We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the (...) new interpretation to theories of classical arithmetic and analysis. (shrink)

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...) schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

We construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with Σ b 1 -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ11-comprehension (...) schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

We show how to interpret intuitionistic propositional logic into a predicative second-order intuitionistic propositional system having only the conditional and the universal second-order quantifier. We comment on this fact. We argue that it supports the legitimacy of using classical logic in a predicative setting, even though the philosophical cast of predicativism is nonrealistic. We also note that the absence of disjunction and existential quantifications allows one to have a process of normalization of proofs that avoids the use of "commuting conversions.".

In "'Yes" and "No'" (2000), Ian Rumfitt proposed bilateralism--a use-based account of the logical words, according to which the sense of a sentence is determined by the conditions under which it is asserted and denied. One of Rumfitt's key claims is that bilateralism can provide a justification of classical logic. This paper raises a techical problem for Rumfitt's proposal, one that seems to undermine the bilateralist programme.

What are the effects of word-by-word predictability on sentence processing times during the natural reading of a text? Although information complexity metrics such as surprisal and entropy reduction have been useful in addressing this question, these metrics tend to be estimated using computational language models, which require some degree of commitment to a particular theory of language processing. Taking a different approach, this study implemented a large-scale cumulative cloze task to collect word-by-word predictability data for 40 passages and compute surprisal (...) and entropy reduction values in a theory-neutral manner. A separate group of participants read the same texts while their eye movements were recorded. Results showed that increases in surprisal and entropy reduction were both associated with increases in reading times. Furthermore, these effects did not depend on the global difficulty of the text. The findings suggest that surprisal and entropy reduction independently contribute to variation in reading times, as these metrics seem to capture different aspects of lexical predictability. (shrink)

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

It has been known for six years that the restriction of Girard's polymorphic system $\text{\bfseries\upshape F}$ to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each $\beta$-reduction step of the full intuitionistic propositional calculus translates into one or more $\beta\eta$-reduction steps in the restricted Girard system. As a consequence, we obtain (...) a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to $\eta$-conversions. (shrink)

We introduce a new typed combinatory calculus with a type constructor that, to each type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, associates the star type σ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*$$\end{document} of the nonempty finite subsets of elements of type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory (...) calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements. (shrink)

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex (...) and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions. (shrink)

In Frege’s logicism, numbers are logical objects in the sense that they are extensions of certain concepts. Frege’s logical system is inconsistent, but Richard Heck showed that its restriction to predicative quantification is consistent. This predicative fragment is, nevertheless, too weak to develop arithmetic. In this paper, I will consider an extension of Heck’s system with impredicative quantifiers. In this extended system, both predicative and impredicative quantifiers co-exist but it is only permissible to take extensions of concepts formulated in the (...) predicative fragment of the language. This system is consistent. Moreover, it proves the principle of reducibility applied to concepts true of only finitely many objects. With the aid of this form of reducibility, it is possible to develop arithmetic in a thoroughly Fregean way. (shrink)

We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles are interpreted while maintaining unmoved the -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.

It is known that there is a sound and faithful translation of the full intuitionistic propositional calculus into the atomic polymorphic system F at, a predicative calculus with only two connectives: the conditional and the second-order universal quantifier. The faithfulness of the embedding was established quite recently via a model-theoretic argument based in Kripke structures. In this paper we present a purely proof-theoretic proof of faithfulness. As an application, we give a purely proof-theoretic proof of the disjunction property of the (...) intuitionistic propositional logic in which commuting conversions are not needed. (shrink)

We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov's principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König's lemma. We discuss some applications, as well as some previous results in the literature.

This paper develops the very basic notions of analysis in a weak second-order theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we (...) show how to interpret the theory BTFA in Robinson's theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. (shrink)

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is animpassable barrierin the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted inbut also what cannot be so (...) interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)

In this article we study applications of the bounded functional interpretation to theories of feasible arithmetic and analysis. The main results show that the novel interpretation is sound for considerable generalizations of weak König’s Lemma, even in the presence of very weak induction. Moreover, when this is combined with Cook and Urquhart’s variant of the functional interpretation, one obtains effective versions of conservation results regarding weak König’s Lemma which have been so far only obtained non-constructively.

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but (...) also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)

We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.

In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...) the interpretation validates and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the dialectica and related interpretations for extracting computational information from ordinary proofs in mathematics. (shrink)

We give an elementary proof of the strong normalization of the atomic polymorphic calculus Fat.

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this (...) extended system. (shrink)

Every model of IΔ0 is the tally part of a model of the stringlanguage theory Th-FO . We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ0 + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models.

I suggest here ecologies of the nascent state, posing the following general questions: what is this state and what is it to live, to fabricate modes of life, in its immanence? I believe populating this state is, by right, ‘ecological’, even if what I offer here is only a sketch or glimpse, playful as it is, of the possibility of such modes of life, of dwelling. As I develop it here, the nascent is in flight of being. It is populated (...) by lesser, minoritarian existences. If it is ‘ecological’, it is because these existences, or modes of becoming, are themselves, in their own right, ‘ecologies’, that is, modes of dwelling, of life, on the ‘other side of existence’, as Antonin Artaud put it once, in exile from Being. The power to return eternally to the nascent state is the power to live, to dwell, in the absolute forgetfulness of Being, in the interstice where philosophy supposedly ends, but where it nevertheless begins again, in oblivion itself, where being is never already the verticality of Being, its difference with beings, but always nascent, in the beginning, eternally so. (shrink)

Let be the fragment of elementary Peano arithmetic in which induction is restricted to -formulas. More than three decades ago, Parsons showed that the provably total functions of are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the -consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.

We formulate schemes and of the “typical” ∀∑ 1 b -sentences that are provable in T 2 1, respectively T 2 2. As an application, we reprove a recent result of Buss and Krajíček which describes witnesses for the ∀∑ 1 b -sentences provable in T 2 1 in terms of solutions to PLS-problems.

We formulate schemes and of the “typical” ∀∑ 1 b -sentences that are provable in T 2 1 , respectively T 2 2 . As an application, we reprove a recent result of Buss and Krajíček which describes witnesses for the ∀∑ 1 b -sentences provable in T 2 1 in terms of solutions to PLS-problems.

This article summarizes the recent evidence on global poverty and inequality, including both developed and developing countries. Section 1 discusses poverty and inequality data and presents evidence on levels and recent trends in poverty and inequality around the world. Section 2 turns to the issues involved in aggregating inequality indices across countries, in order to construct a meaningful measure of global inequality. Section 3 discusses the empirical relationship between economic growth, poverty, and inequality dynamics. Section 4 turns to the likely (...) economic determinants of poverty and inequality changes. Section 5 offers some conclusions, and points to some promising research themes within this general topic. (shrink)

In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, discuss the extra principles that (...) the interpretation validates and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the dialectica and related interpretations for extracting computational information from ordinary proofs in mathematics. (shrink)

We introduce a class of models of the bounded arithmetic theoryPV n . These models, which are generated by their tally part, have a curious feature: they have end-extensions or satisfyB∑ n b only in case they are closed under exponentiation. As an application, we show that if then the polynomial hierarchy does not collapse.

Disfluencies include editing terms such as uh and um as well as repeats and revisions. Little is known about how disfluencies are processed, and there has been next to no research focused on the way that disfluencies affect structure-building operations during comprehension. We review major findings from both computational linguistics and psycholinguistics, and then we summarize the results of our own work which centers on how the parser behaves when it encounters a disfluency. We describe some new research showing that (...) information associated with misarticulated verbs lingers, and which adds to the large body of data on the critical influence of verb argument structures on sentence comprehension. The paper also presents a model of disfluency processing. The parser uses a Tree Adjoining Grammar to build phrase structure. In this approach, filled and unfilled pauses affect the timing of Substitution operations. Repairs and corrections are handled by a mechanism we term “Overlay,” which allows the parser to overwrite an undesired tree with the appropriate, correct tree. This model of disfluency processing highlights the need for the parser to sometimes coordinate the mechanisms that perform garden-path reanalysis with those that do disfluency repair. The research program as a whole demonstrates that it is possible to study disfluencies systematically and to learn how the parser handles filler material and mistakes. It also showcases the power of Tree Adjoining Grammars, a formalism developed by Aravind Joshi which has yielded results in many different areas of linguistics and cognitive science. (shrink)

Any complete theory of lexical access in production must address how words are produced in prosodic contexts. Levelt, Roelofs & Meyer make some progress on this point: for example, they discuss resyllabification in multiword utterances. I present work demonstrating that word articulation takes into account overall prosodic context. This research supports Levelt et al.'s hypothesized separation between metrical and segmental information.

Can the brain become mind? If the question seems strange, aberrant even, it is perhaps due to the way the problem brain–mind is commonly presented, where what's repeatedly asked is whether the brain is or isn’t ‘mind’. Yet, if we take Deleuze and Guattari's provocation seriously, in the Conclusion of What is Philosophy?, the problem is radically recast: even if speculatively, that is, properly philosophically or conceptually, the brain, for them, involves its own becoming, both in terms of its ‘becoming-subject’ (...) and ‘becoming-mind’, as ‘Thought-brain’. I will explore references within Deleuze's philosophy in relation to the brain, namely in The Logic of Sense and his work on cinema, as I try to work out what this fantastic story (with necropolitical undertones) may in fact look like, the one where the brain not simply is but becomes, and in becoming, becomes mind…. (shrink)

Prior research has shown that various types of conventional multiword chunks are processed faster than matched novel strings, but it is unclear whether this processing advantage extends to variant multiword chunks that are less formulaic. To determine whether the processing advantage of multiword chunks accommodates variations in the canonical phrasal template, we examined the robustness of the processing advantage (i.e., predictability) of binomial phrases with non‐canonical conjunctions (e.g.,salt and also pepper; salt as well as pepper). Results from the cloze study (...) (Experiment 1) showed that there was a high tendency of producing the canonical conjunct (pepper), even in the binomials that contained non‐formulaic conjunctions. Consistent with these findings, results from two eye tracking studies (Experiments 2a and 2b) showed that canonical conjuncts were read faster than novel conjuncts that were matched on word length (e.g.,paprika), even in the binomials with variant conjunctions. This robust online processing advantage was replicated in a self‐paced reading study that compared all three conjunction types (Experiment 3). Taken together, these findings show that binomials with variant function words also receive facilitated processing relative to matched novel strings, even though both types of strings are neither conventional nor relatively frequent. Exploratory analyses revealed that this processing speed advantage was driven by the lexical–semantic association between the canonical conjuncts (salt–pepper), rather than lexical and phrasal frequency. Overall, these results highlight flexibility in the processing of multiword chunks that current models of multiword storage and processing must take into account. (shrink)

Frege’s theory is inconsistent. However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak. We also prove the finitistic consistency of the extension of Heck’s theory to $\Delta^{1}_{1}$-comprehension and of Heck’s ramified predicative second-order system.

Este artigo discute o papel da Sociologia na escola básica brasileira, que a despeito do arcabouço científico construído em sua trajetória, ainda consolida o seu espaço enquanto disciplina escolar. Nos últimos anos, com a obrigatoriedade da presença da Sociologia nos currículos escolares, novos desafios vêm sendo enfrentados pela disciplina. Um deles é fazer da Sociologia uma ferramenta para que os estudantes e a comunidade escolar possam ter um papel ativo no debate e na construção da realidade social, no que nos (...) arriscamos a denominar de uma Sociologia Escolar Pública . É com esse objetivo que desenvolvemos o projeto Redes Sociológicas, que articula as dimensões do ensino e da pesquisa na escola básica. (shrink)

We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application. we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.