We consider the theory and its weak fragments in the language of arithmetic expanded with the functional symbol . We prove that and its weak fragments, down to and , are subject to the Tennenbaum phenomenon with respect to , , and . For the last two theories it is still unknown if they may have nonstandard recursive models in the usual language of arithmetic.

We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

A generalization is given of McAloon's result on initial segments ofmodels of GlΔ0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

A generalization is given of McAloon's result on initial segments ofmodels of GlΔ0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

We give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization forκ-saturation, for a cardinal$\kappa \ge \aleph _0 $. Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.

In [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ 0 . However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ 0.

In this paper we prove, modulo Schanuel's Conjecture, that there are algorithms which decide if two exponential polynomials in π are equal in ℝ and if two exponential polynomials in π and i coincide in ℂ.

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r \in R}$$\end{document}, there exists an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i \in I}$$\end{document} so that i ≤ r < i + 1. Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field has an integer part. The procedure of Mourgues and (...) Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, yielding an integer part that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document} —limit computable relative to R. We show that there is a maximal Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} which is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document}. However, this I may not be an integer part for R. By a result of Wilkie (Logic Colloquium ’77), any Z-ring can be extended to an integer part for some real closed field. Using Wilkie’s ideas, we produce a real closed field R with a Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} such that I does not extend to an integer part for R. For a computable real closed field, we do not know whether there must be an integer part in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document}. We know that certain subclasses of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document} are not sufficient. We show that for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}, there is a computable real closed field with no n-c.e. integer part. In fact, there is a computable real closed field with no n-c.e. integer part for any n. (shrink)

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies the (...) property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. (shrink)

In [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in I$\Delta_0$. However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in I$\Delta_0$.

We prove that if G is a Δ 0 -definable function on the natural numbers and F = Π i = 0 n G , then F is also Δ 0 -definable. Moreover, the inductive properties of F can be proved inside the theory IΔ 0.

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We (...) also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)

1. Etica e coscienza. Etica, scienza e tecnica -- 2. Etica e cultura. Etica e politica -- 3. Etica ed economia. Problemi de etica.

In this article, we argue that John Henry Newman was right to think that our passional nature can play a legitimate epistemic role. First, we unpack the standard objection to Newman’s understanding of the relationship between our passional nature and the evidential basis of faith. Second, we argue that the standard objection to Newman operates with a narrow definition of evidence. After challenging this notion, we then offer a broader and more humane understanding of evidence. Third, we survey recent scholarship (...) arguing that emotions, a key aspect of our passional nature, are cognitive. In this light, they plausibly have a proper epistemic role. Fourth, we defend Newman’s reliance on the passional nature in epistemic matters by showing how reasonable it is in light of this recent work on evidence and the nature of emotions. Newman’s insistence that the formation of a right state of heart and mind is crucial for epistemic success is far from untenable. (shrink)

It considers the epistemology of theology and features 42 chapters, divided into 4 sections on 'Theology Relative Epistemic Concepts' and 'General Epistemic Concepts as Related to Theology', and on studies of individual theologians from St Paul through to Hans Urs von Balthasar and of contemporary movements such as Liberation Theology and Feminism.

In recent years, Duncan Pritchard has developed a position in religious epistemology called quasi‐fideism that he claims traces back to John Henry Newman's treatment of the rationality of religious belief. In this paper, we give three reasons to think that Pritchard's reading of Newman as a quasi‐fideist is mistaken. First, Newman's parity argument does not claim that religious and non‐religious beliefs are on a par because both are groundless; instead, for Newman, they are on a par because both often stem (...) from implicit rather than explicit reasoning. Second, pace Pritchard, Newman's distinction between simple and complex assent does not map onto the Wittgensteinian distinction between groundless hinge commitments and beliefs that flow from these hinges. For Newman, simple and complex assent differ in terms of the believer's level of awareness of their grounds. Third, and finally, Newman does not reject Locke's evidentialism in toto. Instead, he argues that certitude is not restricted to beliefs stemming from intuition and demonstration but often rightly includes probabilistically supported (or fallibly evidenced) beliefs. (shrink)

The Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5) uses the conceptualization of psychopathology to make psychiatric diagnoses operational. The use of explicit operational criteria appears to be based on an implicit neo-positivist epistemology. Operationalism involves an excessive focus on quantitative descriptions of behavior manifestations, contesting that psychopathology is understood as a deviation from the normal or the average in a given population. Consequently, the normal and the psychopathological become homogeneous. Our analysis investigates if this neo-positivist epistemology narrows (...) psychopathology conceptualization and endanger integration with the hybrid biopsychosocial model of psychiatry. Based on Georges Canguilhem’s theorization of a qualitative approach to the individual organism who is in a state of morbidity, we show that the (psychiatric) pathology also contains differences in quality. Moreover, that humans are norm-producing organisms that actively respond to changes in their internal and external environment. In this regard, the operationalization of mental disorders could include the normativity in humans, i.e., the ability to produce norms. We argue this will mitigate the one-sided psychopathology conceptualization and strengthen the relationship between psychiatric nosology and psychiatry’s hybrid biopsychosocial model. (shrink)

Searching for better ways to inspire people to pursue wisdom, Frederick D. Aquino argues that teachers and researchers should focus less on state-of-the-art techniques and learning outcomes and instead pay more attention to the intellectual formation of their students. We should, Aquino contends, encourage the development of an integrative habit of mind, which entails cultivating the capacity to grasp how various pieces of data and areas of inquiry fit together and to understand how to apply this information to new situations. (...) To fully explore this notion, _An Integrative Habit of Mind_ brings the work of the great religious figure and educator John Henry Newman into fruitful conversation with recent philosophical developments in epistemology, cognition, and education. Aquino unearths some crucial but neglected themes from Newman’s writings and carries them forward into the contemporary context, revealing how his ideas can help us broaden our horizons, render apt judgments, and better understand our world and how we think about it. (shrink)

The paper discusses the extent to which the availability of unprecedentedly rich data-sets and the need for new approaches – both epistemological and computational – is an emerging issue for Macroeconomics. By adopting an evolutionary approach, we describe the paradigm shifts experienced in the macroeconomic research field and emphasize that the types of data the macroeconomist has to deal with play an important role in the evolutionary process of the development of the discipline. After introducing the current debate over Big (...) Data in social sciences, the paper presents a detailed discussion of possible and existing interactions between Big Data and Computational Behavioral Macroeconomics. We argue that Big Data applied to economic questions can lead to new styles of thinking and research methods, namely to the development of a new research paradigm. (shrink)

In this essay, I show how the virtues, for Maximus the Confessor, contribute to the formation of a positive orientation toward (a deep and abiding desire for) the relevant epistemic goods (e.g., contemplation of God in and through nature, illumination of divine truths, wisdom, and experiential knowledge of God). The first section offers a brief overview of how three character-based virtue epistemologies envision the role of the intellectual virtues in the cognitive life. The second section draws attention to Maximus’s understanding (...) of the relationship between the virtues and the relevant epistemic goods. The third section argues that a relationship of this sort entails a seamless connection (though without confusion) between the practical and contemplative aspects of deiform existence. The fourth section clarifies how select virtues foster within the self a praiseworthy desire for the relevant epistemic goods. (shrink)

John Henry Newman’s philosophical reflection on the nature of faith and its relation to evidence is fascinating, complex, and slightly misleading; yet it shows constructive promise. In particular, I argue that his broader construal of reason should concomitantly play out in a broader construal of evidence. Accordingly, I show how Newman’s distinction between different modes of reasoning informs his understanding of the relationship between faith and evidence. I conclude with three areas that deserve further epistemological attention and development: namely, a (...) more expansive construal of evidence in light of Newman’s broader account of reason, a more constructive understanding of the relationship between his cumulative, though informal, approach and natural theology, and whether his account of faith and evidence operates with a kind of phenomenal conservatism. (shrink)