The editors of the JRE collected short essays from scholars of religion in response to a recent incident at Hamline University that made national headlines. Last fall, Hamline University administrators refused to extend a contract to an adjunct professor of art history after a Muslim student accused her of Islamophobia for showing a 14th‐century image of Mohammad in an online class. The event provoked intense conversations about issues of academic freedom, religious diversity, the status of contingent faculty, and race. These (...) essays bring together scholarly and personal reflections about the incident at Hamline and what it means for the pedagogy of religious studies. (shrink)

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some (...) important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down. (shrink)

In The Anticipatory Corpse, Jeffrey Bishop claims that modern medicine has lost formal and final causality as the dead body has become epistemologically normative, and that a singular focus on efficient and material causality has thoroughly distorted modern medical practice. Bishop implies that the renewal of medicine will require its housing in alternate social spaces. This essay critiques both Bishop’s diagnosis and therapy by arguing, first, that alternate social imaginaries, though perhaps marginalized, are already present within the practice of medicine. (...) And second, the essay argues that alternate social imaginaries in medicine can be reclaimed not through separatist communities but in the re-narration of conceptually underdetermined practices. Given Bishop’s invitation for theology to engage medicine, this essay then draws from theologian Dietrich Bonhoeffer for the kind of diagnosis and therapy currently needed, concluding with a contemporary example of how an alternate social imaginary is being instantiated in modern medicine. (shrink)

In 1957, Gödel proved that completeness for intuitionistic predicate logic HPL implies forms of Markov's Principle, MP. The result first appeared, with Kreisel's refinements and elaborations, in Kreisel. Featuring large in the Gödel-Kreisel proofs are applications of the axiom of dependent choice, DC. Also in play is a form of Herbrand's Theorem, one allowing a reduction of HPL derivations for negated prenex formulae to derivations of negations of conjunctions of suitable instances. First, we here show how to deduce Gödel's results (...) by alternative means, ones arguably more elementary than those of Kreisel. We avoid DC and Herbrand's Theorem by marshalling simple facts about negative translations and Markov's Rule. Second, the theorems of Gödel and Kreisel are commonly interpreted as demonstrating the unprovability of completeness for HPL, if means of proof are confined within strictly intuitionistic metamathematics. In the closing section, we assay some doubts about such interpretations. (shrink)

This essay examines the interconnected nature of salvation and health, and it does so by engaging both recent qualitative research and three scriptural accounts from the life, death, and resurrection of Jesus. In doing so, the essay argues that salvation and health—and their conceptual pairings, sin and disease—are never individualistic. These realities are always cosmic, communal, and interpersonal, even as sin and disease are fundamentally disintegrating and isolating. The salvation and health of people suffering with substance use issues are bound (...) up with the transformations of governing principalities and powers, social realities, and relationships. Health care practitioners and clergy should be wise, communal guides offering care and accompaniment in the pursuit of salvation and health. (shrink)

An interview with Charles McCarty by Piotr Urbańczyk concerning mathematical explanation.

Contemporary conceptions and practices regarding complicity have led to the surprising emergence of citizens who seek rather than flee complicity. The purpose of this is to gain standing to challenge controversial state practices. As in the recent Hobby Lobby decision, such attempts to demonstrate complicity are not motivated by a desire to take ownership over state actions, but to justify institutional reforms or individual opt-outs that would not be legitimized absent such a finding of complicity. This article highlights the danger (...) posed by the proliferation of claims of this sort and works to unpack and critique the logic of such efforts to demonstrate complicity. Such claims rely upon implied political meanings of complicity, which affirm a strong connection between complicity, standing, and sovereignty. These political meanings of complicity are embedded in familiar conceptions of liberal democratic citizenship, but upon examination prove to be in tension with core liberal principles. By bringing to the surface the connection between complicity, standing, and sovereignty, this logic may be reformulated in such a way as to contribute to healthier modes of democratic participation. (shrink)

After distinguishing three criteria of adequacy for any acceptable moral theory's treatment of moral conflict, or conflicts of duties, I explain how Kant's ethics can satisfy all three. Although Kant denies the possibility of conflicting duties, he does allow conflicting "grounds of obligation." I develop a new interpretation of such conflicts, rejecting one proposed earlier by Onora O'Neill.

Drawing on the final report of the Greensboro Truth and Reconciliation Commission and on theology, this essay builds on Miroslav Volf's social Trinitarian account of reconciliation as embrace. Specifically, this essay argues for the necessity of various forms of justice in social and political reconciliation and against the priority of forgiveness in reconciliation argued for by Volf. The heart of this argument is a theological anthropology that claims that to be created in the image of a perichoretic God who is (...) Trinity is necessarily to be interdependent beings. This interdependence is manifest in the interpersonal, social, and political relations that constitute and are constituted by individual humans and the institutions in which they live. Therefore, the creation and maintenance of just institutions is necessary for the formation of persons capable of practicing reconciliation, and for reconciled persons to live within. (shrink)

Provides a guide for starting a "philosophy dinner club," a club that meets to discuss philosophy and cook food from each philosopher's home country.

Straightforwardly and strictly intuitionistic inferences show that the Brouwer– Heyting–Kolmogorov interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. (...) The results of the present article extend and refine those of McCarty, C.. Antirealism and Constructivism: Brouwer’s Weak Counterexamples. The Review of Symbolic Logic. First View. Cambridge University Press. (shrink)

, Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject. I am grateful to (...) Nathan Carter, Gary Ebbs, Janet Folina, Luise Prior McCarty, Stewart Shapiro, Neil Tennant, Christopher Tillman, Beth Tropman, Wen-fang Wang, and two anonymous referees for their comments and suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

What do medicine and war have to do with each other? This question is explored through the writings of James Childress, whose early contributions to just war theory illuminate his work in bioethics. By considering the conceptual influences of just war theory on Childress’s bioethics, the contributions and limits of his approach can be set in relief through normative engagement with certain areas of medicine. In particular, Childress’s just-war-inspired bioethics befits the practice of surgery; but oncology, as a medical analogue (...) to total warfare, requires significant transformation in order to be disciplined by Childress’s approach. Childress offers a coherent schema for navigating moral conflict in a fallen world, but he does not provide a substantive account of the peaceable end toward which medicine as just war aims. (shrink)

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic (...) regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski. Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle. (shrink)

This chapter presents and illustrates fundamental principles of the intuitionistic mathematics devised by L.E.J. Brouwer and then describes in largely nontechnical terms metamathematical results that shed light on the logical character of that mathematics. The fundamental principles, such as Uniformity and Brouwer’s Theorem, are drawn from the intuitionistic studies of logic and topology. The metamathematical results include Gödel’s negative and modal translations and Kleene’s realizability interpretation. The chapter closes with an assessment of anti-realism as a philosophy of intuitionism.

Against the background of current reforms in higher education, we analyze the traditional education of Japanese doctoral students in philosophy of education from Western and Japanese perspectives by focusing on learning as self-education, on being and learning with others, on the socialization into the profession, and on the study of the foreign subject. Imai's explication of the Japanese construction of the adult self as instrumental is compared to Gadamer's ideas on self-education and education with others. A significant element of doctoral (...) education in Japan involves the learning of certain forms ( kata ) of professional conduct: belonging to a group, occupying a clear position in the social hierarchy, and developing a close mentoring relationship that fosters strong feelings of loyalty, harmony and respect. Furthermore, reciprocal dependence ( amae ) is analyzed and contrasted with the Western ideal of autonomy and maturity. Finally, the study of the foreign subject is considered in light of Gadamer's and Hegel's notion of making a home in the alien. Saito affirms a cross-cultural reading of foreign philosophical texts, claiming that they destabilize the native culture and turn a teacher in philosophy of education into a translator and prophet of her own as well as the foreign culture. (shrink)

The use of conventional logical connectives either in logic, in mathematics, or in both cannot determine the meanings of those connectives. This is because every model of full conventional set theory can be extended conservatively to a model of intuitionistic set plus class theory, a model in which the meanings of the connectives are decidedly intuitionistic and nonconventional. The reasoning for this conclusion is acceptable to both intuitionistic and classical mathematicians. En route, I take a detour to prove that, given (...) strictly intuitionistic principles, classical negation cannot exist. (shrink)

Satisfiability or Sat\ is the metatheoretic statementEvery formally intuitionistically consistent set of first-order sentences has a model.The models in question are the Tarskian relational structures familiar from standard first-order model theory, but here treated within intuitionistic metamathematics. We prove that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat\ to be false outright. Following the lead of Carter :75–95, 2008), we then generalize this result to some provably intermediate first-order logics, including the Rose logic. These metatheorems (...) distinguish the intuitionistic foundational significance of Sat\ sharply from that of Sat\, the satisfiability claim for intuitionistic propositional logic. At the same time, they establish intuitionistic connections with and between Test, COMP\, and \. Here, Test is the scheme of Testability, and COMP\ and \ are completeness for intuitionistic propositional logic and predicate logic, respectively. (shrink)

This essay engages Miguel De La Torre’s proposal to “embrace hopelessness” and argues that Christians should hold on to hope. The author places De La Torre’s argument in conversation with others who have written on hope and hopelessness and excavates two main weaknesses in his argument: first, a definition of hope that does not stand up to a review of the literature on the topic, especially as advocated by scholars from oppressed communities, and, second, a proposal for hopelessness that does (...) not address how it can contribute to sustainable social transformation. The author then defends hope by drawing on the theological labor and lived experience of oppressed people who utilize hope to empower transformative social action. (shrink)

We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.

Rather than “embracing hopelessness,” many marginalized communities understand their practices of political resistance as exercises in hope. One space of contemporary activism where this is evident is in transformative justice movements. Utilizing the idea of moral imagination as articulated in peacebuilding and conflict transformation literature, and the idea of hope as a social practice as articulated by Keri Day, I argue that a close examination of transformative justice organizing reveals hope as a social practice of embodied moral imagination practiced by (...) communities on the margins. (shrink)

Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to _Test_, the recognition condition, and the _BHK_ interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the _BHK_, then every relevant instance of the classical principle _Test_ is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is (...) seen, in this instance, to fail. (shrink)

McCarty, Douglas Morality and ethics would be recognised by virtually everyone as a necessity in human society. In fact morality and ethics are the very foundations on which human society is built. Humans are complex individuals, with personal needs, desires, inclinations, dispositions and preferences, but are also naturally predisposed to live in social groups large and small with similarly complex individuals.

One’s first impression is that Brouwer’s Continuity Theorem of intuitionistic analysis, that every total, real-valued function of a real variable is continuous, stands in straightforward contradiction to a simple theorem of conventional real analysis, that there are discontinuous, real-valued func­tions. Here we argue that, despite philosophical views to the contrary, first impressions are not misleading; the Brouwer Theorem, together with its proof, presents mathematicians and philosophers of mathematics with an antimony, one that can only be resolved by a close, foundational (...) study of the structure of the intuitive continuum. (shrink)

This article offers an interpretation of Vladimir Nabokov’s unique contribution to political theory as seen primarily through the lens of his novel Invitation to a Beheading. Although most frequently interpreted as an indictment of totalitarianism, the novel depicts a form of cruelty practiced not only by totalitarians, but also by the rulers and citizens of milder political orders, including liberalism. The novel suggests that such cruelty is more insidious than that familiar to readers of dystopian novels precisely because of its (...) universality. This article demonstrates that Nabokov’s contribution to liberalism may be found in the surprising coherence between his aesthetic principles and his art, both of which critique the imposition of “general ideas” on either persons or books. What emerges is a picture of aesthetic liberalism in which Nabokov’s model for the ideal liberal citizen is neither the sensitive artist nor the apolitical aesthete, but rather the careful reader. (shrink)