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$G_{\delta \sigma }$ GAMES AND INDUCTION ON REALS
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- J. P. AGUILERA, P. D. WELCH
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- Journal:
- The Journal of Symbolic Logic / Volume 86 / Issue 4 / December 2021
- Published online by Cambridge University Press:
- 13 September 2021, pp. 1676-1690
- Print publication:
- December 2021
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It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.
The crystal structures of the mixed-valence tellurium oxysalts tlapallite, (Ca,Pb)3CaCu6[Te4+3Te6+O12]2(Te4+O3)2(SO4)2·3H2O, and carlfriesite, CaTe4+2Te6+O8
- Owen P. Missen, Anthony R. Kampf, Stuart J. Mills, Robert M. Housley, John Spratt, Mark D. Welch, Mark F. Coolbaugh, Joe Marty, Marek Chorazewicz, Cristiano Ferraris
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- Journal:
- Mineralogical Magazine / Volume 83 / Issue 4 / August 2019
- Published online by Cambridge University Press:
- 12 February 2019, pp. 539-549
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The crystal structure of tlapallite has been determined using single-crystal X-ray diffraction and supported by electron probe micro-analysis, powder diffraction and Raman spectroscopy. Tlapallite is trigonal, space group P321, with a = 9.1219(17) Å, c = 11.9320(9) Å and V = 859.8(3) Å3, and was refined to R1 = 0.0296 for 786 reflections with I > 2σ(I). This study resulted from the discovery of well-crystallised tlapallite at the Wildcat prospect, Utah, USA. The chemical formula of tlapallite has been revised to (Ca,Pb)3CaCu6[Te4+3Te6+O12]2(Te4+O3)2(SO4)2·3H2O, or more simply (Ca,Pb)3CaCu6Te4+8Te6+2O30(SO4)2·3H2O, from H6(Ca,Pb)2(Cu,Zn)3(TeO3)4(TeO6)(SO4). The tlapallite structure consists of layers containing distorted Cu2+O6 octahedra, Te6+O6 octahedra and Te4+O4 disphenoids (which together form the new mixed-valence phyllotellurate anion [Te4+3Te6+O12]12−), Te4+O3 trigonal pyramids and CaO8 polyhedra. SO4 tetrahedra, Ca(H2O)3O6 polyhedra and H2O groups fill the space between the layers. Tlapallite is only the second naturally occurring compound containing tellurium in both the 4+ and 6+ oxidation states with a known crystal structure, the other being carlfriesite, CaTe4+2Te6+O8. Carlfriesite is the predominant secondary tellurium mineral at the Wildcat prospect. We also present an updated structure for carlfriesite, which has been refined to R1 = 0.0230 for 874 reflections with I > 2σ(I). This updated structural refinement improves upon the one reported previously by refining all atoms anisotropically and presenting models of bond valence and Te4+ secondary bonding.
Phase transitions in hydroxide perovskites: a Raman spectroscopic study of stottite, FeGe(OH)6, to 21 GPa
- A. K. Kleppe, M. D. Welch, W. A. Crichton, A. P. Jephcoat
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- Journal:
- Mineralogical Magazine / Volume 76 / Issue 4 / August 2012
- Published online by Cambridge University Press:
- 05 July 2018, pp. 949-962
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The effect of pressure on the naturally occurring hydroxide-perovskite stottite, FeGe(OH)6, has been studied in situ by micro-Raman spectroscopy to 21 GPa at 300 K. The ambient spectrum contains six OH-stretching bands in the range 3064 3352 cm–1. The presence of six non-equivalent OH groups is inconsistent with space group P42/n. In view of this inconsistency a new ambient structure determination of stottite from Tsumeb was carried out, but this did not allow the clear rejection of P42/n symmetry. However, a successful refinement was also carried out in space group P2/n, a subgroup of P42/n, which allows for six non-equivalent O atoms. The two refinements are of comparable quality and do not allow a choice to be made based purely on the X-ray data. However, taken with the ambient and 150 K Raman spectra, a good case can be made for stottite having P2/n symmetry at ambient conditions. On this basis, the pressure induced spectroscopic changes are interpreted in terms of a reversible phase transition P2/n ↔ P42/n.
The crystal structure of seeligerite, Pb3IO4Cl3, a rare Pb-I-oxychloride from the San Rafael mine, Sierra Gorda, Chile
- L. Bindi, M. D. Welch, P. Bonazzi, G. Pratesi, S. Menchetti
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- Journal:
- Mineralogical Magazine / Volume 72 / Issue 3 / June 2008
- Published online by Cambridge University Press:
- 05 July 2018, pp. 771-783
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The crystal structure of seeligerite, Pb3IO4Cl3, from the San Rafael mine, Sierra Gorda, Chile, was solved in the space group Cmm2, and refined to R = 3.07%. The unit-cell parameters are: a = 7.971(2), b = 7.976(2), c = 27.341(5) Å, V = 1738.3(6) Å3 and Z = 8. The crystal structure consists of a stacking sequence along [001] of square-net layers of O atoms and square-net layers of Cl atoms with Pb+ and I+ cations located in the voids of the packing. As is typical of cations with a stereoactive lone-pair of electrons, Pb2+ and I5+ adopt strongly-asymmetrical configurations. Pb2+ cations occur in a variety of coordination polyhedra, ranging from anticubes and monocapped anticubes to pyramidal ‘one-sided’ coordinations. I5+ is coordinated by a square of four oxygen atoms: I1 and I3 exhibit a ‘one-sided’ coordination, whereas I2 has square-planar coordination.
The TEM investigation has revealed additional superlattice reflections (which were not registered by X-ray diffraction (XRD)) in the hk0 diffraction pattern of seeligerite based upon a 0.158 Å-1 square net, which can be interpreted as arising from a 20-cation super-sheet motif (12.6 Å x 12.6 Å), likely related to a further level of Pb-I order superimposed upon the 8-site motif identified by XRD.
Neutron diffraction at simultaneous high temperatures and pressures, with measurement of temperature by neutron radiography
- Y. Le Godec, M. T. Dove, D. J. Francis, S. C. Kohn, W. G. Marshall, A. R. Pawley, G. D. Price, S. A. T. Redfern, N. Rhodes, N. L. Ross, P. F. Schofield, E. Schooneveld, G. Syfosse, M. G. Tucker, M. D. Welch
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- Journal:
- Mineralogical Magazine / Volume 65 / Issue 6 / December 2001
- Published online by Cambridge University Press:
- 05 July 2018, pp. 737-748
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The commissioning and operation of apparatus for neutron diffraction at simultaneous high temperatures and pressures is reported. The basic design is based on the Paris-Edinburgh cell using opposed anvils, with internal heating. Temperature is measured using neutron radiography. The apparatus has been shown in both on-line and off-line tests to operate to a pressure of 7 GPa and temperature of 1700°C. The apparatus has been used in a neutron diffraction study of the crystal structure of deuterated brucite, and results for 520°C and 5.15 GPa are presented. The diffraction data that can be obtained from the apparatus are of comparable quality to previous high-pressure studies at ambient temperatures, and are clearly good enough for Rietveld refinement analysis to give structural data of reasonable quality.
Paratacamite-(Mg), Cu3(Mg,Cu)Cl2(OH)6; a new substituted basic copper chloride mineral from Camerones, Chile
- A. R. Kampf, M. J. Sciberras, P. Leverett, P. A. Williams, T. Malcherek, J. Schlüter, M. D. Welch, M. Dini, A. A. Molina Donoso
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- Journal:
- Mineralogical Magazine / Volume 77 / Issue 8 / December 2013
- Published online by Cambridge University Press:
- 05 July 2018, pp. 3113-3124
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Paratacamite-(Mg) (IMA 2013-014), Cu3(Mg, Cu)Cl2(OH)6, is the new Mg-analogue of paratacamite. It was found near the village of Cuya in the Camarones Valley, Arica Province, Chile. The mineral is a supergene secondary phase occurring in association with anhydrite, atacamite, chalcopyrite, copiapite, dolomite, epsomite, haydeeite, hematite, magnesite and quartz. Paratacamite-(Mg) crystals are rhombs and thick to thin prisms up to 0.3 mm in size exhibiting the forms {201} and {001}. Twinning by reflection on {10} is common. The mineral is transparent with a vitreous lustre, with medium to deep-green colour and light-green streak. Mohs hardness is 3–3½, the tenacity is brittle and the fracture is conchoidal. Paratacamite-(Mg) has one perfect cleavage on {201}. The measured and calculated densities are 3.50(2) and 3.551 g cm–3, respectively. The mineral is optically uniaxial (–) with ε = 1.785(5) and ω > 1.8 and slight pleochroism: O (bluish green) > E (green). Electron-microprobe analyses provided the empirical formula Cu3(Mg0.60Cu0.38Ni0.01Mn0.01)Cl2(OH)6. The mineral is easily soluble in dilute HCl. Paratacamite-(Mg) is trigonal, R, with cell parameters a = 13.689(1), c = 14.025(1) Å, V = 2275.8(3) Å3 and Z = 12. There is a pronounced sub-cell corresponding to a' ≈ ½a, c' ≈ c in space group Rm. The eight strongest lines in the X-ray powder diffraction pattern are [dobs Å(I)(hkl)]: 5.469(87)(021), 4.686(26)(003), 2.904(34)(401), 2.762(100)(22,042), 2.265(81)(404), 1.819(26)(603), 1.710 (34)(440) and 1.380(19)(446). The structure was refined to R1 = 0.039 for 480 Fo > 4σF reflections. Refinement using interlayer Mg-Cu site scattering factors indicated that Mg is distributed statistically between both interlayer octahedra M1O6 and M2O6. A comparison of the distortions associated with M1O6 and M2O6 octahedra suggest that the sample is near the upper compositional limit for stability of the R phase.
The crystal-structure determination and redefinition of eztlite, Pb22+ Fe33+(Te4+O3)3(SO4)O2Cl
- Owen P. Missen, Stuart J. Mills, John Spratt, Mark D. Welch, William D. Birch, Michael S. Rumsey, Jan Vylita
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- Journal:
- Mineralogical Magazine / Volume 82 / Issue 6 / December 2018
- Published online by Cambridge University Press:
- 15 May 2018, pp. 1355-1367
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The crystal structure of eztlite has been determined using single-crystal synchrotron X-ray diffraction and supported using electron microprobe analysis and powder diffraction. Eztlite, a secondary tellurium mineral from the Moctezuma mine, Mexico, is monoclinic, space group Cm, with a = 11.466(2) Å, b = 19.775(4) Å, c = 10.497(2) Å, β = 102.62(3)° and V = 2322.6(9) Å3. The chemical formula of eztlite has been revised to ${\rm Pb}_{\rm 2}^{2 +} {\rm Fe}_3^{3 +} $(Te4+O3)3(SO4)O2Cl from that stated previously as ${\rm Fe}_6^{3 +} {\rm Pb}_{\rm 2}^{2 +} $(Te4+O3)3(Te6+O6)(OH)10·nH2O. This change has been accepted by the Commission on New Minerals, Nomenclature and Classification of the International Mineralogical Association, Proposal 18-A. Eztlite was reported originally to be a mixed-valence Te oxysalt; however the crystal structure, bond-valence analysis and charge balance considerations clearly show that all Te is tetravalent. Eztlite contains a unique combination of elements and is only the second Te oxysalt to contain both sulfate and chloride. The crystal structure of eztlite contains mitridatite-like layers, with a repeating triangular nonameric [${\rm Fe}_9^{3 +} $O36]45– arrangement formed by nine edge-sharing Fe3+O6 octahedra, decorated by four trigonal pyramidal Te4+O3 groups, compared to PO4 or AsO4 tetrahedra in mitridatite-type minerals. In eztlite, all four tellurite groups associated with one nonamer are orientated with the lone pair of the Te atoms pointing in the same direction, whereas in mitridatite the central tetrahedron is orientated in the opposite direction to the others. In mitridatite-type structures, interlayer connections are formed exclusively via Ca2+ and water molecules, whereas the eztlite interlayer contains Pb2+, sulfate tetrahedra and Cl–. Interlayer connectivity in eztlite is achieved primarily by connections via the long bonds of Pbφ8 and Pbφ9 groups to sulfate tetrahedra and to Cl–. Secondary connectivity is via Te–O and Te–Cl bonds.
Millsite, CuTeO3·2H2O: a new polymorph of teineite from Gråurdfjellet, Oppdal, Norway
- Michael S. Rumsey, Mark D. Welch, Frode Mo, Annette K. Kleppe, John Spratt, Anthony R. Kampf, Morten P. Raanes
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- Journal:
- Mineralogical Magazine / Volume 82 / Issue 2 / April 2018
- Published online by Cambridge University Press:
- 28 February 2018, pp. 433-444
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Millsite, CuTeO3·2H2O, is a new mineral from Gråurdfjellet in Oppdal, Norway. It occurs as a minor secondary phase alongside teineite, other copper secondaries and relict primary tellurides in a boulder of quartz-rich granite, which is probably a glacial erratic. Millsite is bright cyan to royal blue in colour. The mineral is transparent to slightly translucent with a vitreous lustre and has a perfect (100) cleavage. It is brittle, has a conchoidal fracture and a pale green streak. Millsite is optically biaxial (+), α = 1.756(5), β = 1.794(5), γ = 1.925calc and 2Vmeas = 60(1)°. Millsite has monoclinic space group P21/c, with a = 7.4049(2) Å, b = 7.7873(2) Å, c = 8.5217(2) Å, β = 110.203(3)°, V = 461.17(2) Å3 and Z = 4. The empirical formula is Cu0.99(Te0.98Se0.02)O3(H2O)2. The five strongest reflections in the powder X-ray diffraction pattern are [dhkl in Å (hkl, Irel%)]: 6.954 (100, 100), 3.558 (012, 64), 2.838 (12$\bar 2$, 47), 2.675 (211, 43) and 3.175 (210, 39). The crystal structure has been determined to R1 = 0.016, wR2 = 0.036 and GooF = 1.049. The diagnostic structural unit of millsite consists of a Cu2O6(H2O)4 dimer that is decorated with four TeO3 groups connecting adjacent dimers and defining (100) heteropolyhedral sheets. These heteropolyhedral sheets are only connected by layers of structurally significant hydrogen bonds and correlate with the (100) cleavage. Millsite is a polymorph of teineite with a unique configuration of the M2O6(H2O)4 dimer that leads to a sheet topology. No isostructural selenium or tellurium analogue exists. The monoclinic polymorph (P21/c) of chalcomenite ‘monoclinic-CuSeO3·2H2O’ hereafter, ahlfeldite and MgSeO3·2H2O have M2O6(H2O)4 dimers, but their configuration differs significantly from that of millsite and leads to a framework topology rather than a sheet. Teineite does not have a dimeric structure and so is fundamentally different from millsite. The sheet topology of millsite appears to be unique among tellurites.
Visuospatial Functioning in the Primary Progressive Aphasias
- Christa L. Watson, Katherine Possin, I. Elaine Allen, H. Isabel Hubbard, Marita Meyer, Ariane E. Welch, Gil D. Rabinovici, Howard Rosen, Katherine P. Rankin, Zachary Miller, Miguel A. Santos-Santos, Joel H. Kramer, Bruce L. Miller, Maria Luisa Gorno-Tempini
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- Journal:
- Journal of the International Neuropsychological Society / Volume 24 / Issue 3 / March 2018
- Published online by Cambridge University Press:
- 17 October 2017, pp. 259-268
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Objectives: The aim of this study was to identify whether the three main primary progressive aphasia (PPA) variants would show differential profiles on measures of visuospatial cognition. We hypothesized that the logopenic variant would have the most difficulty across tasks requiring visuospatial and visual memory abilities. Methods: PPA patients (n=156), diagnosed using current criteria, and controls were tested on a battery of tests tapping different aspects of visuospatial cognition. We compared the groups on an overall visuospatial factor; construction, immediate recall, delayed recall, and executive functioning composites; and on individual tests. Cross-sectional and longitudinal comparisons were made, adjusted for disease severity, age, and education. Results: The logopenic variant had significantly lower scores on the visuospatial factor and the most impaired scores on all composites. The nonfluent variant had significant difficulty on all visuospatial composites except the delayed recall, which differentiated them from the logopenic variant. In contrast, the semantic variants performed poorly only on delayed recall of visual information. The logopenic and nonfluent variants showed decline in figure copying performance over time, whereas in the semantic variant, this skill was remarkably preserved. Conclusions: This extensive examination of performance on visuospatial tasks in the PPA variants solidifies some previous findings, for example, delayed recall of visual stimuli adds value in differential diagnosis between logopenic variant PPA and nonfluent variant PPA variants, and illuminates the possibility of common mechanisms that underlie both linguistic and non-linguistic deficits in the variants. Furthermore, this is the first study that has investigated visuospatial functioning over time in the PPA variants. (JINS, 2018, 24, 259–268)
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- By Francesco Acerbi, Ayca Akgoz, Matthew R. Amans, Ramsey Ashour, Mohammed Ali Aziz-Sultan, H. Hunt Batjer, Donnie Bell, Bernard R. Bendok, Giovanni Broggi, Morgan Broggi, Charles A. Bruno, Steven D. Chang, In Sup Choi, Omar Choudhri, Douglas J. Cook, William P. Dillon, Peter Dirks, Rose Du, Travis M. Dumont, Tarek Y. El Ahmadieh, Najib E. El Tecle, Mohamed Samy Elhammady, Paolo Ferroli, Alana M. Flexman, John C. Flickinger, Kai U. Frerichs, Sasikhan Geibprasert, Adrian W. Gelb, Y. Pierre Gobin, Bradley A. Gross, Seunggu J. Han, Tomoki Hashimoto, Juha Hernesniemi, Roberto C. Heros, Steven W. Hetts, Randall T. Higashida, Joshua A. Hirsch, Nikolai J. Hopf, L. Nelson Hopkins, Maziyar A. Kalani, M. Yashar S. Kalani, Hideyuki Kano, Syed Aftab Karim, Robert M. Koffie, Douglas S. Kondziolka, Timo Krings, Aki Laakso, Giuseppe Lanzino, Michael T. Lawton, Elad I. Levy, L. Dade Lunsford, Adel M. Malek, Michael P. Marks, George A. C. Mendes, Philip M. Meyers, Jacques Morcos, Nitin Mukerji, Christian Musahl, Ludmila Pawlikowska, Matthew B. Potts, Ross Puffer, James D. Rabinov, Jonathan J. Russin, Mina G. Safain, Duke Samson, Marco Schiariti, R. Michael Scott, Jason P. Sheehan, Paul Singh, Edward R. Smith, Scott G. Soltys, Robert F. Spetzler, Gary K. Steinberg, Philip E. Stieg, Hua Su, Karel terBrugge, Kiron Thomas, Tarik Tihan, Babu Welch, Jonathan White, H. Richard Winn, Chun-Po Yen, Jacky T. Yeung, Byron Yip, Samer G. Zammar
- Edited by Robert F. Spetzler, Douglas S. Kondziolka, Randall T. Higashida, University of California, San Francisco, M. Yashar S. Kalani
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- Comprehensive Management of Arteriovenous Malformations of the Brain and Spine
- Published online:
- 05 January 2015
- Print publication:
- 08 January 2015, pp x-xiv
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- By Venkataraman Anantharaman, Philip D. Anderson, Christopher W. Baugh, J. Stephen Bohan, Kirsten Boyd, Matthias Brachmann, Peter R. Brown, Shelley Calder, David Callaway, Peter Cameron, Jody Crane, Meaghan Cussen, Christina Dempsey, Jonathan A. Edlow, Thomas Fleischmann, Robert L. Freitas, John D. Halamka, Manuel Hernandez, Cherri Hobgood, Jock Hoffman, Steven Horng, Kirk B. Jensen, Jennifer R. Johnson, Stephanie Kayden, Tasnim Khan, Daniel G. Kirkpatrick, James Lennon, Mary Leupold, Thom Mayer, J. Lawrence Mottley, Scott B. Murray, Deirdre Mylod, Larry A. Nathanson, Michael P. Pietrzak, Elke Platz, Nadeem Qureshi, Matthew M. Rice, Andrew Schenkel, Chet Schrader, Puneet Seth, Richard B. Siegrist, David Smith, Robert E. Suter, Carrie Tibbles, Sebastian N. Walker, Lee A. Wallis, Julie Welch, Leana S. Wen
- Edited by Stephanie Kayden, Philip D. Anderson, Robert Freitas, Elke Platz
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- Book:
- Emergency Department Leadership and Management
- Published online:
- 05 December 2014
- Print publication:
- 27 November 2014, pp ix-xii
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SOME OBSERVATIONS ON TRUTH HIERARCHIES
- P. D. WELCH
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- Journal:
- The Review of Symbolic Logic / Volume 7 / Issue 1 / March 2014
- Published online by Cambridge University Press:
- 02 January 2014, pp. 1-30
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- March 2014
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We show how in the hierarchies ${F_\alpha }$ of Fieldian truth sets, and Herzberger’s ${H_\alpha }$ revision sequence starting from any hypothesis for ${F_0}$ (or ${H_0}$) that essentially each ${H_\alpha }$ (or ${F_\alpha }$) carries within it a history of the whole prior revision process.
As applications (1) we provide a precise representation for, and a calculation of the length of, possible path independent determinateness hierarchies of Field’s (2003) construction with a binary conditional operator. (2) We demonstrate the existence of generalized liar sentences, that can be considered as diagonalizing past the determinateness hierarchies definable in Field’s recent models. The ‘defectiveness’ of such diagonal sentences necessarily cannot be classified by any of the determinateness predicates of the model. They are ‘ineffable liars’. We may consider them a response to the claim of Field (2003) that ‘the conditional can be used to show that the theory is not subject to “revenge problems”.’
TRUTH, LOGICAL VALIDITY AND DETERMINATENESS: A COMMENTARY ON FIELD’S SAVING TRUTH FROM PARADOX
- P. D. WELCH
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- Journal:
- The Review of Symbolic Logic / Volume 4 / Issue 3 / September 2011
- Published online by Cambridge University Press:
- 10 October 2011, pp. 348-359
- Print publication:
- September 2011
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We consider notions of truth and logical validity defined in various recent constructions of Hartry Field. We try to explicate his notion of determinate truth by clarifying the path-dependent hierarchies of his determinateness operator.
Determinacy in strong cardinal models
- P. D. Welch
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- Journal:
- The Journal of Symbolic Logic / Volume 76 / Issue 2 / June 2011
- Published online by Cambridge University Press:
- 12 March 2014, pp. 719-728
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- June 2011
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We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:
Theorem A. Det(-IND) ⇒ there exists an inner model with a strong cardinal.
Theorem B. Det(AQI) ⇒ there exist type-l mice and hence inner models with proper classes of strong cardinals.
where -IND(AQI) is the pointclass of boldface -inductive (respectively arithmetically quasi-inductive) sets of reals.
Ramsey-like cardinals II
- Victoria Gitman, P. D. Welch
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- Journal:
- The Journal of Symbolic Logic / Volume 76 / Issue 2 / June 2011
- Published online by Cambridge University Press:
- 12 March 2014, pp. 541-560
- Print publication:
- June 2011
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This paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.
We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.
Hypermachines
- Sy-David Friedman, P. D. Welch
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- Journal:
- The Journal of Symbolic Logic / Volume 76 / Issue 2 / June 2011
- Published online by Cambridge University Press:
- 12 March 2014, pp. 620-636
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- June 2011
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The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a “Σ2-machine” in that it uses a Σ2Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σn rule. Such machines either halt or enter an infinite loop by stage , again generalising precisely the ITTM case.
The collection of such machines taken together computes precisely those reals of the least model of analysis.
Weak systems of determinacy and arithmetical quasi-inductive definitions
- P. D. Welch
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- Journal:
- The Journal of Symbolic Logic / Volume 76 / Issue 2 / June 2011
- Published online by Cambridge University Press:
- 12 March 2014, pp. 418-436
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- June 2011
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We locate winning strategies for various -games in the L-hierarchy in order to prove the following:
Theorem 1. KP + Σ2-Comprehension -Determinacy.”
Alternatively: “there is a β-model of -Determinacy.” The implication is not reversible. (The antecedent here may be replaced with instances of Comprehension with only -lightface definable parameters—or even weaker theories.)
Theorem 2. KP + Δ2-Comprehension + Σ2-Replacement + -Determinacy.
(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: -Determinacy.
Hence the theories: , and are in strictly descending order of strength.
Contributors
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- By Basem Abdelmalak, Joseph Abdelmalak, Alaa A. Abd-Elsayed, David L. Adams, Eric E. Adelman, Maged Argalious, Endrit Bala, Gene H. Barnett, Sheron Beltran, Andrew Bielaczyc, William Bingaman, James M. Blum, Alina Bodas, Vera Borzova, Richard Bowers, Adam Brown, Chad M. Brummett, Alexandra S. Bullough, James F. Burke, Juan P. Cata, Neeraj Chaudhary, Michael J. Claybon, Miguel Cruz, Milind Deogaonkar, Vikram Dhawan, Thomas Didier, D. John Doyle, Zeyd Ebrahim, Hesham Elsharkawy, Wael Ali Sakr Esa, Ehab Farag, Ryen D. Fons, Joseph J. Gemmete, Matt Giles, Phil Gillen, Goodarz Golmirzaie, Marcos Gomes, Lisa Grilly, Maged Guirguis, David W. Healy, Heather Hervey-Jumper, Shawn L. Hervey-Jumper, Paul E. Hilliard, Samuel A. Irefin, George K. Istaphanous, Teresa L. Jacobs, Ellen Janke, Greta Jo, James W. Jones, Rami Karroum, Allen Keebler, Stephen J. Kimatian, Colleen G. Koch, Robert Scott Kriss, Andrea Kurz, Jia Lin, Michael D. Maile, Negmeldeen F. Mamoun, Mariel Manlapaz, Edward Manno, Donn Marciniak, Piyush Mathur, Nicholas F. Marko, Matthew Martin, George A. Mashour, Marco Maurtua, Scott T. McCardle, Julie McClelland, Uma Menon, Paul S. Moor, Laurel E. Moore, Ruairi Moulding, Dileep R. Nair, Todd Nelson, Julie Niezgoda, Edward Noguera, Jerome O’Hara, Aditya S. Pandey, Mauricio Perilla, Paul Picton, Marc J. Popovich, J. Javier Provencio, Venkatakrishna Rajajee, Mohit Rastogi, Stacy Ritzman, Lauryn R. Rochlen, Leif Saager, Vivek Sabharwal, Oren Sagher, Kenneth Saliba, Milad Sharifpour, Lesli E. Skolarus, Paul Smythe, Wolf H. Stapelfeldt, William R. Stetler, Peter Stiles, Vijay Tarnal, Khoi D. Than, B. Gregory Thompson, Alparslan Turan, Christopher R. Turner, Justin Upp, Sumeet Vadera, Jennifer Vance, Anthony C. Wang, Robert J. Weil, Marnie B. Welch, Karen K. Wilkins, Erin S. Williams, George N. Youssef, Asma Zakaria, Sherif S. Zaky, Andrew Zura
- Edited by George A. Mashour, Ehab Farag
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- Book:
- Case Studies in Neuroanesthesia and Neurocritical Care
- Published online:
- 03 May 2011
- Print publication:
- 03 February 2011, pp x-xvi
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- Book:
- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
- Print publication:
- 20 September 2010, pp xi-xliv
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Games for Truth
- P. D. Welch
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- Journal:
- Bulletin of Symbolic Logic / Volume 15 / Issue 4 / December 2009
- Published online by Cambridge University Press:
- 15 January 2014, pp. 410-427
- Print publication:
- December 2009
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We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of all sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games considered here are simple, those over the natural model of arithmetic being all within the arithmetical class of .