This paper initiates the reverse mathematics of social choice theory, studying Arrow’s impossibility theorem and related results including Fishburn’s possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in ${\mathsf {RCA}}_0$. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in ${\mathsf {RCA}}_0$. This approach yields a proof of Arrow’s theorem in ${\mathsf {RCA}}_0$, and thus in $\mathrm {PRA}$, since Arrow’s theorem can be formalised as a $\Pi ^0_1$ sentence. Finally we show that Fishburn’s possibility theorem for countable societies is equivalent to ${\mathsf {ACA}}_0$ over ${\mathsf {RCA}}_0$.

The majority of studies on ‘faith’ (fides) in the thought of Thomas Aquinas consider it in a religious or theological context: fides as the theological virtue by which one assents to the truths of divine revelation. The focus on theological faith is appropriate, given its central importance as a theological virtue, but this is not the only sense of fides that Thomas identifies. The present study investigates two non-theological senses formulated in his commentary on the De Trinitate of Boethius: first, fides as the proximate cause of assenting to principles within a given science (‘epistemic faith’) and, second, fides as an indispensable element of society (‘societal faith’). These senses have been largely overlooked in secondary literature but, I argue, might help to dispel mischaracterizations of faith as fundamentally unreasonable.

Tertiary weathered sediments located immediately to the west of the harbor at Newhaven, Sussex, UK, were investigated by examination of major and trace elements by scanning electron microscope (SEM), microprobe, and inductively coupled plasma mass spectrometer (ICP-MS), and the mineralogy was studied by optical petrography, X-ray diffraction (XRD), transmission electron microscope (TEM), selective leaching, and thermodynamic modeling. Studied outcrops experienced acid leaching by sulfuric acid percolating downward through Tertiary sediments overlying Cretaceous chalk. The progressive neutralization of the percolating acid fluids resulted in “sequentially” layered neoformation of minerals: jarosite, iron oxides, aluminous minerals (sulfates, oxyhydroxides), gypsum, and Fe-Mn oxides. Substantial agreement was found between field observations and mineral assemblages obtained by modeling with the program CHILLER. These results suggest that the initial assumptions on the weathering process and mechanisms are correct. The relevance and implications of this study in the modeling of future denudation and weathering processes of radioactive waste-disposal sites (both deeply buried sites for high-leveI waste and surface sites for low-level waste) are discussed. Neoformed phases, such as jarosite, aluminous minerals, and silico-aluminous gels may play a significant role in the efficient trapping of mobilized pollutant radionuclides.

Selected Brazilian Oxisols were sampled and submitted to high-gradient magnetic separation (HGMS) to study the iron oxide-clay mineral associations. The soils, derived from four different parent materials, have mineralogy dominated mostly by hematite, goethite, and kaolinite. Gibbsite appears in most soil samples. The high-gradient magnetic separation showed good separation for some soils, as indicated by color differentiation and iron oxide segregation between magnetic and nonmagnetic fractions. Soils that showed a somewhat low surface area for the iron oxides associated with high phosphate adsorption were poorly separated by HGMS. This suggests a strong interaction between kaolinite and iron oxides, which would indicate a low estimation of their surface areas obtained by the difference method using BET-N2 data. A relative concentration of anatase and rutile in the magnetic portion of some of the samples was attributed to the presence of Fe, either as coatings on the crystals or within the structure of these two minerals.

The importance of precipitation rate as an effective control on illite and kaolanite formation during diagenesis has been examined by measuring precipitation rates, from Al fluid concentration, in a Dickson fluid-sampling vessel at 160°-250° and 500 bars (50 MPa). These experiments are considered to be analogues of the precipitation of clays in sandstones from porewaters containing dissolved carboxylic acids, which have a transient stability and may influence aluminosilicate solubility. Precipitated illite had a lath-shaped morphology and its composition was consistent with authigenic illite in sandstones. Kaolinite formed tabular rather than vermicular shaped crystals. Kaolinite precipitation rate was two orders of magnitude faster than illite precipitation and was rate-limited by the decomposition of oxalate; kaolinite formation should be equilibrium-controlled at virtually all stages of burial. Extrapolation of illite precipitation rate to burial temperatures indicates that the first appearance of illite in a burial sequence may be kinetically controlled. A model of illite precipitation based on these experimental results has been used to predict the time required to precipitate illite during burial of a sandstone, taking into account temperature changes during burial. For northern North Sea examples, a predicted illitization threshold of -60°C occurring at 60–80 Ma corresponds to the observed initiation of authigenic illite precipitation. Times of around 2–5 Ma would be required to reach a 98% approach to equilibrium at this threshold. The main phase of illite precipitation in the northern North Sea basin is a later, hydrologically controlled event (30-50 Ma). Equilibrium would be approached in around 0.1 Ma during this phase, which is consistent with the narrow illite K/Ar age range (1-5 Ma) recorded for some sequences.

The fine silt (2–5 μm) coarse clay (0.1–2 μm) and fine clay (<0.1 μm) fractions of a Vertisol from South Italy were studied with X-ray diffraction. The most reactive fine clay (<0.1 μm) fraction was investigated in detail using a curve decomposition method analysis of X-ray diffraction diagrams, FTIR spectroscopy and chemical analysis. In the soil parent material, the fine clay fraction was dominated by low-charge smectites (beidellite and montmorillonite) whereas, high-charge beidellite was the dominant clay mineral in the fine clay from the upper soil horizons. This suggested that high-charge beidellite was produced through alteration of the preexisting low-charge smectites and was the stable clay phase in this soil environment, characterized by high pH (>8.0). After K-saturation and 25 wetting and drying cycles, the high-charge beidellite from the soil horizons lost expandability far more than the original low-charge smectites.

The dissolution of two Ca2+-exchanged nontronite samples has been studied in 10% HCl. Early acid-dissolution studies (Osthaus, 1954) have indicated that after two hours of dissolution most of the octahedral Fe3+ (VIFe3+) would be removed leaving mainly tetrahedral Fe3+ (IVFe3+) in the nontronite structure. In the present study, 57Fe Mössbauer spectra of acid-treated samples were recorded and fitted with two octahedral Fe3+ (2 × VIFe3+) and two octahedral and one tetrahedral (2 × VIFe3+, 1 × IVFe3+) doublet models. The Mössbauer spectra of acid-treated Garfield nontronite samples could be adequately fitted with two-doublet models but acid-treated Hohen Hagen nontronite samples could not. Isomer shift and quadrupole splitting values obtained from the two-doublet models corresponded to VIFe3+ and not IVFe3+, as was suggested by the Osthaus (1954) experiment. When an IVFe3+ doublet was included in the model used to fit the Mössbauer spectra of acid-treated Garfield nontronite samples, a slight increase in the intensity of the IVFe3+ doublet occurred with increasing dissolution, but this was much lower than indicated by Osthaus (1954). No trend in the intensity of the IVFe3+ doublet was observed for acid-treated Hohen Hagen nontronite. Therefore, acid treatment appears to remove VIFe3+ and IVFe3+ from the nontronite structure at about the same rate. Mössbauer spectroscopy, infrared spectroscopy and X-ray powder diffraction data indicate that the nontronite that remains undissolved following acid treatment is structurally similar to the untreated nontronite.

Monte Carlo (MC) simulations of interlayer molecular structure in monolayer hydrates of Na-saturated Wyoming-type montmorillonites and vermiculite were performed. Detailed comparison of the stimulation results with experimental diffraction and thermodynamic data for these clay-water systems indicated good semiquantitative to quantitative agreement. The MC simulations revealed that, in the monolayer hydrate, interlayer water molecules tend to increase their occupation of the midplane as layer charge increases. As the percentage of tetrahedral layer charge increases, water molecules are induced to interact with the siloxane surface O atoms through hydrogen bonding and Na+ counter-ions are induced to form inner-sphere surface complexes. These results suggest the need for careful diffraction experiments on a series of monolayer hydrates of montmorillonite whose layer charge and tetrahedral isomorphic substitution charge vary systematically.

The clay mineral textures, assemblages, formation mechanisms, and controlling geological parameters relating to alteration of silicic volcanic rocks by hydrothermal solutions, in core samples from the Broadlands-Ohaaki hydrothermal system, New Zealand, were investigated using X-ray diffraction (XRD), scanning electron microscopy (SEM), and transmission and analytical electron microscopy (TEM/AEM). Mineralogical and textural relations of this active hydrothermal system, for which temperatures and fluid relations are well known, are equivalent to those in the Golden Cross hydrothermal gold deposit as described in Part 1.

XRD data show a sequence of clay minerals from smectite to a range of interstratified I-S to mica with increasing depth and temperature, on average. TEM observations are in general agreement with XRD data, especially with respect to relative proportions of illite (I)- and smectite (S)- like layers. TEM data also show that: (1) Smectite packets contain no discrete illite-like layers in samples identified as (Reichweite, R = 0) I-S by XRD. They coexist with separate packets of (R = 1) I-S. (2) A continuous range in I-S occurs from (R = 1) I-S with increasing proportion of illite-like layers, but at high illite-like layer contents there is a gap between I-S and illite. (3) 1M and 2M1 polytypes of mica coexist in separate packets, but the rare 1M polytype has a larger VIMg content.

The data imply that clay minerals formed by dissolution and neocrystallization directly from volcanic phases, although multiple reaction events can not be ruled out. Such “episodic” alteration produces a sequence of clay minerals identical to those of prograde diagenesis of pelitic sediments. This result implies that the presence of a continuous sequence is not definitive proof of continuous sequences of transformation as a function of time and continuous burial. Reaction progress of the clay-mineral sequence is in general accord with the known temperature gradient, but with significant and common exceptions. High porosity and permeability, both inherent in rock texture and local structure, are inferred to foster local reaction progress, as consistent with metastability of phases and the Ostwald step rule.

This paper describes structural models of four pyrophyllite edge faces: {010}, {110}, {100}, and {130}. Water molecules chemisorbed to Lewis acid sites stabilize edge faces both crystallochemically and electrostatically. The detailed assignment of protons to surface oxygens and the orientation of OH bond-vectors both influence the surface Coulomb energy.

The geometry chosen for the electrostatic calculations was infinite pyrophyllite ribbon the thickness of a single phyllosilicate layer and the width of 50 to 70 unit cells. Such a phyllosilicate ribbon has only two edges, a top and bottom, which were simulated using the edge-face models mentioned above. About 94% of the surface Coulomb energy is confined to the edge-face repeat unit. The surface Coulomb energies of the four edge faces are on the order of 2–3 nJ/m, varying ± 1 nJ/m with proton assignment. The Coulomb potential, measured either within the layer or parallel to the layer, has a distinct negative trend near the edge face that can be traced to chemisorbed water molecules. Finally, the correlation between proton Coulomb potentials at the edge face and the coordination environment of the protons is poor, obscured by long-range interactions.

There is a lack of comparative and quantitative research on how poverty manifests itself in the economic wellbeing of older people across European countries. In this study, we focus in on two central dimensions of economic wellbeing: the ability to pay for usual expenses and unexpected expenses. Our aim is to find out how often older people living at risk of poverty experience hardship on these dimensions, how these dimensions overlap, and whether the incidence of hardship differs between the poor and the non-poor. The study is based on the cross-sectional component of the European Union Statistics on Income and Living Conditions (EU-SILC) 2018 survey, involving 29 countries and 148,432 respondents aged 65+ years. The analysis builds on both descriptive statistics and multinomial logistic regression, which takes into account differences in household characteristics between the poor and the non-poor. The results reveal that for the poor, meeting unexpected expenses is a more common problem than meeting usual expenses, although they typically experience hardship on both dimensions. Hardship among the poor is more frequent in Central Eastern and some Southern European countries, while poor people living in Continental and Nordic countries tend to fare better, even though relatively large numbers in these countries lack cash margin. The non-poor do also experience hardship, but to a lesser extent. The poor experience combined hardship relatively often in Continental European countries. Based on the results, we conclude that studies should pay closer attention to the different dimensions of economic wellbeing in old age.

The use of Derjaguin-Landau-Verwey-Overbeek (DLVO) theory in colloid science has continued to the present day in spite of the inability of this theory to explain, even qualitatively, numerous phenomena exhibited by clays and other colloidal materials. An alternative description of the fundamental forces involved in the formation of dispersions and gels was presented in 1938 by Langmuir, but was never widely accepted among colloid scientists. Recent experimental and theoretical work, particularly in Japan, has revived this alternative view, in which a long-range Coulombic attraction force appears to explain several phenomena, particularly transitions among ordered and disordered phases of colloidal particles in dilute salt solutions.

Examples are given from surface chemistry where rather complex models based on diffuse double layer theory are used to explain chemical adsorption behavior that often has a simpler explanation. It is argued that the rule of parsimony (Ockham's Razor) should be applied to complex models that appear to explain the data at hand before such models are taken to be generally valid. A satisfactory fit of model predictions to experimental data obtained under a very limited range of conditions does not prove the validity of the model. Thus, it is concluded that the DLVO theory and its modifications fail to meet the 2 criteria necessary to the acceptance of a theory: agreement with observations and simplicity.

Progressively buried sandstones and shales of the Mahakam Delta Basin in Indonesia were studied. Mineralogical, morphological and K-Ar isotopic data were obtained for clay, mica and feldspar minerals. The data indicate that K necessary for the illitization of illite/smectite mixed-layer minerals was supplied mainly from K-feldspar alteration within the sandstones and from mica within the shales. Most of the K-feldspar alteration for both the shale and sandstone samples were observed outside the main zone of illitization, which is restricted to the upper 2000 m of sediment. The feldspar grains were altered below this depth for both lithologies. Therefore, illitization requires an open sedimentary system. This is in contrast to the illitization model for deeply buried shales of the Gulf Coast. That system is commonly assumed to be a closed K system.