen. However, Quine is given the last word, responding to the essays in the final contribution. (shrink)

Suppose you’re certain that a claim—say, ‘Frida is tall’—does not have a determinate truth value. What attitude should you take towards it? This is the question of the cognitive role of indeterminacy. This paper presents a puzzle for theories of cognitive role. Many of these theories vindicate a seemingly plausible principle: if you are fully certain that A, you are rationally required to be fully certain that A is determinate. Call this principle ‘Certainty’. We show that Certainty, in combination with (...) some minimal side premises, entails a very implausible claim: whenever you’re certain that it’s indeterminate whether A, it is rationally required that you reject A. This is a surprising result, which requires abandoning at least some intuitive views about indeterminacy and cognitive role. (shrink)

The interpretation of Parmenides’ Περί Φύσεως is a fascinating topic to which philosophers, historians of philosophy and scientists have dedicated many studies along the history of Western thought. The aim of this paper is to present the reading of Parmenides’s work offered by Federigo Enriques. It is based on several original theses: (1) Parmenides was the discoverer of abstract geometry; (2) his critics was addressed against the Pythagoreans rather than against Heraclitus; (3) Parmenides discovered and applied the contradiction and the (...) third excluded principles in the context of his research on foundation of geometry; (4) Parmenides’s metaphysical and physical conceptions have their bases in his speculation on geometry; (5) Parmenides used the principle of sufficient reason. Enriques’s reading is worth being expounded and discussed within the historical, philosophical and scientific context in which it is inserted. Since Enriques’s ideas are not widely known and discussed, my research has the purpose to fill this gap. The article also aims to provide elements to illustrate the discussion on Parmenides in the first half of the last century. (shrink)

This article recounts how Neo-Scholastic international lawyers navigated the complex political landscape of the 1920s and 30s, combining universalism, nationalism and religious belief. Participating in the contemporary re-engagement of Catholics with modern politics, they re-imagined the international legal order in Catholic terms. They argued that a universal morality, overruling the extremes of state sovereignty, was the only solid basis for just and stable global legal relations. While the contribution of Catholics to the establishment of the post-war world order and the (...) rise of human rights is widely acknowledged, the interwar genealogy of these developments is not. Reading these thinkers’ universalist legal tenets in conjunction with their political trajectories provides a relevant frame to understand the moral charge characterizing international law in the early years of the Cold War. (shrink)

Since the early 1990s, an increasing number of troops have been deployed in peacekeeping missions all around the world. The mixed success and high-profile failures of several missions have provided peacekeepers and scholars with a wealth of experience from which to generate knowledge and understand key lessons. In this article I use the Rwandan case to explore the issue of the use of force to protect unarmed civilians that have become the target of violence. In particular, I focus on the (...) moral responsibilities implied in such operations. I contend that in order for peacekeepers to be effective, they need to subscribe to, and be motivated by a peacekeeper ethos. The core element of this ethos is the protection of human life. Those peacekeepers who, through experience and education, internalize this ethos will be the best prepared for the difficult task they face. This article emphasizes the importance of sharpening moral autonomy among peacekeepers, in order to provide them with the skills they need to be more effective in saving lives. To illustrate this point, I analyze the slaughter of more than 2000 refugees at the ETO school in Kigali, Rwanda; a UN site protected by Belgian peacekeepers in 1994. (shrink)

According to McDowell, in Empiricism and the Philosophy of Mind the myth of Jones has the purpose of completing the account of experience that Sellars needs to argue against traditional empiricism. In particular, on McDowell's view the myth of Jones should explain how to conceive of non-inferentially knowable experiences as containing propositional claims. This article argues that the myth of Jones does not succeed in providing such an account, especially on McDowell's own terms: assuming McDowell's epistemological distinction between inferential and (...) non-inferential knowledge, it turns out that in Sellars' thought experiment perceptual experiences can contain propositional claims only at the price of being known inferentially rather than non-inferentially. Therefore McDowell's Sellars' attack fails against traditional empiricism. (shrink)

In this article I suggest that in the Soviet period Central Asians cultivated and conceptualized Islam as an episteme. They did this by reaching beyond alienating categories offered to them by the state. I argue that the constitution of an Islamic culture was made possible, among other things, by Central Asians’ encounters with the past, most notably with what they perceived as an Islamic past. We observe the curious phenomenon of Central Asians’ continuous interaction with the Islamic historical sites that (...) escaped the bulldozers of the Soviet campaigns of religious repression. For some, encounters with the past might be accidental. For many others, the exploration of the past represented a purposive, self-conscious, and reiterated emotional act. I show that Central Asians in the Soviet period—even if at school they were taught little about, and were usually offered a distorted vision of, the Islamic history of their region—were still able to access their past through the surviving architectural presence of Islam. Monumental sites, however, were not enough for Muslims to understand the past and use it to construct their own identity. Such artifacts acquired meaning through an interpretive framework provided by Sufi narratives about saints and their miracles. Therefore, shrines represented for Central Asia a collective memory space, i.e., a place in which the past was preserved for mobilization in the present through narrative. (shrink)

We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order‐theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies a finiteness condition. (...) We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case. (shrink)

Most human thinking is thoroughly informed by context but, until recently, theories of reasoning have concentrated on abstract rules and generalities that make no reference to this crucial factor. _Perspectives on Contexts_ brings together essays from leading cognitive scientists to forge a vigorous interdisciplinary understanding of the contextual phenomenon. Applicable to human and machine cognition in philosophy, artificial intelligence, and psychology, this volume is essential to the current renaissance in thinking about context.

This paper investigates the concept of étendue intelligible, as used by Malebranche to frame a metaphysical theory on top of the concept of space, a new perspective on ideas and a solution to Cartesian issues about the nature of thought and its relationships with extension. In particular, attention is paid to the reasons why the étendue intelligible is, for Malebrance, the joining link between the human mind and God. Such a connection is emphasized not only for its metaphysical meaning but (...) also for the intrinsic relevance of a topological view of ideas. (shrink)

The specific complexity of the question of genres depends on the plurality of the problems to which genre is pertinent and relevant. When Croce proclaimed, at the beginning of the century, the non-existence of the genre, he reduced the entire range of problems to the sole question relative to a work's belonging to a genre. In more recent times, there has been an attempt to reduce genre to the property of the linguistic act, in a methodological framework that understands poetics (...) exclusively as the science of literature. Against such reductionism, it is important to keep in mind a wider, more articulated idea of poetics as a pragmatic meditation, to investigate genres according to the multiple lines of systematization that are pertinent to them. (shrink)