This article addresses the use of foreign law in constitutional adjudication. We draw on the ideas of wide reflective equilibrium and public reason in order to defend an engagement model of comparative adjudication. According to this model, the judicial use of foreign law is justified if it proceeds by testing and mutually adjusting the principles and rulings of our constitutional doctrines against reasonable alternatives, as represented by the principles and rulings of other reasonable doctrines. By this, a court points to (...) a wide reflective equilibrium, justifying its own interpretations with reasonable arguments, i.e. arguments that are acceptable from the perspectives defined by other constitutional doctrines, as endorsed by other courts. The point of a judicial engagement of this sort is to work out an overlap between different, reasonable, doctrines in the judicial forum, as part of a liberal forum of public reason. Here, the exercise of public reason filters out the premises of comprehensive doctrines so as to leave us in the region of an overlapping consensus: a region of mid-level principles that can be shared, notwithstanding the fact of legal pluralism. (shrink)

Ronald Preston defended the middle axiom approach to doing Christian social ethics developed by J. H. Oldham for the 1937 ‘Life and Work’ conference. Preston argued that middle axioms continue to offer the churches a relevant ecumenical method. Middle axions has since been subject to fundamental criticism by ethicists such as Duncan Forrester. It will be argued that a case study of the Church of Scotland's contribution to the devolution debate, as part of Scottish civil society, (...) supports Preston's defence of the middle axiom approach as a relevant form of political engagement in the new context of local-global politics. (shrink)

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

This chapter contains sections titled: Aristotle's Declaration of a General Science of Being qua Being A Problem for the Science of Being The Content of the General Science of Being Including Axioms in the General Science of Being The Notion of the Firmest Principle Proving Something about an Axiom: the Indubitability Proof of PNC PNC as the Ultimate Principle Defending an Axiom: the Elenctic Proof of PNC Theology and the General Science of Being Notes Bibliography.

Criticizing psychologism about logic in the Foreword of Grundgesetze, Frege examines an answer to the question of how we can justify our acknowledgment of logical axioms as true—the logical laws that cannot be proved from other laws. The answer he entertains states that we cannot reject logical axioms if we do not want to give up our judgment altogether. Suspending his judgment about this answer, Frege points out that it is still compatible with his anti-psychologist conception of logic. (...) There are two issues with this paragraph. First, it is difficult to read this paragraph consistently. Second, it is not clear why Frege brings up this particular view regarding the justification for logical axioms in the middle of his criticism of psychologism. This paper develops a consistent interpretation of this paragraph. The view on the justification for logical axioms, which Frege is dealing with, is neo-Kantian. In the paragraph, Frege is criticizing a psychologistic misunderstanding of the neo-Kantian justification for logical axioms. The way Frege explains away this misunderstanding shows that he takes the neo-Kantian justification for logical axioms to be a non-evidential epistemic warrant if it is an epistemic warrant. It is this notion of non-evidential epistemic warrant that Frege suspends his judgment about. Thus, the paragraph shows that Frege has a reason to have reservations about neo-Kantianism. Furthermore, it turns out that the sole issue between Frege and psychological logicians is whether truth is objective, which provides a different way to explain away a much-discussed tension between Begriffsschrift and Frege's anti-psychologism. (shrink)

It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of (...) CEM are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional. (shrink)

In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule ) for intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of \ and \ which are closed under \ but do not derive the independence of premiss axiom.

The logic of the weak law of excluded middleKC p is obtained by adding the formula A A as an axiom scheme to Heyting's intuitionistic logicH p . A cut-free sequent calculus for this logic is given. As the consequences of the cut-elimination theorem, we get the decidability of the propositional part of this calculus, its separability, equality of the negationless fragments ofKC p andH p , interpolation theorems and so on. From the proof-theoretical point of view, the formulation presented (...) in this paper makes clearer the relations betweenKC p ,H p , and the classical logic. In the end, an interpretation of classical propositional logic in the propositional part ofKC p is given. (shrink)

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA ${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. (...) These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis. (shrink)

In this article, I argue that while Christians should share Engelhardt’s disappointment in how bioethics functions in the world, they should not share his exasperation. I begin by outlining the general argument in After God, its understanding of secularism, and of how such secularism has impacted bioethics. Next, I suggest that Englehardt appears to lean toward disengagement or at least an extremely suspicious sectarianism. Rather, I claim that it is possible for Christians to morally engage in a useful way with (...) others and still base their own moral activity on proper life telos. Such cooperation is not to serve a commonly-held ultimate purpose, but shared proximate ends. That said, the moral discourse with those outside the Church may be necessarily limited by shallow moral acquaintanceship. I argue that the concept of “middle axioms” can be used as a mechanism for engagement with others in this society. Importantly, this approach does not require abandoning distinctive Christian commitments and foundational assumptions. (shrink)

It is well known that the extensional axiom of choice implies the law of excluded middle. We here prove that the converse holds as well if we have the intensional axiom of choice ACint, which is provable in Martin-Löf's type theory, and a weak extensionality principle, which is provable in Martin-Löf's extensional type theory. In particular, EM is equivalent to ACext in extensional type theory.

This article examines the changing scope and method of ecumenical public theology from the World Missionary Conference of 1910 until the present. Most changes were made in response to the changing ideological and political contexts. The collapse of liberalism and the social gospel was followed by a type of confessional ethics which arose directly out of the German Church Struggle. In opposition to this there emerged a realist ecumenical social ethics, much indebted to Reinhold Niebuhr, and of Ronald Preston. This (...) type of public theology and its distinctive ‘middle axiom’ method are examined and contrasted with the more recent public theologies which were influenced by liberation theology and grassroots movements. It is suggested that the ending of the Cold War and the fundamental changes that have taken place since September 11 2001 present a radically new kind of challenge to public theology. (shrink)

The purpose of this article is to examine and challenge the assumption that the theological legacy of Archbishop William Temple is best continued in the work of Ronald Preston. Preston's concerns in the areas of social ethics and ecumenical relations, as well as his championing of middle axioms, demonstrate his indebtedness to Temple's influence. However, a closer examination of the doctrinal foundations of Preston's social and ecumenical thought did not display a deep understanding of Temple's thought. This is (...) most noticeable in the area of ecclesiology. If Preston had not dismissed Temple's earlier works in favour of the 1942 Christianity and Social Order, he might have avoided developing a theory of the church whose being does not support the tasks Preston requires it to do. (shrink)

Recent years have witnessed a concern among theological bioethicists that secular debate has grown increasingly "thin," and that "thick" religious traditions and their spokespersons have been correspondingly excluded. This essay disputes that analysis. First, religious and theological voices compete for public attention and effectiveness with the equally "thick" cultural traditions of modern science and market capitalism. The distinctive contribution of religion should be to emphasize social justice in access to the benefits of health care, challenging the for-profit global marketing of (...) research and biotechnology to wealthy consumers. Second, religion and theology have been and are still socially effective in sponsoring activism for practical change, both locally and globally. This claim will be supported with specific examples; with familiar concepts like subsidiarity and "mid- dle axioms"; and with recent analyses of "participatory democracy" and of emerging, decentralized forms of global governance. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Methodist Morals: Social Principles in the Public Church's Witness by Darryl W. StephensWonchul ShinMethodist Morals: Social Principles in the Public Church's Witness Darryl W. Stephens knoxville: university of tennessee press, 2016. 320 pp. $48.00Darryl W. Stephens's Methodist Morals presents a historical, theological, and ethical analysis of a particular form of social witness in the United Methodist Church (UMC): the Social Principles. Stephens mainly argues that the Social Principles (...) are not only an institutionalized "document" on contemporary social issues but also an "ongoing moral discourse" within a set of democratic, legislative, and participatory practices of the UMC as a "moral community" (6–8). This ongoing moral discourse illustrates the public church as an authentic social witness in the world and the process of moral formation of its members.Stephens's case study of the public church proceeds in three parts. The first part (chapters 1–3) explores the historical development of the Social Principles and examines the distinctive nature and function of it. From its inception, the Social Principles were to function as "middle axioms," a level of ethical deliberation placed between "universal principles and specific policy recommendations" (22). They were intended to promote "dialogue" among different opinions and "conscientious discernment" on social issues (22). The Social Principles' commitment to the dialogue and discernment is further bolstered by representative democratic and legislative procedures in its quadrennial revision at the General Conference.The second part (chapters 4–7) examines the texts and contents of the Social Principles. Through a comparison of the Social Principles to secular human rights documents, Stephens reveals a "deep-seated commitment to human rights" with a theological emphasis on "the universality of God's grace" in the texts of Social Principles (72). On the other hand, Stephens points out a lack of engagement with the broader Methodist social teaching tradition and the undeveloped theological foundations and languages for the commitments in the Social Principles (92–94). Specifically, Stephens calls for developing a "theology of marriage," which has been neglected in the history of Methodist teachings on marriage (140). In addition, Stephens proposes the need of theological guidance for "ecclesial disobedience" as an alternative paradigm for clergy ethics in response to the issue of homosexuality (171–73).In the final part (chapter 8 and the conclusion), Stephens sketches "the public church ecclesiology implied by the Social Principles" (178). He points out the sectarian tendency in the ecclesiologies of Paul Ramsey and Stanley [End Page 205] Hauerwas and argues for a different form of public engagement: the public church is "sometimes distinct from but always part of and partner with the world" (194). In other words, the public church should embrace "the challenges of worldliness," and the Social Principles illustrates "the messiness of genuine moral engagement in life here and now" (197–99).Stephens presents a rich and well-balanced analysis of the UMC Social Principles as a distinctive form of social witness in the public church. Eschewing a biased analysis praising his own denomination's social teaching, Stephens critically points out "the messiness of actual moral meaning-making" in the UMC (3). Chapter 5 highlights his critical evaluation of the Social Principles: the sin of "US-centrism" and the marginalization of the European, African, and Asian UMC bodies (107–9). However, Stephens's critical evaluation could be developed further. As he points out that the US-centrism is about a "perceived hegemony and uniformity of a white, US, middle-class elite," he could also critically address the dominance of a certain racial/ethnic group's perspective within the US context. Some readers might wonder why Stephens does not substantially cover the issue of racism and white privilege that lies at the heart of the US-centrism critiqued in this book.Nevertheless, Methodist Morals has added rich insight into the public church's social witness and its members' moral formation. This book would be a great resource for theological educators in their courses on Christian ethics, public theology, and Methodism.Wonchul ShinEmory UniversityCopyright © 2018 Society of Christian Ethics... (shrink)

It is well known that the extensional axiom of choice implies the law of excluded middle . We here prove that the converse holds as well if we have the intensional axiom of choice ACint, which is provable in Martin-Löf's type theory, and a weak extensionality principle , which is provable in Martin-Löf's extensional type theory. In particular, EM is equivalent to ACext in extensional type theory.

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on (...) the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory. (shrink)

: As a form of in vivo knowledge, the transdisciplinary methodology suggests going beyond disciplines. According to Nicolescu, this methodology occurs at different levels of reality, different levels of perception, and within the logic of the included middle axioms that exist simultaneously. In addressing these levels, the researcher is the interlocutor between the external world of the Object and the internal world of the Subject. In architecture, this knowledge emerges through a variety of disciplines that need to be (...) fused with an approach that is rhizomatic and nomadic, leading to thought without an image as characterized by Deleuze and Guattari. By following their “Pink Panther” metaphor for an imageless thought and approach in TR, this article aims to understand the relationship between the TR methodology and the theory of thought without an image - an approach which can enable the comprehension of ambiguous architecture and urban design problems. Resumo: Como um conhecimento in vivo, a metodologia transdisciplinar sugere ir além das disciplinas. Segundo Nicolescu, essa metodologia ocorre em distintos níveis de realidade, em diferentes níveis de percepção e na lógica do terceiro incluído, que existem simultaneamente. Ao abordar esses níveis, o pesquisador é a correspondência entre o mundo externo do Objeto e o mundo interno do Sujeito. Na arquitetura, esse conhecimento surge por meio de uma variedade de disciplinas que precisam ser combinadas e fundidas com uma abordagem rizomática e nômade, levando ao pensamento sem imagem, como tratado por Gilles Deleuze e Félix Guattari. Seguindo a metáfora da “pantera rosa” para um pensamento sem imagem e abordagem em TR, que não imita ou reproduz outra coisa, o objetivo deste artigo é compreender a relação entre a metodologia TR, a teoria do pensamento sem imagem - uma abordagem que pode permitir a compreensão de arquitetura ambígua e problemas de projeto urbano. (shrink)

Stephen Turner complains about weaknesses of Robert K. Merton's teachings without noticing that these are common. He puts down Merton's ideas despite his innovations, on the ground that they are not successful and not sufficiently revolutionary. The criteria by which he condemns Merton are too vague and too high. Merton's merit is in his having put the sociology of science on the map and drawn attention to the egalitarianism that was prominent in classical science and that is now diminished. Key (...) Words: American sociology • intellectual frameworks • Merton • middle axioms • time lag. (shrink)

Since the middle of this century, the dominant prescriptive approach to decision theory has been a deductive viewpoint which is concerned with axioms of rational preference and their consequences. After summarizing important problems with the preference primitive, this paper argues for a constructive approach in which information is the foundation for decision-making. This approach poses comparability of uncertain acts as a question rather than an assumption. It is argued that, in general, neither preference nor subjective probability can be (...) assumed given, and that these need to be generated by using the relevant information available to the decision-agent in a given situation. A specific constructive model is discussed and illustrated with a real example from this viewpoint. (shrink)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...) by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

The axiom Extra Ecclesiam nulla salus was drawn up by Cyprian of Carthage in the middle of the third century. In the Middle Ages it was essential for the pontifical theocracy with popes like Gregory VII (Dictatus papae, 1075) Inocent III (IV Lateran Council, 1215) and Boniface VIII (Bull Unam Sanctam , 1302).With the necessary adaptations and nuances (St. ThomasAquinas in the XIIIth century; the theologians of Salamanca in the XVIth century, the II Vatican Council in the XXth (...) century) the controversy around this maxim has reached our days. (shrink)

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL-chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL-chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak (...) cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (shrink)

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis ('SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ('CA') without resort to the method of limits. Formally, however, unlike Robinsonian 'nonstandard analysis', SIA conflicts with CA, deriving, e.g., 'not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this 'change of (...) logic', arguing that standard arguments based on 'smoothness' requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism. (shrink)

Our aim is to propose several itineraries which follow the scenario having as a first step Euclid’s Fifth Postulate; as a second step the Bolyai–Lobachevsky’s non-Euclidean geometries and as a third step Einstein’s relativity theory. The role of Euclid’s fifth postulate is successively assumed by Archimedes’ axiom; Zermelo’s choice axiom; Cantor’s continuum hypothesis; von Neumann’s foundation axiom for set theory; Church–Turing thesis and Turing’s computability; the validity of classical logic under the form of the principles of identity, non-contradiction and excluded (...) middle; the macroscopic universe assumed as form of existence, including the fact that humans are macroscopic beings; human language and semiosis assumed as the unique way to approach understanding and communication; the old Greek ideal of order, symmetry and harmony, adopted for representation of the world in Art and Poetry. The common denominator of these starting steps is the fact that they all correspond to the sensorial-empirical perception of the world. We point out the strong interaction among the steps corresponding, in these different itineraries, to Einstein’s relativity theory. We infer in this way the need of a transdisciplinary approach. (shrink)

'The god wanted everything to be good, marred by as little imperfection as possible.'Timaeus, one of Plato's acknowledged masterpieces, is an attempt to construct the universe and explain its contents by means of as few axioms as possible. The result is a brilliant, bizarre, and surreal cosmos - the product of the rational thinking of a creator god and his astral assistants, and of purely mechanistic causes based on the behaviour of the four elements. At times dazzlingly clear, at (...) times intriguingly opaque, this was state-of-the-art science in the middle of the fourth century BC. The world is presented as a battlefield of forces that are unified only by the will of God, who had to do the best he could with recalcitrant building materials.The unfinished companion piece, Critias, is the foundational text for the story of Atlantis. It tells how a model society became corrupt, and how a lost race of Athenians defeated the aggression of the invading Atlanteans. This new edition combines the clearest translation yet of these crucial ancient texts with an illuminating introduction and diagrams. (shrink)

In the middle of myMathematical logicI defined a certain class of formulae as “stratified,” and conjectured that exclusion from this class is a feature “shared, presumably, by all the untenable statements(p. 157). This ushered in a set of axioms of class-membership which Rosser has since shown to be inconsistent. Accordingly, inElement and numberI dropped the principle*200, in which had been assembled axioms to the effect, roughly, that “stratified functions of elements are elements.” In lieu of*200 I set (...) forth alternatives in which no appeal is made to stratification. The system ofMathematical logicexclusive of*200 carries over as an unchanging framework; and this framework admits, we know, of a simple consistency proof. My concern in the present paper is to draw attention to certain relationships between this framework and earlier theories. (shrink)

The author uses a series of examples to illustrate two versions of a new, nonprobabilist principle of epistemic rationality, the special and general versions of the metacognitive, expected relative frequency principle. These are used to explain the rationality of revisions to an agent’s degrees of confidence in propositions based on evidence of the reliability or unreliability of the cognitive processes responsible for them—especially reductions in confidence assignments to propositions antecedently regarded as certain—including certainty-reductions to instances of the law of excluded (...) middle or the law of noncontradiction in logic or certainty-reductions to the certainties of probabilist epistemology. The author proposes special and general versions of the MERF principle and uses them to explain the examples, including the reasoning that would lead to thoroughgoing fallibilism—that is, to a state of being certain of nothing. The author responds to the main defenses of probabilism: Dutch Book arguments, Joyce’s potential accuracy defense, and the potential calibration defenses of Shimony and van Fraassen by showing that, even though they do not satisfy the probability axioms, degrees of belief that satisfy the MERF principle minimize expected inaccuracy in Joyce’s sense; they can be externally calibrated in Shimony and van Fraassen’s sense; and they can serve as a basis for rational betting, unlike probabilist degrees of belief, which, in many cases, human beings have no rational way of ascertaining. The author also uses the MERF principle to subsume the various epistemic akrasia principles in the literature. Finally, the author responds to Titelbaum’s argument that epistemic akrasia principles require that we be certain of some epistemological beliefs, if we are rational. (shrink)

The completeness w.r.t. Kripke frames with equality (or, equivalently, w.r.t. Kripke sheaves, [ 8 ] or [4, Sect. 3.6]) is established for three superintuitionistic predicate logics: ( Q - H + D *), ( Q - H + D *&K), ( Q - H + D *& K & J ). Here Q - H is intuitionistic predicate logic, J is the principle of the weak excluded middle, K is Kuroda’s axiom, and D * (cf. [ 12 ]) is (...) a weakened version of the well-known constant domains principle D . Namely, the formula D states that any individual has ancestors in earlier worlds, and D * states that any individual has $${\neg\neg}$$ -ancestors (i.e., ancestors up to $${\neg\neg}$$ -equality) in earlier worlds. In particular, the logic ( Q - H + D *& K & J ) is the Kripke sheaf completion of ( Q - H + E & K & J ), where E is a version of Markov’s principle (cf. [ 12 ]). On the other hand, we show that the logic ( Q - H + D *& J ) is incomplete w.r.t. Kripke sheaves. (shrink)

We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann (...) constant. An overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)

A purely deductive medical ethics cannot properly account for the varieties of circumstances which arise in medical practice. By contrast, a purely inductive medical ethics lacks sufficient guidance from ethical principles. In resolving ethical dilemmas in medicine, most often an appeal is made to middle-level axioms and methodological rules to mediate between theory and practice. I argue that this appeal must be augmented by considerations of context, such considerations, in effect, constituting a moral rule based on the social (...) structure of medical practice. A contextual grid is proposed which assists the process of weighing values in resolving cases. (shrink)

NADOT M. Nursing Inquiry 2010; 17: 118–127The world’s first secular autonomous nursing school against the power of the churchesSecular healthcare practices were standardized well before the churches’ established their influence over the nursing profession. Indeed, such practices, resting on the tripartite axiom of domus, familia, hominem, were already established in hospitals during the middle ages. It was not until the last third of the eighteenth century that the Catholic Church imposed its culture on secular health institutions; the Protestant church (...) followed suit in 1836. In reaction to the encroachment of religious orders on civil society and the amalgam of religious denominations favored, for example, by the devout Florence Nightingale (supported, in 1854, by Sir Sidney Herbert, the influential Puseyite), it is on 20 July 1859 that the great Swiss nineteenth century pedagogue and recipient of the Académie française Gold Medal, Valérie de Gasparin-Boissier (1813–94), proposed a model of secular healthcare training for nurses that would become a counter-model set in opposition to religious health institutions. Forerunner of later schools, the world’s first secular autonomous Nursing School was founded in Lausanne, Switzerland. Its mission was to bring decisive changes to the statutes of nurses’ training, which were then still based on six principles not far removed from those of religious communities at the time: commitment for life, the Rule of St Augustine, obedience, celibacy, the renouncement of salary, and the uniform. (shrink)

This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 presents (...) the model-theoretic concepts, based on those in [7], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. Box and Diamond are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0-version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [7]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.). (shrink)

Although the theory of the assertoric syllogism was Aristotle's great invention, one which dominated logical theory for the succeeding two millenia, accounts of the syllogism evolved and changed over that time. Indeed, in the twentieth century, doctrines were attributed to Aristotle which lost sight of what Aristotle intended. One of these mistaken doctrines was the very form of the syllogism: that a syllogism consists of three propositions containing three terms arranged in four figures. Yet another was that a syllogism is (...) a conditional proposition deduced from a set of axioms. There is even unclarity about what the basis of syllogistic validity consists in. Returning to Aristotle's text, and reading it in the light of commentary from late antiquity and the middle ages, we find a coherent and precise theory which shows all these claims to be based on a misunderstanding and misreading. (shrink)

We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$. Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$. In addition, we show that this conservation theorem is optimal in (...) the sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$. In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles. (shrink)

, which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property.More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following (...) Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages.There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property.In the sequel, we shall extend the λ c -calculus to the Zermelo-Frænkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF ɛ. (shrink)

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

This paper offers an overview of the history of the axiom forma dat esse, which was commonly quoted during the Middle Ages to describe formal causality. The first part of the paper studies the origin of this principle, and recalls how the ambiguity of Boethius’s first formulation of it in the De Trinitate was variously interpreted by the members of the School of Chartres. Then, the paper examines the various declensions of the axiom that existed in the late (...) class='Hi'>Middle Ages, and shows how its evolution significantly follows the progressive decline of the Aristotelian model of formal causality. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS The Immutability of God in the Theology of Hans Urs von Balthasar. By GERARD F. O'HANLON, S.J. Cambridge: Cambridge University Press, 1990. Pp. 246. $59.95 (cloth). O'Hanlon unfolds Balthasar's theology in four main chapters, which treat the question of immutability in terms, respectively, of Christ· ology; creation; time and eternity; and inner trinitarian life in God. In Chapter 5, O'Hanlon compares Balthasar's approach with some English-speaking authors (...) (B. Davies, A. Kenny, E. Stump, Kretzmann, P. Geach, J. Kvanvig, D. H. Mellor, A. E. Williams). Finally, in Chapter 6, O'Hanlon ofiers a summary and assessment of Balthasar's proposals. (1) Regarding his treatment of Christology in Chapter 1, O'Hanlon discusses how Balthasar's attempt to find a "middle way" between simple mutability and a traditional sense of immutability in God lies in uncovering the trinitarian presuppositions of Christology. Whether God is open to being afiected by dialogue with creatures hinges decisively on how we understand Jesus Christ or, more precisely, on how we understand the humanity of Jesus to reveal what God is like (9-10). This is an approach difierent from.that of those who would criticize the traditional axiom of divine immutability on the basis of certain philosophical assumptions regarding the priority of becoming over being (10-11). Balthasar's approach is first theological: when we con· sider the soteriological and eschatological dimensions of Christ's work for us, we are forced to ask the question about the nature of God: "does he too not sufier, is he not afiected by sin, can he he simply apathetic to the prospect of any final loss of his creatures to hell? " (9). At the heart of Ba1thasar's position, then, lies an affirmation of a kenosis within the inner life of God, which already provides a foundation for the possibility of the (second) kenosis revealed in the incarnation and the cross (11-14). That is, the "kind of emptying and self. giving we see in the incarnation is both the image and the effect of the eternal ' externalization ' of God that is involved in the intra-trinitarian life" (14). It is imperative that one see the nuances of this affirmation. On the one hand, Balthasar is careful to avoid a simple identification of the second with the first kenosis: this would make the incarna· 335 336 BOOK REVIEWS tion and cross natural and necessary to God, thus entailing theopaschism (14-15). On the other hand, the suffering love of Christ does truly reveal God's sovereign power to be absolute love and self-giving, because there " is a certain analogy between the divine and human natures of Christ which is due to the identity of the person, the pre-existent Logos" (14). Thus Balthasar is led to qualify the traditional understanding of divine immutability (15) : there is " some real kenosis within God which has ontological status and is not merely functional, soteriological, or a simple addition which does not affect God" {16). O'Hanlon makes clear how Balthasar is at pains always to respect the mysterious nature of the issue, as well as the limits set by the great Christian tradition. At the heart of the matter lies a return to the person -nature distinction as set by Chalcedon. Traditionally, the affirmation has been of a divine nature in Christ which remains immutable and a human nature which changes and suffers. Balthasar goes beyond this affirmation. On the basis of Philippians 2:5-11, he arrives at the necessity of positing a real kenosis in God ; and from his recognition of the ontological, personal identity of the Logos as the subject who unites the two distinct natures in Christ, he refuses to limit the change and suffering which Christ experiences to his human nature alone. In fact, the tendency to understand the human nature of Christ as an instrumentum conjunctum which does not affect the divine person he considers as Nestorian in character. Yet, at the same time, Balthasar affirms the incommensurable difference between God and the world, between the divine and human " unmixed " natures of Christ. The relation between the two natures is precisely analogous (after the manner... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hypatia 17.4 (2002) 238-243 [Access article in PDF] Book Review Feminist Interpretations of Aristotle Feminist Interpretations of Aristotle. Edited by Cynthia A. Freeland. University Park: Pennsylvania State University Press, 1998. This volume consists of twelve essays, mostly newly published, on a variety of topics in Aristotelian scholarship ranging from the theoretical to the practical and productive parts of the corpus. The volume divides the papers into one group addressing (...) topics in Aristotle's metaphysics, physics, epistemology, biology, and logic on one hand, and his ethics, politics, poetics, and rhetoric on the other. The contributors include established scholars in ancient philosophy, such as Cynthia Freeland, Deborah Modrak, Martha Nussbaum, and Charlotte Witt, and younger scholars such as Angela Curran, as well as those in disciplines outside ancient philosophy, including literature, law, and political science. The latter group of essays includes a chapter by Luce Irigaray on Book IV of Aristotle's Physics from her work, An Ethics of Sexual Difference (1993), Freeland's interpretation of Irigaray's essay, as well as papers on Aristotelian political emotion, the historiography of Aristotle's rhetoric, and his political anthropology from Texas Law Review (1992). The very range of methodological perspective that lends breadth to the volume presents difficulties for an overview, in light of which four papers were selected for detailed comment, those on Aristotle's logic, Freeland's essay on Irigaray's reading of Physics IV, Aristotelian virtue ethics, and Aristotelian political emotion. Feminism and Aristotelian Logic While it is familiar to cite the inherent biases of theoretical disciplines like the social sciences and philosophy, it is less common to read how areas like mathematics and logic are similarly biased. It seems hard to see how a system of proof with rules, axioms, and formal notation is inherently biased against any one group. Yet feminists like Andrea Nye, Valerie Plumwood, and Luce Irigaray have criticized formal logic on the ground that it is antithetical to women's experience and interests. 1 In "Feminist Readings of Aristotelian Logic" (1998), Marjorie Hass takes on criticisms of Aristotle's logic, including three main objections: his logic is motivated by unequal relations of power, uses binary [End Page 238] truth-value, and ignores the concrete and the subjective in experience. In being thus characterized, Aristotle's logical theory represents a "gendered" way of reasoning, and should be avoided. Hass does well in replying to these objections with one exception. The first objection, that logic masks unequal power relations, is answered by Hass noting that logic and logical systems (in mathematics and philosophy) have arisen among various cultures and classes worldwide, and are not the domain of an elite group of men. Hass's further point that attaining logical skill can be liberatory to women needs to be underscored: it is surely politically loaded for feminists to argue that women cannot, or should not, do logic. Hass rightly responds to the third objection concerning the need for concrete experience on two grounds: first, for Aristotle the end of logical argument is in part the illumination of observed experience, not its dissolution, and second, that it is impossible to conceive of a language without any abstraction. We cannot construct a feasible language without abstract terms or categories, as Nye's criticism suggests.Hass's response to the second criticism, that bivalence is noxious, is correct but incomplete. The ground of this objection stems from the idea that in classical logic the laws of excluded middle and noncontradiction obtain such that given two contradictory propositions, only one can be true, and this fact implies inequality to feminist critics. But, as Hass points out, this objection does not acknowledge Aristotle's notion of contrariety in which both sentences are false—the point being that contrariety does not yield the same results as negation. Still, Hass admits, the "fluid" form of negation sought for by thinkers like Irigaray is not forthcoming in Aristotle's logic. 2 Here Hass overlooks what Aristotle has to say about future contingents in De Interpretatione 9, that statements about the future are neither true nor false: "it is not necessary that of every affirmation... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Historical Violence, Censorship, and the Serial Killer: The Case of American PsychoCarla Freccero (bio)R.L.: Do you believe in God?B.E.E.: Are you asking me if I was raised in a religious family or if I go to church? I was raised an agnostic. I don’t know—I hate to fly, I have a fear of flying. That means either that I have no faith in air traffic controllers or that I’ve (...) done something really bad, and this is God’s way of getting at me. Maybe I’m caught in the middle.... But no, I don’t believe in God. That’s such a strange thing to admit in an interview.—Robert Love, “Psycho Analysis,” Rolling Stone interview with Bret Easton EllisBefore Capitol Hill acts on these dimwitted proposals, we should remember what the writer Bret Easton Ellis might have said, but didn’t: “Doing bad stuff is wrong.”—Ted Rall, “Meet the Cusp Kids”US mass media has become a much-publicized target of censorious commentary within American public culture in recent years. Censorship, as E. S. Burt notes, may take at least two forms: philosophical censorship, such as that discussed by Leo Strauss’s Persecution and the Art of Writing, and the more commonly applied form of US censorship enacted against “pornography” or “obscenity.” Philosophical censorship, in this reading, entails the state’s persecution of message-bearing texts or utterances that counter the doxa and threaten to disrupt the state, while the second form of censorship involves “a language use that falls outside the interpretive scheme defining the work as the intentional act of a moral being” [Burt 25]. In the United States, state censorship of the first kind may perhaps be practiced through the “incitement to riot” escape clause provision in the First Amendment, while obscenity censorship—in certain forms—is explicitly written into legislation as excluded from speech protection.Extending Strauss’s argument to the liberal state, Burt suggests in effect that these two forms of censorship become related insofar as the state’s censorious intent is directed not so much at the content of the expression as at the degree to which it confirms the axiom, what Burt calls the “very fundamental, convening assumption of the... liberal state” [25] that “virtue is knowledge” [see Strauss 25]. Thus obscenity is a form of expression that cannot be interpreted as productive knowledge but instead as a certain repetitive excess, “dirt for dirt’s sake” [Burt, quoting Judge Woolsey 28]. The problem here, of course, is that such distinctions do not apply comfortably to that category of utterance termed the literary, and especially to the use of figurative language, where “art for art’s sake” can be [End Page 44] seen precisely to resemble the nonproductivity of obscenity. The text I am going to discuss in depth here, Bret Easton Ellis’s American Psycho, has been attacked on both grounds—as an active (and violent) disruption of the doxa that is also obscenely nonproductive of knowledge—and as such, it serves as an illustrative, and I will argue, noncoincidental, case of “censorship” in the liberal state.It may be useful here to begin by surveying and making distinctions among the various popular and popularized censorships in recent US media: Ice-T’s “Cop Killer” was probably the most authentic case of “philosophical” censorship, even though its silencing did not derive directly from the state (the liberal state often does not function in some of the ways supposed when the definition of censorship is restricted to state censorship). “Cop Killer” is a song about killing policemen, and thus came closest—in the list I am going to propose of silenced works—to tempting the state’s retaliation. It was written and sung by a black man (a black man who often plays a gangster and a cop in popular films), in a genre already regarded as threateningly disruptive to the state: rap. The result is that the CD containing this song was pulled from the shelves of music stores and can no longer be purchased. There was no trial. The author withdrew his work from public circulation in the face of... (shrink)

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of (...) some familiar results in the theory of (complete) Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

There are many studies and theories in the literature on the source of power in the formative years of the Mamlūk state. In these studies, the importance of power and influence as a form of becoming a sultan in the Mamlūks has been emphasized, although the oligarchic power of the umara has been indirectly mentioned in terms of the source of power by discussing the distinction between a puppet sultan and an authoritarian sultan. However, this axiom causes a number of (...) problems and ambiguities within itself and leads to the supersession of the determinism of the elite amirs. This work will try to overcome this problem by focusing on the political developments during the foundation years of the state and discussing how powerful and influential amirs were restricted by the umara on their way to becoming sultans. At the same time, by focusing on the chronicles of the period in question, the administrative determinism of the Mamlūk factions as a decision-making mechanism in the Mamlūk state will be pointed out. On the other hand, the statement that power emanated from both the sultan’s household faction and the military support of the elite amirs or khushdāshs who held the Mamlūk oligarchy in their hands leads to overlooking the tension or conflict between the two throughout Mamlūk political history. Considering the state of conflict between the two sides, it is a logical contradiction for the two opposing poles to agree on the same candidate. For instance, considering Muʿizz Aybak’s accession to power, the Sālihiyya-Bahriyya amirs unanimously endorsed his sultanate, but in the process, Muʿizz Aybak prevailed over the tensions he experienced with the elite that brought him to power and managed to become the absolute sultan, albeit for a short time. Therefore, it is essential to keep in mind that there was no uniform relationship between the sultan and the elite amirs, but rather a dynamic functioning. In other words, the future sultan, who had to establish good relations with the amirs until he became sultan, had to go against the amirs who brought him to power if he wanted to become an absolute ruler after he became sultan. In this study, the source of Mamlūk power is traced to the collective interests of the Mamlūk factions. It is emphasized that these interests were usually centered on candidates who were seen as more moderate among the factions. At the same time, by revealing the shortcomings of the understandings presented as a rule such as “the one who kills the sultan becomes the sultan” and “the powerful and influential become the sultan”, the analysis of the period from the founding years of the Mamlūks to Baybars’ sultanate mostly emphasizes the continuity of the status quo of the elite amirs by highlighting the “sultanate of weak and middle-ranking amirs”. The analysis has pointed out that those who killed sultans or usurped the throne were punished rather than usurping the throne as a legitimate right. Similarly, the consensus of the umara on the candidate for sultanate was pointed out by mentioning that those who were powerful and influential were being challenged rather than made sultan. (shrink)

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, (...) i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)