John Foxal was an English Franciscan active primarily in Italy in the latter half of the fifteenth century. His philosophical and theological positions were Scotist, and in his works he comments upon various texts of John Duns Scotus with reference to the Scotist school of thought active in the early fourteenth century. In this article we supplement the list of extant manuscripts, provide a revised list of Foxal’s treatises and offer an edition of one of these, the Tractatus de propositione (...) per se nota. (shrink)

In the post-colonial era, the incarnation has become an important model for cross-cultural missionary presence. Though this model improves on Eurocentric and colonial models, it is deficient because it is unrealistic, potentially paternalistic, inappropriate in the light of globalization and post-modern understandings of culture, and because it doesn’t sufficiently respect the particularity of the incarnation of Christ. This article proposes an alternative model of the role of the cross-cultural missionary as a guest and argues that it is more appropriate on (...) precisely these counts. (shrink)

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory. In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as (...) more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned. (shrink)

Until recently, only two Arabic translations of Ptolemy’s Almagest were thought to have survived, one by al-Ḥajjāj made under the caliph al-Maʾmūn and one by Isḥāq b. Ḥunayn, later revised by Thābit b. Qurra, prepared in the second half of the third/ninth century. In the 2010s, a third Arabic version of the Almagest, authored by Thābit b. Qurra alone, was first shown by Dirk Grupe to be extant in a partial Latin translation in a Dresden manuscript, and then in an (...) Arabic manuscript in Jaipur that breaks off after chapter VI.2. We introduce here a second manuscript of this Thābit version, extant in the Khuda Bakhsh Library in Patna, which is around four centuries older than the Jaipur manuscript, was copied from a manuscript by the hand of al-Bīrūnī, and extends the text till the middle of Book VIII. After a general introduction, we provide a detailed description of this manuscript. (shrink)

A nucleus is an operation on the collection of truth values which, like double negation in intuitionistic logic, is monotone, inflationary, idempotent and commutes with conjunction. Any nucleus determines a proof-theoretic translation of intuitionistic logic into itself by applying it to atomic formulas, disjunctions and existentially quantified subformulas, as in the Gödel–Gentzen negative translation. Here we show that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda’s negative translation. The key is (...) to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well. (shrink)

Though the Christian faith has in recent years increasingly shown itself to be a truly African religion, a variety of African authors such as Kä Mana, George Kinoti, Hannah Kinoti, August Shutte and Efoé Julien Penoukou have noted that sub-Saharan Africa is facing a moral crisis. This article explores this crisis in as far as it is caused by difficulties in the reception of the Christian ethic by African Christian communities. It points out that this crisis is visible in double (...) morality, immorality and legalism. It shows that it is both caused by rapid social change in contemporary Africa and by the way the Christian ethic was introduced with a lack of attention for the relationship between worldview and ethics, the social impact of changing cultural practices and the importance of virtue ethics. In this way it also points to the shape Christian moral education for present-day Africa should take. (shrink)

In earlier work we introduced two systems for nonstandard analysis, one based on classical and one based on intuitionistic logic; these systems were conservative extensions of first-order Peano and Heyting arithmetic, respectively. In this paper we study how adding the principle of countable saturation to these systems affects their proof-theoretic strength. We will show that adding countable saturation to our intuitionistic system does not increase its proof-theoretic strength, while adding it to the classical system increases the strength from first- to (...) full second-order arithmetic. (shrink)

The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on finitely complete and cocomplete categories. As an instance of this result, we build the final coalgebra for the powerclass functor, in the context of a Heyting pretopos with a class of small maps. This is then proved to provide models for various non-well-founded set theories, depending on the chosen axiomatisation (...) for the class of small maps. (shrink)

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Gödel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of E-HAω and E-PAω, strengthening earlier results by Moerdijk and Palmgren, and Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. To conclude the (...) paper, we will point out some open problems and directions for future research, including some initial results on saturation principles. (shrink)

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...) maps and show that it provides a sound and complete semantics for constructive set theories like IZF and CZF. The main technical contribution of this paper is that it shows in detail that such categories can always be conservatively embedded in categories that are exact. These exactness properties play a crucial rôle in showing that predicative categories with small maps contain models of set theory and that they are closed under sheaves and realizability. We will prove the former statement in this paper as well, leaving a proof of the closure properties to the papers on realizability and sheaves as mentioned above. (shrink)

We present a version of arithmetic in all finite types based on a systematic use of an internally definable notion of observational equivalence for dealing with equalities at higher types. For this system both intensional and extensional models are possible, the deduction theorem holds and the soundness of the Dialectica interpretation is provable inside the system itself.

Using the theory of exact completions, I construct a certain class of pretoposes, consisting of what one might call “predicative realizability toposes”, that can act as categorical models of certain predicative type theories, including Martin-Löf Type Theory.

We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call “non-deterministic inductive definitions”. We give applications to formal topology as well as a predicative justification of this principle.

Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with M-types under various topos-theoretic constructions; namely, slicing, formation of coalgebras , and sheaves for an (...) internal site. (shrink)

We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category EFF. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as such provide a context in which one can interpret many notions from homotopy theory and Homotopy Type Theory. Within the path category EFF one can identify a class of discrete fibrations which is closed under push forward along arbitrary fibrations (in other words, this (...) class is polymorphic or closed under impredicative quantification) and satisfies propositional resizing. This class does not have a univalent representation, but if one restricts to those discrete fibrations whose fibres are propositions in the sense of Homotopy Type Theory, then it does. This means that, modulo the usual coherence problems, it can be seen as a model of the Calculus of Constructions with a univalent type of propositions. We will also build a more complicated path category in which the class of discrete fibrations whose fibres are sets in the sense of Homotopy Type Theory has a univalent representation, which means that this will be a model of the Calculus of Constructions with a univalent type of sets. (shrink)

The science and faith debate is dominated by Western voices. In order to enrich this debate, the authors study the discourses of different groups of Christian academics and master's students in francophone Africa. This article describes the process of reconstructing and analyzing the discourse of a group of master's students from Abidjan (Ivory Coast) with the help of group model building and focus groups. Three characteristic features that emerge from this discourse include the foundational position of faith, the central role (...) of truth, and the ambiguous connotations of the term “science” in this context. The reconstructed discourse is then brought into conversation with the North Atlantic debate, with a special focus on the concept of scientism. (shrink)

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of (...) the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

Analisi critica d'opera, incentrata sulla recezione dell'opera di Tolomeo durante il Medioevo e sul pensiero scientifico-filosofico dell' Età Medievale e Rinascimentale.

Ptolemaic Tradition and Islamic Innovation: The Astronomical Tables of Kūshyār ibn Labbān. By Benno Van Dalen. Ptolemaeus Arabus et Latinus, Texts, vol. 2. Turnhout, Belgium: Brepols, 2021. Pp. xviii + 595, 16 color pls. €120; https://ptolemaeus.badw.de/publications.

Classical apologetic strategies are in dire straits, according to some experts, particularly in a postmodern, post-Western, post-Enlightenment, and post-Christian world. Benno van den Toren confronts these questions head-on while proposing a methodologically sophisticated and theologically robust rationale for a dialogically oriented Christian apologetics. Christian Apologetics as Cross-Cultural Dialogue invites conversation about epistemological, dialogical, ecclesiological, eschatological, and christological matters, all of which suggest that a new paradigm of apologetics may be on the horizon for evangelical philosophy and theology. Those who (...) disagree will need to make their case against the arguments in this book. (shrink)

Keuze van artikelen uit het werk van de Vlaamse filosoof over wetenschap, milieu, en medische en biologische ethiek.

Dit jaarboek van het Studium Generale van de Hogeschool Gent bevat essays van de sprekers die tijdens het academiejaar 2013-2014 een lezing hebben gehouden over het jaarthema "Tijd? Altijd!". In deze bundel wordt stilgestaan bij de vraag hoe allerlei vormen van en opvattingen over tijd aan de orde zijn in de maatschappij, in de kunsten en in het wetenschappelijk onderzoek.0Deze lezingenreeks werd samengesteld en georganiseerd door Sofie Vandamme i.s.m. Jeroen Cluckers. Het boek bevat bijdragen van Sofie Vandamme, Benno Barnard, (...) Mark Fisher, Pedro De Bruyckere, Els Stuyven, Joke J. Hermsen, Marli Huijer, Dieter Ceustermans, Carl Devos, Petra Van Brabandt en Lisaboa Houbrechts"--Page 4 of cover. (shrink)

Welke rol mogen levensbeschouwingen spelen in het onderwijs? Waar liggen de grenzen van het multiculturele samenleven? Hoe tolerant kun je zijn voor intolerantie? Wat is de rol van het actief pluralisme? Hoe verhoudt het humanisme zich tot ecologie? Is een moraal mogelijk zonder god? De Vrijzinnige dienst van de Universiteit Antwerpen nodigde tijdens de lezingencyclus "Onze waarden? Welke toekomst voor het humanisme?" Dirk Verhofstadt, Wim van Rooy, Floris van den Berg, Raymonda Verdyck, Magda Michielsens, Ronald Commers en Benno Barnard (...) uit om zich hierover uit te spreken. Jurgen Slembrouck bundelde hun lezingen. (shrink)