We provide a new and elementary proof of strong normalization for the lambda calculus of intersection types. It uses no strong method, like for instance Tait-Girard reducibility predicates, but just simple induction on type complexity and derivation length and thus it is obviously formalizable within first order arithmetic. To obtain this result, we introduce a new system for intersection types whose rules are directly inspired by the reduction relation. Finally, we show that not only the set of strongly normalizing terms (...) of pure lambda calculus can be characterized in this system, but also that a straightforward modification of its rules allows to characterize the set of weakly normalizing terms. (shrink)

By using some classical reasoning we show that any countably presented formal topology, namely, a formal topology with a countable axiom set, is spatial.

Abstract.Formal topologies are today an established topic in the development of constructive mathematics. One of the main tools in formal topology is inductive generation since it allows to introduce inductive methods in topology. The problem of inductively generating formal topologies with a cover relation and a unary positivity predicate has been solved in [CSSV]. However, to deal both with open and closed subsets, a binary positivity predicate has to be considered. In this paper we will show how to adapt to (...) this framework the method used to generate inductively formal topologies with a unary positivity predicate; the main problem that one has to face in such a new setting is that, as a consequence of the lack of a complete formalization, both the cover relation and the positivity predicate can have proper axioms. (shrink)

We prove a generalization of the hyper-game theorem by using an abstract version of inductively generated formal topology. As applications we show proofs for Cantor theorem, uncountability of the set of functions from N to N and Gödel theorem which use no diagonal argument.

In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.

Formal topology is today an established topic in the development of constructive mathematics and constructive proofs for many classical results of general topology have been obtained by using this approach. Here we analyze one of the main concepts in formal topology, namely, the notion of formal point. We will contrast two classically equivalent definitions of formal points and we will see that from a constructive point of view they are completely different. Indeed, according to the first definition the formal points (...) of the formal topology of the real numbers can be indexed by a set whereas this is not possible according to the second one. (shrink)

In this paper we analyze an extension of Martin-Löf s intensional set theory by means of a set contructor P such that the elements of P are the subsets of the set S. Since it seems natural to require some kind of extensionality on the equality among subsets, it turns out that such an extension cannot be constructive. In fact we will prove that this extension is classic, that is “ true holds for any proposition A.

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal (...) topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales. (shrink)

In this paper a proof of the normal form theorem for the closed terms of Girard's system F is given by using a computability method à la Tait. It is worth noting that most of the standard consequences of the normal form theorem can be obtained using this version of the theorem as well. From the proof-theoretical point of view the interest of the proof is that the definition of computable derivation here used does not seem to be well founded. (...) MSC: 03F05, 03B15. (shrink)

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B prop [x : A], i. e. to require that the predicate ∨ ¬ B) is provable, is equivalent, when working within the framework of Martin-Löf's Intuitionistic Type Theory, to require that there exists a decision function ψ: A → Boole such that = Booletrue) ↔ B). Since we will also show that the proposition x = Booletrue [x: Boole] is decidable, we can (...) alternatively say that the main result of this paper is a proof that the decidability of the predicate Bprop [x : A] can be effectively reduced by a function ψ A → Boole to the decidability of the predicate ψ = Booletrue [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function ψ, together with a proof of its correctness, is effectively constructed as a function of the proof of ∨ ¬ B). (shrink)

We analyze some extensions of Martin-Löf 's constructive type theory by means of extensional set constructors and we show that often the most natural requirements over them lead to classical logic or even to inconsistency.

We characterise the collections of infinite binary sequences that, when barred by a set of finite binary sequences, are also barred by a finite subset of such a set.

In this paper we prove that any quantale Q is a quantale of suitable relations on Q. As a consequence two isomorphism theorems are also shown with suitable sets of functions of Q into Q. These theorems are the mathematical background one needs in order to give natural and complete semantics for Linear Logic using relations.

In 1996, Krivine applied Friedman's A-translation in order to get an intuitionistic version of Gödel completeness result for first-order classical logic and countable languages and models. Such a result is known to be intuitionistically underivable 559), but Krivine was able to derive intuitionistically a weak form of it, namely, he proved that every consistent classical theory has a model. In this paper, we want to analyze the ideas Krivine's remarkable result relies on, ideas which where somehow hidden by the heavy (...) formal machinery used in the original proof. We show that the ideas in Krivine's proof can be used to intuitionistically derive some crucial mathematical results, which were supposed to be purely classical up to now: the Ultrafilter Theorem for countable Boolean algebras, and the maximal ideal theorem for countable rings. (shrink)

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal (...) topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales. (shrink)

We propose a formal representation of objects , those being mathematical or empirical objects. The powerful framework inside which we represent them in a unique and coherent way is grounded, on the formal side, in a logical approach with a direct mathematical semantics in the well-established field of constructive topology, and, on the philosophical side, in a neo-Kantian perspective emphasizing the knowing subject’s role, which is constructive for the mathematical objects and constitutive for the empirical ones.

This work deals with the exponential fragment of Girard's linear logic without the contraction rule, a logical system which has a natural relation with the direct logic . A new sequent calculus for this logic is presented in order to remove the weakening rule and recover its behavior via a special treatment of the propositional constants, so that the process of cut-elimination can be performed using only “local” reductions. Hence a typed calculus, which admits only local rewriting rules, can be (...) introduced in a natural manner. Its main properties — normalizability and confluence — has been investigated; moreover this calculus has been proved to satisfy a Curry-Howard isomorphism with respect to the logical system in question. MSC: 03B40, 03F05. (shrink)

In this short note we show that any proof of a general spatiality theorem for inductively generated formal topologies requires full classical logic. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

An intuitionistic version of Cantor's theorem, which shows that there is no surjective function from the type of the natural numbers N into the type N → N of the functions from N into N, is proved within Martin-Löf's Intuitionistic Type Theory with the universe of the small types.

Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in (...) a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined. (shrink)

Este texto reflete sobre o memoricídio, consequente à presunção racial branca, como vestígio de operações para o genocídio dos corpos e das mentes negras a partir da Peleja de Inácio da Catingueira contra Romano. Sob o jogo discursivo da humilhação, já se encontrava a vontade brutal de destruição de qualquer traço, perspectiva, visão remanescente de saberes e culturas africanas ou descendentes, fossem de cunho material, simbólico, reflexivo, imagético, expressivo, na língua, no corpo. No embate, Inácio, negro, escravizado, já rebaixado sob (...) a égide do sistema escravocrata, embate-se contra a tentativa presunçosa de aniquilamento perpetrada por seu oponente. (shrink)

Is democracy a requirement of justice or an instrument for realizing it? The correct answer to this question, I argue, depends on the background circumstances against which democracy is defended. In the presence of thin reasonable disagreement about justice, we should value democracy only instrumentally (if at all); in the presence of thick reasonable disagreement about justice, we should value it also intrinsically, as a necessary demand of justice. Since the latter type of disagreement is pervasive in real-world politics, I (...) conclude that theories of justice designed for our world should be centrally concerned with democracy. (shrink)

When and why do socially constructed norms—including the laws of the land, norms of etiquette, and informal customs—generate moral obligations? I argue that the answer lies in the duty to respect others, specifically to give them what I call “agency respect.” This is the kind of respect that people are owed in light of how they exercise their agency. My central thesis is this: To the extent that (i) existing norms are underpinned by people’s commitments as agents and (ii) they (...) do not conflict with morality, they place moral demands on us on agency-respect grounds. This view of the moral force of socially constructed norms, I suggest, is superior to views that deny the moral force of such norms, and it elegantly explains certain instances of wrongdoing that would otherwise remain unaccounted for. (shrink)

In his recent bookThe Idea of Justice, Amartya Sen suggests that political philosophy should move beyond the dominant, Rawls-inspired, methodological paradigm – what Sen calls ‘transcendental institutionalism’ – towards a more practically oriented approach to justice: ‘realization-focused comparison’. In this article, I argue that Sen's call for a paradigm shift in thinking about justice is unwarranted. I show that his criticisms of the Rawlsian approach are either based on misunderstandings, or correct but of little consequence, and conclude that the Rawlsian (...) approach already delivers much of what Sen himself wants from a theory of justice. (shrink)

A globalized world, some argue, needs a global democracy. But there is considerable disagreement about whether global democracy is an ideal worth pursuing. One of the main grounds for scepticism is captured by the slogan: “No global demos, no global democracy.” The fact that a key precondition of democracy—a demos—is absent at the global level, some argue, speaks against the pursuit of global democracy. The paper discusses four interpretations of the skeptical slogan—each based on a specific account of the notion (...) of “the demos”—and concludes that none of them establishes that the global democratic ideal must be abandoned. In so doing, the paper (i) systematizes different types of objections against global democracy, thus bringing some clarity to an otherwise intricate debate and (ii) offers a robust but qualified defense of the global democratic ideal. (shrink)

This article provides a conceptual map of the debate on ideal and non‐ideal theory. It argues that this debate encompasses a number of different questions, which have not been kept sufficiently separate in the literature. In particular, the article distinguishes between the following three interpretations of the ‘ideal vs. non‐ideal theory’ contrast: (i) full compliance vs. partial compliance theory; (ii) utopian vs. realistic theory; (iii) end‐state vs. transitional theory. The article advances critical reflections on each of these sub‐debates, and highlights (...) areas for future research in the field. (shrink)

In this article, I develop a new account of the liberal view that principles of justice are meant to justify state coercion, and consider its implications for the question of global socioeconomic justice. Although contemporary proponents of this view deny that principles of socioeconomic justice apply globally, on my newly developed account this conclusion is mistaken. I distinguish between two types of coercion, systemic and interactional, and argue that a plausible theory of global justice should contain principles justifying both. The (...) justification of interactional coercion requires principles regulating interstate interference; that of systemic coercion requires principles of global socioeconomic justice. I argue that the proposed view not only helps us make progress in the debate on global justice, but also offers an independently compelling and systematic account of the function and conditions of applicability of justice. -/-. (shrink)

A prominent view in contemporary political theory, the ‘associative view’, says that duties of justice are triggered by particular cooperative relations between morally significant agents, and that ‘therefore’ principles of justice apply only among fellow citizens. This view has been challenged by advocates of global justice, who point to the existence of a world-wide cooperative network to which principles of justice apply. Call this the challenge from geographical extension. In this paper, I pose a structurally similar challenge to the associative (...) view: the challenge from species extension. This says that the existing network of cooperation extends beyond the human species, to encompass some non-human animals, particularly domesticated dogs. In light of this, if one believes that (i) certain non-human animals are morally significant (i.e. objects of moral concern), and that (ii) justice applies to fellow cooperators, one should also hold that domesticated dogs are owed justice in much the same way our human fellow citizens are. I conclude by considering the implications of this argument for the associative view, and animal-rights theory. (shrink)

Which standards should we employ to evaluate the global order? Should they be standards of justice or standards of legitimacy? In this article, I argue that liberal political theorists need not face this dilemma, because liberal justice and legitimacy are not distinct values. Rather, they indicate what the same value, i.e. equal respect for persons, demands of institutions under different sets of circumstances. I suggest that under real-world circumstances – characterized by conflicts and disagreements – equal respect demands basic-rights protection (...) and democratic participation, which I here call ‘political justice’. I conclude the article by considering three possible configurations of the global order – the ‘democratic world-state’, ‘independent democratic states’, and ‘mixed’ models – and argue that a commitment to political justice speaks in favour of the latter. (shrink)

Principles of distributive justice bind macro-level institutional agents, like the state. But what does justice require in non-ideal circumstances, where institutional agents are unjust or do not e...

This article suggests developing the concept of spiritual ecology into that of balanced spiritual ecosystems. Philosophies, theologies, education systems, political parties, and gender-based and ethnic identity politics need to be critiqued both from within and without so that they can finally contribute to the creation, maintenance and flourishing of balanced spiritual ecosystems.Contribution: Spiritual ecology is a concept on which converge different worldviews. This article recommends using balanced spiritual ecosystems, instead. The new concept could provoke further reflection on how our -ontologies (...) and communities co-exist and whether they do it organically and spiritually. This article befits HTS’s interest in ‘theology and nature’. (shrink)