Kevin Madigan
- Published in print:
- 2007
- Published Online:
- May 2007
- ISBN:
- 9780195322743
- eISBN:
- 9780199785407
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195322743.003.0008
- Subject:
- Religion, History of Christianity
This concluding chapter begins with a discussion of the major focus of this study, which is what Cardinal Newman — thinking of the difference between the earliest expression of a dogma and its ...
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This concluding chapter begins with a discussion of the major focus of this study, which is what Cardinal Newman — thinking of the difference between the earliest expression of a dogma and its developed form — called a “prima facie dissimilitude” between the two. It is argued that whatever one might think of the great Cardinal Newman's theory of dogmatic development and whatever of the history of Christian thought it might be believed to elucidate, it simply cannot begin to do justice to the evidence considered in this study. Time and again, “prima facie” dissimilitudes have been seen between an ancient Christological doctrine and its medieval development. However, closer inspection and analysis have revealed, that in virtually all these cases, the “dissimilitude” was very far from superficial.Less
This concluding chapter begins with a discussion of the major focus of this study, which is what Cardinal Newman — thinking of the difference between the earliest expression of a dogma and its developed form — called a “prima facie dissimilitude” between the two. It is argued that whatever one might think of the great Cardinal Newman's theory of dogmatic development and whatever of the history of Christian thought it might be believed to elucidate, it simply cannot begin to do justice to the evidence considered in this study. Time and again, “prima facie” dissimilitudes have been seen between an ancient Christological doctrine and its medieval development. However, closer inspection and analysis have revealed, that in virtually all these cases, the “dissimilitude” was very far from superficial.
Michael Potter
- Published in print:
- 2004
- Published Online:
- September 2011
- ISBN:
- 9780199269730
- eISBN:
- 9780191699443
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269730.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an ...
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This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.Less
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.
Geoffrey Hellman
- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198240341
- eISBN:
- 9780191597664
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198240341.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to ...
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Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to abstract entities. Modal logic is combined with notions of part/whole (mereology) enabling a systematic interpretation of ordinary mathematical statements as asserting what would be the case in any (suitable) structure there (logically) might be, e.g. for number theory, functional analysis, algebra, pure geometry, etc. Structures are understood as comprising objects, whatever their nature, standing in suitable relations as given by axioms or defining conditions in mathematics proper. The characterization of structures is aided by the addition of plural quantifiers, e.g. ‘Any objects of sort F’ corresponding to arbitrary collections of Fs, achieving the expressive power of second‐order logic, hence a full logic of relations. (See the author's ‘Structuralism without Structures’, Philosophia Mathematica 4 (1996): 100–123.) Claims of absolute existence of structures are replaced by claims of (logical) possibility of enough structurally interrelated objects (modal‐existence postulates). The vast bulk of ordinary mathematics, and scientific applications, can thus be recovered on the basis of the possibility of a countable infinity of atoms. As applied to set theory itself, these ideas lead to a ‘many worlds’—– as opposed to the standard ‘fixed universe’—view, inspired by Zermelo (1930), respecting the unrestricted, indefinite extendability of models of the Zermelo–Fraenkel axioms. Natural motivation for (‘small’) large cardinal axioms is thus provided. In sum, the vast bulk of abstract mathematics is respected as objective, while literal reference to abstracta and related problems with Platonism are eliminated.Less
Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to abstract entities. Modal logic is combined with notions of part/whole (mereology) enabling a systematic interpretation of ordinary mathematical statements as asserting what would be the case in any (suitable) structure there (logically) might be, e.g. for number theory, functional analysis, algebra, pure geometry, etc. Structures are understood as comprising objects, whatever their nature, standing in suitable relations as given by axioms or defining conditions in mathematics proper. The characterization of structures is aided by the addition of plural quantifiers, e.g. ‘Any objects of sort F’ corresponding to arbitrary collections of Fs, achieving the expressive power of second‐order logic, hence a full logic of relations. (See the author's ‘Structuralism without Structures’, Philosophia Mathematica 4 (1996): 100–123.) Claims of absolute existence of structures are replaced by claims of (logical) possibility of enough structurally interrelated objects (modal‐existence postulates). The vast bulk of ordinary mathematics, and scientific applications, can thus be recovered on the basis of the possibility of a countable infinity of atoms. As applied to set theory itself, these ideas lead to a ‘many worlds’—– as opposed to the standard ‘fixed universe’—view, inspired by Zermelo (1930), respecting the unrestricted, indefinite extendability of models of the Zermelo–Fraenkel axioms. Natural motivation for (‘small’) large cardinal axioms is thus provided. In sum, the vast bulk of abstract mathematics is respected as objective, while literal reference to abstracta and related problems with Platonism are eliminated.
Hagit Borer
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780199263905
- eISBN:
- 9780191718182
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263905.003.0007
- Subject:
- Linguistics, Semantics and Pragmatics
This chapter investigates the properties of the Modern Hebrew (henceforth Hebrew) singular in its different indefinite and definite instantiations. It is here that much of this variation is best ...
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This chapter investigates the properties of the Modern Hebrew (henceforth Hebrew) singular in its different indefinite and definite instantiations. It is here that much of this variation is best illustrated. The investigation of the singular will further serve as a springboard to discussing the properties of Hebrew cardinal determiners in general, the status of specifier cardinals vs. head cardinals, and the interaction, in Hebrew, between the assignment of range to d and to #.Less
This chapter investigates the properties of the Modern Hebrew (henceforth Hebrew) singular in its different indefinite and definite instantiations. It is here that much of this variation is best illustrated. The investigation of the singular will further serve as a springboard to discussing the properties of Hebrew cardinal determiners in general, the status of specifier cardinals vs. head cardinals, and the interaction, in Hebrew, between the assignment of range to d and to #.
Owen Chadwick
- Published in print:
- 1980
- Published Online:
- November 2003
- ISBN:
- 9780198269199
- eISBN:
- 9780191600487
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198269196.003.0004
- Subject:
- Religion, Church History
An account is given of the position of the office of the Pope in the Catholic Church in Europe in the eighteenth century, before the Enlightenment. The following aspects and popes are covered: ...
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An account is given of the position of the office of the Pope in the Catholic Church in Europe in the eighteenth century, before the Enlightenment. The following aspects and popes are covered: authority, election, the veto, Pope Clement XI Albani (1700–21), the bull Unigenitus of 1713 condemning the French Oratorian Pasquier Quesnel's Moral Reflexions on the New Testament, the debate on infallibility, the Chinese rites, Pope Innocent XIII Conti (1721–4) and Benedict XIII Orsini (1724–30), Pope Clement XII Corsini (1730–40), Pope Benedict XIV Lambertini (1740–58), the secretary of state, the cardinal‐nephew, the college of cardinals, nuncios, the Congregations (curial offices commonly known as the dicasteries), the Inquisition (the Congregation of the Holy Office) and Index, the Spanish Inquisition, the Congregation of the Council, the action of the Pope, and the jubilee (a holy year).Less
An account is given of the position of the office of the Pope in the Catholic Church in Europe in the eighteenth century, before the Enlightenment. The following aspects and popes are covered: authority, election, the veto, Pope Clement XI Albani (1700–21), the bull Unigenitus of 1713 condemning the French Oratorian Pasquier Quesnel's Moral Reflexions on the New Testament, the debate on infallibility, the Chinese rites, Pope Innocent XIII Conti (1721–4) and Benedict XIII Orsini (1724–30), Pope Clement XII Corsini (1730–40), Pope Benedict XIV Lambertini (1740–58), the secretary of state, the cardinal‐nephew, the college of cardinals, nuncios, the Congregations (curial offices commonly known as the dicasteries), the Inquisition (the Congregation of the Holy Office) and Index, the Spanish Inquisition, the Congregation of the Council, the action of the Pope, and the jubilee (a holy year).
Gerd‐Rainer Horn
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199204496
- eISBN:
- 9780191708145
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199204496.003.0006
- Subject:
- Religion, Theology
Perhaps the most influential of all attempts to missionise industrial Europe remained organizationally independent from Catholic Action. Assisted by the protective support of Cardinal Emmanuel ...
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Perhaps the most influential of all attempts to missionise industrial Europe remained organizationally independent from Catholic Action. Assisted by the protective support of Cardinal Emmanuel Suhard, a domestic mission for secularized France was set up, complete with its own seminary, run by Louis Augros, located in Lisieux. Team building and the construction of Christian communities, also termed base communities, gave the Mission de France its singular cohesion and dynamism. Spin‐off ventures included the founding of the Mission de Paris, as well as the ill‐fated ventures to create a Mission de Belgique. The internationally renowned phenomenon of the French and Belgian worker priests likewise grew out of this missionary enterprise. The taming of the worker priests and various efforts to sustain the Mission de France in the 1950s form the closing section of this chapter, which carries the story forward to the eve of Vatican II.Less
Perhaps the most influential of all attempts to missionise industrial Europe remained organizationally independent from Catholic Action. Assisted by the protective support of Cardinal Emmanuel Suhard, a domestic mission for secularized France was set up, complete with its own seminary, run by Louis Augros, located in Lisieux. Team building and the construction of Christian communities, also termed base communities, gave the Mission de France its singular cohesion and dynamism. Spin‐off ventures included the founding of the Mission de Paris, as well as the ill‐fated ventures to create a Mission de Belgique. The internationally renowned phenomenon of the French and Belgian worker priests likewise grew out of this missionary enterprise. The taming of the worker priests and various efforts to sustain the Mission de France in the 1950s form the closing section of this chapter, which carries the story forward to the eve of Vatican II.
IAIN FENLON
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780197265055
- eISBN:
- 9780191754166
- Item type:
- chapter
- Publisher:
- British Academy
- DOI:
- 10.5871/bacad/9780197265055.003.0015
- Subject:
- Music, History, Western
Although the desirability for Italian princes and those who emulated their social practices to construct rooms specifically dedicated to the performance of music was specified by Paolo Cortesi in his ...
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Although the desirability for Italian princes and those who emulated their social practices to construct rooms specifically dedicated to the performance of music was specified by Paolo Cortesi in his treatise De cardinalatu at the beginning of the sixteenth century, the identification of these dedicated spaces and in particular of the repertories performed within them is fraught with difficulty. Beginning with Isabella d'Este's various studioli, this chapter considers what can be reconstructed about the provision of such rooms within the ducal palace in Mantua. In so doing, it draws upon inventories, archival documents and the iconography of a number of decorative schemes, finishing with the work of the architect Giovan Battista Bertani, to suggest what can be determined about the shape and internal disposition of the rooms themselves.Less
Although the desirability for Italian princes and those who emulated their social practices to construct rooms specifically dedicated to the performance of music was specified by Paolo Cortesi in his treatise De cardinalatu at the beginning of the sixteenth century, the identification of these dedicated spaces and in particular of the repertories performed within them is fraught with difficulty. Beginning with Isabella d'Este's various studioli, this chapter considers what can be reconstructed about the provision of such rooms within the ducal palace in Mantua. In so doing, it draws upon inventories, archival documents and the iconography of a number of decorative schemes, finishing with the work of the architect Giovan Battista Bertani, to suggest what can be determined about the shape and internal disposition of the rooms themselves.
ARNALDO MORELLI
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780197265055
- eISBN:
- 9780191754166
- Item type:
- chapter
- Publisher:
- British Academy
- DOI:
- 10.5871/bacad/9780197265055.003.0019
- Subject:
- Music, History, Western
This chapter investigates the locations and modes of musical performance in the residences of the nobility in seventeenth-century Rome, indicating the differences between this period and the ...
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This chapter investigates the locations and modes of musical performance in the residences of the nobility in seventeenth-century Rome, indicating the differences between this period and the Renaissance. In particular, instances of music-making in the courts of princes and cardinals are identified and described, in relation to considerations of etiquette, social conventions and anthropology. This research, based on first-hand documentary research in the archives of Roman noble families, has revealed unexpected locations for music-making, which cannot always be justified in terms of acoustic or aesthetic criteria. Particular attention is paid to the places where instruments were stored, as recorded in inventories, and their typology.Less
This chapter investigates the locations and modes of musical performance in the residences of the nobility in seventeenth-century Rome, indicating the differences between this period and the Renaissance. In particular, instances of music-making in the courts of princes and cardinals are identified and described, in relation to considerations of etiquette, social conventions and anthropology. This research, based on first-hand documentary research in the archives of Roman noble families, has revealed unexpected locations for music-making, which cannot always be justified in terms of acoustic or aesthetic criteria. Particular attention is paid to the places where instruments were stored, as recorded in inventories, and their typology.
Edward A. Siecienski
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780195372045
- eISBN:
- 9780199777297
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195372045.003.0006
- Subject:
- Religion, History of Christianity
Although an uneasy peace was maintained in the years after Photius, during the tenth and eleventh centuries political, cultural, and religious factors rapidly drove East and West further part. The ...
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Although an uneasy peace was maintained in the years after Photius, during the tenth and eleventh centuries political, cultural, and religious factors rapidly drove East and West further part. The mutual excommunications of Cardinal Humbert and Patriarch Michael Cerularius in 1054, often called the beginning of the “Great Schism” between East and West, reignited the filioque debate, as its omission from/addition to the creed came to be seen as sign of the other’s heretical ways. While the Greek-speaking East continued to rely heavily on the claims put forward in the Mystagogia, Latin scholastic theologians like Anselm and Thomas Aquinas advanced an entirely new series of arguments in favor of the doctrine. Theological encounters between the two sides (with some notable exceptions) only exacerbated the tension, and following the Fourth Crusade there seemed little chance of healing the breach that had grown up between Christian East and West.Less
Although an uneasy peace was maintained in the years after Photius, during the tenth and eleventh centuries political, cultural, and religious factors rapidly drove East and West further part. The mutual excommunications of Cardinal Humbert and Patriarch Michael Cerularius in 1054, often called the beginning of the “Great Schism” between East and West, reignited the filioque debate, as its omission from/addition to the creed came to be seen as sign of the other’s heretical ways. While the Greek-speaking East continued to rely heavily on the claims put forward in the Mystagogia, Latin scholastic theologians like Anselm and Thomas Aquinas advanced an entirely new series of arguments in favor of the doctrine. Theological encounters between the two sides (with some notable exceptions) only exacerbated the tension, and following the Fourth Crusade there seemed little chance of healing the breach that had grown up between Christian East and West.
Karin Vélez
- Published in print:
- 2018
- Published Online:
- May 2019
- ISBN:
- 9780691174006
- eISBN:
- 9780691184494
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174006.003.0003
- Subject:
- Religion, History of Christianity
This chapter reexamines Loreto's historical “brand” by looking at some of the first authors of Loreto's origin story. It opens with the architects of Loreto's shrine seal, Cardinal Antonio Maria ...
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This chapter reexamines Loreto's historical “brand” by looking at some of the first authors of Loreto's origin story. It opens with the architects of Loreto's shrine seal, Cardinal Antonio Maria Gallo and Cardinal Rutilio Benzoni. Moving down the ranks and back in time, there are two earlier sixteenth-century chroniclers who laid the groundwork for the high clergy's selective mythohistory: a shrine governor, Pietro di Giorgio Tolomei, and a local secretary, Girolamo Angelitta. Like Virgin and house, and seal and medal, these men are first introduced in pairs and then grouped in larger company because, sometimes unbeknownst to each other, they worked in tandem to reconstruct Loreto's past. Their overlaps and contradictions illuminate the often inadvertently collaborative project of building official shrine history.Less
This chapter reexamines Loreto's historical “brand” by looking at some of the first authors of Loreto's origin story. It opens with the architects of Loreto's shrine seal, Cardinal Antonio Maria Gallo and Cardinal Rutilio Benzoni. Moving down the ranks and back in time, there are two earlier sixteenth-century chroniclers who laid the groundwork for the high clergy's selective mythohistory: a shrine governor, Pietro di Giorgio Tolomei, and a local secretary, Girolamo Angelitta. Like Virgin and house, and seal and medal, these men are first introduced in pairs and then grouped in larger company because, sometimes unbeknownst to each other, they worked in tandem to reconstruct Loreto's past. Their overlaps and contradictions illuminate the often inadvertently collaborative project of building official shrine history.
Gabriel Flynn
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199216451
- eISBN:
- 9780191712173
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199216451.003.0033
- Subject:
- Religion, Theology, Religion and Society
Cardinal Yves Congar (1904-95) and Bishop Basil Christopher Butler (1902-86) were both dedicated to the renewal of Catholic Learning and to the promotion of Christian unity. Butler, a convert to ...
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Cardinal Yves Congar (1904-95) and Bishop Basil Christopher Butler (1902-86) were both dedicated to the renewal of Catholic Learning and to the promotion of Christian unity. Butler, a convert to Catholicism, followed a more ‘conservative’ line than Congar. For his part, Congar, the leading figure of the Catholic ecumenical movement in France and a member of the Catholic–Lutheran Commission of Dialogue since 1965, was profoundly influenced by Lutheran theology in the formulation of his later ‘progressive’ stance on ecumenism. This chapter explores the themes of Receptive Ecumenism and Catholic Learning as they relate to the thought of Congar and Butler, respectively. It also seeks to extrapolate ethical implications for the present-day ecumenical movement. Specifically, it attempts to draw Congar and Butler into dialogue on the central doctrine of the incarnation, regarded by the former as ‘the key to the whole mystery of the Church’.Less
Cardinal Yves Congar (1904-95) and Bishop Basil Christopher Butler (1902-86) were both dedicated to the renewal of Catholic Learning and to the promotion of Christian unity. Butler, a convert to Catholicism, followed a more ‘conservative’ line than Congar. For his part, Congar, the leading figure of the Catholic ecumenical movement in France and a member of the Catholic–Lutheran Commission of Dialogue since 1965, was profoundly influenced by Lutheran theology in the formulation of his later ‘progressive’ stance on ecumenism. This chapter explores the themes of Receptive Ecumenism and Catholic Learning as they relate to the thought of Congar and Butler, respectively. It also seeks to extrapolate ethical implications for the present-day ecumenical movement. Specifically, it attempts to draw Congar and Butler into dialogue on the central doctrine of the incarnation, regarded by the former as ‘the key to the whole mystery of the Church’.
Jerome P. Baggett
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195326956
- eISBN:
- 9780199870301
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195326956.003.0001
- Subject:
- Religion, Religion and Society
This chapter begins with a discussion of the dramatic changes in American Catholicism that have occurred within the past century. Topics covered include the theological tumult caused by the 1899 ...
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This chapter begins with a discussion of the dramatic changes in American Catholicism that have occurred within the past century. Topics covered include the theological tumult caused by the 1899 encyclical Testem benevolentiae (A Testament of Esteem), which Pope Leo XIII addressed to Cardinal James Gibbons; Catholicism in the 1950s, and the ideological impact of Vatican II reforms. The chapter then details the rationale and research methods used in this book.Less
This chapter begins with a discussion of the dramatic changes in American Catholicism that have occurred within the past century. Topics covered include the theological tumult caused by the 1899 encyclical Testem benevolentiae (A Testament of Esteem), which Pope Leo XIII addressed to Cardinal James Gibbons; Catholicism in the 1950s, and the ideological impact of Vatican II reforms. The chapter then details the rationale and research methods used in this book.
John L. Bell
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.003.0002
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter presents a brief overview of the concepts from axiomatic set theory, then constructs the Boolean-valued universe V(B) and establishes its basic properties. The idea of a mixture of ...
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This chapter presents a brief overview of the concepts from axiomatic set theory, then constructs the Boolean-valued universe V(B) and establishes its basic properties. The idea of a mixture of Boolean-valued sets is introduced and used to prove the Maximum Principle. The truth of the axioms of set theory in V(B) is then established. The chapter concludes with a discussion of ordinals, cardinals, and constructible sets V(B).Less
This chapter presents a brief overview of the concepts from axiomatic set theory, then constructs the Boolean-valued universe V(B) and establishes its basic properties. The idea of a mixture of Boolean-valued sets is introduced and used to prove the Maximum Principle. The truth of the axioms of set theory in V(B) is then established. The chapter concludes with a discussion of ordinals, cardinals, and constructible sets V(B).
John L. Bell
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.003.0006
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of ...
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This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of Boolean-valued models in proving theorems about complete Boolean algebras.Less
This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of Boolean-valued models in proving theorems about complete Boolean algebras.
Michael Potter
- Published in print:
- 2004
- Published Online:
- September 2011
- ISBN:
- 9780199269730
- eISBN:
- 9780191699443
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269730.003.0015
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter studies the elementary consequences of the following definitions: If a = card(A) and b = card(B), we let a + b = card(A ? B); ab = card(A x B); ab = card(BPA). The definitions are ...
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This chapter studies the elementary consequences of the following definitions: If a = card(A) and b = card(B), we let a + b = card(A ? B); ab = card(A x B); ab = card(BPA). The definitions are formally consistent, but this does not explain why they have been chosen. A partial explanation is supplied in this chapter, where it is shown that for finite cardinals they simply reproduce the everyday operations of addition, multiplication, and exponentiation which are already familiar. But this is only a partial explanation: other definitions are no doubt possible which coincide with these ones for finite cardinals but come apart from them in the infinite case. The fruitfulness of the definitions that has been given is thus not something that can be judged straightaway but emerges only once the theory to which they give rise has been developed.Less
This chapter studies the elementary consequences of the following definitions: If a = card(A) and b = card(B), we let a + b = card(A ? B); ab = card(A x B); ab = card(BPA). The definitions are formally consistent, but this does not explain why they have been chosen. A partial explanation is supplied in this chapter, where it is shown that for finite cardinals they simply reproduce the everyday operations of addition, multiplication, and exponentiation which are already familiar. But this is only a partial explanation: other definitions are no doubt possible which coincide with these ones for finite cardinals but come apart from them in the infinite case. The fruitfulness of the definitions that has been given is thus not something that can be judged straightaway but emerges only once the theory to which they give rise has been developed.
Thomas Streicher
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter discusses a notion of universe in toposes, which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH). In this way, one can construct ...
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This chapter discusses a notion of universe in toposes, which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH). In this way, one can construct families of types in the universe by structural recursion and quantify over such families. Further, it shows that (hierarchies of) such universes do exist in all sheaf and realizability toposes. They do not exist instead either in the free topos or in the Vω+ω model of Zermelo set theory. Though universes in the category Set are necessarily of strongly inaccessible cardinality, it remains an open question as to whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF).Less
This chapter discusses a notion of universe in toposes, which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH). In this way, one can construct families of types in the universe by structural recursion and quantify over such families. Further, it shows that (hierarchies of) such universes do exist in all sheaf and realizability toposes. They do not exist instead either in the free topos or in the Vω+ω model of Zermelo set theory. Though universes in the category Set are necessarily of strongly inaccessible cardinality, it remains an open question as to whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF).
John C. Olin (ed.)
- Published in print:
- 2000
- Published Online:
- March 2011
- ISBN:
- 9780823219902
- eISBN:
- 9780823236572
- Item type:
- book
- Publisher:
- Fordham University Press
- DOI:
- 10.5422/fso/9780823219902.001.0001
- Subject:
- Religion, Church History
In 1539, Cardinal Jacopo Sadoleto, Bishop of Carpentras, addressed a letter to the magistrates and citizens of Geneva, asking them to return to the Roman Catholic faith. John Calvin replied to ...
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In 1539, Cardinal Jacopo Sadoleto, Bishop of Carpentras, addressed a letter to the magistrates and citizens of Geneva, asking them to return to the Roman Catholic faith. John Calvin replied to Sadoleto, defending the adoption of the Protestant reforms. Sadoleto's letter and Calvin's reply constitute one of the most interesting exchanges of Roman Catholic/Protestant views during the Reformation and an excellent introduction to the great religious controversy of the 16th century. These statements are not in vacuo of a Roman Catholic and Protestant position. They were drafted in the midst of the religious conflict that was then dividing Europe. And they reflect too the temperaments and personal histories of the men who wrote them. Sadoleto's letter has an irenic approach, an emphasis on the unity and peace of the Church, highly characteristic of the Christian Humanism he represented. Calvin's reply is in part a personal defense, an apologia pro vita sua, that records his own religious experience. Its taut, comprehensive argument is characteristic of the disciplined and logical mind of the author of The Institutes of the Christian Religion.Less
In 1539, Cardinal Jacopo Sadoleto, Bishop of Carpentras, addressed a letter to the magistrates and citizens of Geneva, asking them to return to the Roman Catholic faith. John Calvin replied to Sadoleto, defending the adoption of the Protestant reforms. Sadoleto's letter and Calvin's reply constitute one of the most interesting exchanges of Roman Catholic/Protestant views during the Reformation and an excellent introduction to the great religious controversy of the 16th century. These statements are not in vacuo of a Roman Catholic and Protestant position. They were drafted in the midst of the religious conflict that was then dividing Europe. And they reflect too the temperaments and personal histories of the men who wrote them. Sadoleto's letter has an irenic approach, an emphasis on the unity and peace of the Church, highly characteristic of the Christian Humanism he represented. Calvin's reply is in part a personal defense, an apologia pro vita sua, that records his own religious experience. Its taut, comprehensive argument is characteristic of the disciplined and logical mind of the author of The Institutes of the Christian Religion.
Hagit Borer
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780199263905
- eISBN:
- 9780191718182
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263905.003.0005
- Subject:
- Linguistics, Semantics and Pragmatics
This chapter argues that the properties of the Quantity Phrase (#P) in conjunction with the properties of DP are responsible for derivation of strong vs. weak readings for quantifiers, cardinals, and ...
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This chapter argues that the properties of the Quantity Phrase (#P) in conjunction with the properties of DP are responsible for derivation of strong vs. weak readings for quantifiers, cardinals, and indefinites. In lieu of an operation of type-shifting, the key element in deriving distinct types involves the assignment of range by the same determiner to #, the value heading #P, and to d, the value heading DP.Less
This chapter argues that the properties of the Quantity Phrase (#P) in conjunction with the properties of DP are responsible for derivation of strong vs. weak readings for quantifiers, cardinals, and indefinites. In lieu of an operation of type-shifting, the key element in deriving distinct types involves the assignment of range by the same determiner to #, the value heading #P, and to d, the value heading DP.
Mark S. Massa
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199734122
- eISBN:
- 9780199866373
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199734122.003.0005
- Subject:
- Religion, Religion and Society
The chapter examines the famous case of the IHM nuns of Los Angeles in their battle with Cardinal McIntyre, their bishop. The IHMs represent the single largest group of American nuns to leave the ...
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The chapter examines the famous case of the IHM nuns of Los Angeles in their battle with Cardinal McIntyre, their bishop. The IHMs represent the single largest group of American nuns to leave the Catholic Church en masse, which they did as a result of their rough handling by McIntyre. The IHMs were initially excited by the prospect of reforming their order along the lines mandated by the Second Vatican Council in its famous document (Perfectae Caritatis): getting rid of their old habits, introducing more democracy into their governance, and so on. The archbishop of Los Angeles (who understood the historical significance of these things in a radically different way than the nuns) was, as a result, responsible for one of the most infamous events in American Catholic history.Less
The chapter examines the famous case of the IHM nuns of Los Angeles in their battle with Cardinal McIntyre, their bishop. The IHMs represent the single largest group of American nuns to leave the Catholic Church en masse, which they did as a result of their rough handling by McIntyre. The IHMs were initially excited by the prospect of reforming their order along the lines mandated by the Second Vatican Council in its famous document (Perfectae Caritatis): getting rid of their old habits, introducing more democracy into their governance, and so on. The archbishop of Los Angeles (who understood the historical significance of these things in a radically different way than the nuns) was, as a result, responsible for one of the most infamous events in American Catholic history.
James W. Underhill
- Published in print:
- 2011
- Published Online:
- March 2012
- ISBN:
- 9780748643158
- eISBN:
- 9780748651566
- Item type:
- chapter
- Publisher:
- Edinburgh University Press
- DOI:
- 10.3366/edinburgh/9780748643158.003.0003
- Subject:
- Linguistics, English Language
The explosion of work on metaphor in recent decades has its roots in the ground-breaking book Metaphors We Live By, written by Lakoff and Johnson in 1980. Though the two authors have modified their ...
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The explosion of work on metaphor in recent decades has its roots in the ground-breaking book Metaphors We Live By, written by Lakoff and Johnson in 1980. Though the two authors have modified their position in separate and co-written works since then, and though cognitive approaches have moved on to other fields of linguistics, semantics and epistemology, and though they have introduced new paradigms for analysing metaphor, it is worth quoting the fundamental claims made in this work, since these claims have influenced the terms of the debate that revolves around the representation of conceptual constructs in language. The pith of these fundamental claims can be summed up in seven points: metaphors live; metaphors form systematic constructs; metaphors highlight and hide; conceptual metaphors often contradict one another; metaphors are grounded in experience; metaphors create similarity; and metaphor is the cardinal trope.Less
The explosion of work on metaphor in recent decades has its roots in the ground-breaking book Metaphors We Live By, written by Lakoff and Johnson in 1980. Though the two authors have modified their position in separate and co-written works since then, and though cognitive approaches have moved on to other fields of linguistics, semantics and epistemology, and though they have introduced new paradigms for analysing metaphor, it is worth quoting the fundamental claims made in this work, since these claims have influenced the terms of the debate that revolves around the representation of conceptual constructs in language. The pith of these fundamental claims can be summed up in seven points: metaphors live; metaphors form systematic constructs; metaphors highlight and hide; conceptual metaphors often contradict one another; metaphors are grounded in experience; metaphors create similarity; and metaphor is the cardinal trope.
