The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. (...) This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU. (shrink)

In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory $GPK^+ \infty$.

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, (...) axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc. (shrink)

In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it (...) is not known if these implications reverse, BGS also showed that those theorems imply finite choice (in some cases, with additional induction assumptions). This motivated us to study the proof-theoretic strength of finite choice. Using a variant of Steel forcing with tagged trees, we show that finite choice is not provable from the $\Delta ^1_1$ -comprehension scheme (even over $\omega $ -models). We also show that finite choice is a consequence of the arithmetic Bolzano–Weierstrass theorem (introduced by Friedman and studied by Conidis), assuming $\Sigma ^1_1$ -induction. Our results were used by BGS to show that several theorems in graph theory cannot be proved using $\Delta ^1_1$ -comprehension. Our results also strengthen results of Conidis. (shrink)

The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of (...) as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences. (shrink)

We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...) (second-order number theory with a -comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a operator can be introduced in other settings. (shrink)

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 (...) has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles. (shrink)

We present a way out of Russell’s paradox for sets in the form of a direct weakening of the usual inconsistent full comprehension axiom scheme, which, with no additional axioms, interprets ZFC. In fact, the resulting axiomatic theory 1) is a subsystem of ZFC + “there exists arbitrarily large subtle cardinals”, and 2) is mutually interpretable with ZFC + the scheme of subtlety.

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is (...) a “strong” theory since “On is ramifiable” implies the existence of a proper class of inaccessible cardinals. (shrink)

We study the proof-theoretic strength of the Π11-separation axiom scheme, and we show that Π11-separation lies strictly in between the Δ11-comprehension and Σ11-choice axiom schemes over RCA0.

This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are $${\Delta_1}$$ provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether $${I\Delta_0 + \neg \mathit{exp}}$$ implies $${B\Sigma_1}$$ to a purely recursion-theoretic question.

One of the earliest applications of transfinite numbers is in the construction of derived sequences by Cantor [2]. In [6], the existence of derived sequences for countable closed sets is proved in ATR0. This existence theorem is an intermediate step in a proof that a statement concerning topological comparability is equivalent to ATR0. In actuality, the full strength of ATR0 is used in proving the existence theorem. To show this, we will derive a statement known to be equivalent to ATR0, (...) using only RCA0 and the assertion that every countable closed set has a derived sequence. We will use three of the subsystems of second order arithmetic defined by H. Friedman , which can be roughly characterized by the strength of their set comprehension axioms. RCA0 includes comprehension for Δmath image definable sets, ACA0 includes comprehension for arithmetical sets, and ATR0 appends to ACA0 a comprehension scheme for sets defined by transfinite recursion on arithmetical formulas. MSC: 03F35, 54B99. (shrink)

The comprehensive nature of information and insight available in the Upanishads, the Indian philosophical systems like the Advaita Philosophy, Sabdabrahma Siddhanta, Sphota Vaada and the Shaddarsanas, in relation to the idea of human consciousness, mind and its functions, cognitive science and scheme of human cognition and communication are presented. All this is highlighted with vivid classification of conscious-, cognitive-, functional- states of mind; by differentiating cognition as a combination of cognitive agent, cognizing element, cognized element; formation; form and structure (...) of cognition, instruments and means of cognition, validity of cognition and the nature of energy/matter which facilitates and also is the medium of cognition- cognizing process; and also as the container and content of cognition. The human communication process which is just the reverse of cognizing process is also presented with necessary description and detail. The sameness of cognitive / communicative process during language acquisition and communication processes and the modes of language acquisition and communication are also given. The hardware and software of human cognition, language acquisition and communication as envisaged in Indian spiritual and philosophical expressions are given. In the light of the information and insight obtained as above, the axioms for human cognition / communication are formed, framed and presented. A brain-wave frequency modulation / demodulation model of human cognition, communication and language acquisition and communication process based on Upanishadic expressions, Shaddarsanas and Sabdabrahma Siddhhanta is defined and discussed here. The use of these theories and models and axioms in mind-machine modelling and natural language comprehension branch of artificial intelligence will be put forward and highlighted. (shrink)

We study the theories I∇n, L∇n and overspill principles for ∇n formulas. We show that IEn ⇒ L∇n ⇒ I∇n, but we do not know if I∇n L∇n. We introduce a new scheme, the growth scheme Crγ, and we prove that L∇n ⇒ Cr∇n⇒ I∇n. Also, we analyse the utility of bounded collection axioms for the study of the above theories.

The socially responsible investment industry (SRI) is slowly changing from a screening, avoidance paradigm to a comprehensive paradigm that seeks to affect corporate behavior. Credible rating systems are a key component of this sea change. Reliable and recognizable social and environmental metrics are critical to this progress. The Total Social Impact (TSI) rating approach is a new social metric scheme based on a comprehensive rating of stakeholder issues. This paper describes the evolution of SRI ratings and the role that (...) TSI hopes to play in affecting business behavior by promoting principled business leadership. (shrink)

This article is an extended promenade strolling along the winding roads of identity, equality, nameability and completeness, looking for places where they converge. We have distinguished between identity and equality; the first is a binary relation between objects while the second is a symbolic relation between terms. Owing to the central role the notion of identity plays in logic, you can be interested either in how to define it using other logical concepts or in the opposite scheme. In the (...) first case, one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts in terms of the identity relation, using also abstraction. The present paper investigates whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic a reliable definition of identity is possible. However, the definition needs the standard semantics and we know that with this semantics completeness is lost. We have also studied the relationship of equality with comprehension and extensionality and pointed out the relevant role played by these two axioms in Henkin’s completeness method. We finish our paper with a section devoted to general semantics, where the role played by the nameable hierarchy of types is the key in Henkin’s completeness method. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Buddhist-Christian Studies 21.1 (2001) 157-161 [Access article in PDF] Book Review Toward a Christian Theology of Religious Pluralism Toward a Christian Theology of Religious Pluralism. By Jacques Dupuis, S.J. Maryknoll, New York: Orbis Books. 1997. xiv + 433 pp. There may not be another individual more qualified than Jacques Dupuis to write this book. He has not only spent a lifetime teaching and serving in a part of the (...) world where the encounter between Christianity and other faiths has been particularly pressing (India), but he has also previously reflected intensively and published widely on this topic. Even more impressive, however, is the fact that the scholarship that emerges in this volume reflects Dupuis as a faithful Catholic Christian, both in the sense of paying sincere heed to the wide range of the Roman Church's official teachings on this and related topics (he taught for years at the Gregorian University in Rome and edited, with Josef Neuner, the massive The Christian Faith in the Doctrinal Documents of the Catholic Church 1 ) and in the sense of thinking along with others in the universal community of faithful Christians. Here we have a master theologian practicing his craft, working within a community of theological inquirers, bringing out, displaying, and putting to work old things and new in the effort to understand one of the most crucial issues of our time.The result is a comprehensive and lucid discussion of Christian theology of religions that is indispensable as an introduction to students without any knowledge of the landscape even while it also serves as a handy status questionis to those already [End Page 157] familiar with the terrain. The first half of the book (part 1) is an overview of the history of Christian approaches to other faiths, beginning with the biblical material, continuing through the Logos spermatikos theology of the early Fathers, the developments surrounding the Extra ecclesiam nulla salus ("no salvation outside the Church") axiom over the centuries, and the pre- and post-Vatican II attempts to develop a theology of religions in the Catholic Church. This part concludes by summarizing the paradigm shifts from ecclesiocentrism to (in order) christocentrism, theocentrism, and regnocentrism/soteriocentrism. Throughout the discussion--especially of the pre-twentieth-century history that focused much more so on the fate of the unevangelized--Dupuis manages to provide important connections to developments in Christian thinking about the role of the religions in the divine scheme of things.The second half of the volume (part 2) presents seven of the most difficult theological questions in Christian theologia religionum today. First, concerning the covenant: is there only one, or are there many? Is salvation history particular, or universal? Dupuis' provisional response is that there is one cosmic covenant that is never revoked. This covenant is revealed according to what he calls a "trinitarian rhythm" whereby God covenants through his Word, in his Spirit, and this covenant includes the witnesses of God left in other religions as well. Second, concerning revelation: does revelation come to us from "without," or emerge from "within" ourselves? What about the sacred texts of other traditions? How are Christians to understand the fullness of the revelation in Jesus Christ vis-à-vis other faiths? Dupuis' answer is that divine revelation is indeed scattered among the various religions, even if fully captured only in Christ, and that these revelations are thereby differentiated but complementary. 2 Third, concerning the divine mystery: is the "Real" beyond the categories of "personal" and "impersonal"? Is God one, or many (triune)? Dupuis responds that the religions have complementary and converging insights into the divine reality. Fourth, concerning Jesus Christ: is he one among many savior figures? What does the uniqueness of Christ mean? Is there a distinction or an absolute identity between Jesus and Christ? Is the Christ particular, or universal? Dupuis posits a "searching christology" whereby God reveals Godself in and through the images and symbols of the religions and through which humankind therefore responds to the divine graciousness. Jesus' uniqueness is thereby both "constitutive" (of universal significance) and "relational" (inserted in the overall plan of... (shrink)

Unrestricted use of the axiom schema of comprehension, ?to every mathematically (or set-theoretically) describable property there corresponds the set of all mathematical (or set-theoretical) objects having that property?, leads to contradiction. In set theories of the Zermelo?Fraenkel?Skolem (ZFS) style suitable instances of the comprehension schema are chosen ad hoc as axioms, e.g.axioms which guarantee the existence of unions, intersections, pairs, subsets, empty set, power sets and replacement sets. It is demonstrated that a uniform syntactic description may be (...) given of acceptable instances of the comprehension schema, which include all of the axioms mentioned, and which in their turn are theorems of the usual versions of ZFS set theory. Well then, shall we proceed as usual and begin by assuming the existence of a single essential nature or Form for every set of things which we call by the same name? Do you understand? (Plato, Republic X.596a6; cf. Cornford 1966, 317). (shrink)

In this paper we deal with some new axiom schemes for Peano's Arithmetic that can substitute the classical induction, least-element, collection and strong collection schemes in the description of PA.

We introduce a new axiom scheme for constructive set theory, the Relation Reflection Scheme . Each instance of this scheme is a theorem of the classical set theory ZF. In the constructive set theory CZF–, when the axiom scheme is combined with the axiom of Dependent Choices , the result is equivalent to the scheme of Relative Dependent Choices . In contrast to RDC, the scheme RRS is preserved in Heyting-valued models (...) of CZF– using set-generated frames. We give an application of the scheme to coinductive definitions of classes. (shrink)

This book provides a systematic analysis of many common argumentation schemes and a compendium of 96 schemes. The study of these schemes, or forms of argument that capture stereotypical patterns of human reasoning, is at the core of argumentation research. Surveying all aspects of argumentation schemes from the ground up, the book takes the reader from the elementary exposition in the first chapter to the latest state of the art in the research efforts to formalize and classify the schemes, outlined (...) in the last chapter. It provides a systematic and comprehensive account, with notation suitable for computational applications that increasingly make use of argumentation schemes. (shrink)

Argumentation schemes [1–3] are a relatively recent notion that continues an extremely ancient debate on one of the foundations of human reasoning, human comprehension, and obviously human argumentation, i.e., the topics. To understand the revolutionary nature of Walton’s work on this subject matter, it is necessary to place it in the debate that it continues and contributes to, namely a view of logic that is much broader than the formalistic perspective that has been adopted from the 20th century until (...) nowadays. With his book Argumentation schemes for presumptive reasoning, Walton attempted to start a dialogue between three different fields or views on human reasoning – one (argumentation theory) very recent, one (dialectics) very ancient and with a very long tradition, and one (formal logic) relatively recent, but dominating in philosophy. Argumentation schemes were proposed as dialectical instruments, in the sense that they represented arguments not only as formal relations, but also as pragmatic inferences in the sense that they depend on what the interlocutors share and accept in a given dialogical circumstance and affect their dialogical relation. In this introduction, the notion of argumentation scheme will be analyzed in detail, showing its different dimensions and its defining features which make them an extremely useful instrument in Artificial Intelligence. This theoretical background will be followed by a literature review on the uses of the schemes in computing, aimed at identifying the most important areas and trends, the most promising proposals, and the directions of future research. (shrink)

This book provides a systematic analysis of many common argumentation schemes and a compendium of 96 schemes. The study of these schemes, or forms of argument that capture stereotypical patterns of human reasoning, is at the core of argumentation research. Surveying all aspects of argumentation schemes from the ground up, the book takes the reader from the elementary exposition in the first chapter to the latest state of the art in the research efforts to formalize and classify the schemes, outlined (...) in the last chapter. It provides a systematic and comprehensive account, with notation suitable for computational applications that increasingly make use of argumentation schemes. (shrink)

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false (...) in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down. (shrink)

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every (...) decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC00m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${AB_{1/2,0}^{2^{\mathbb{n}}}}$$\end{document}. We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd. (shrink)

Consciousness is more important than the Higgs-Bosen particle. Consciousness has emerged as a term, and a problem, in modern science. Most scientists believe that it can be accomodated and explained, by existing scientific principles. I say that it cannot, that it calls all existing principles into question, and so I propose a New Copernican Revolution among our fundamental terms. I say that consciousness points completely beyond present day science, to a whole new view of the universe, where consciousness, and not (...) matter or matter/energy is the true basis of the universe and the true fundamental term for the universe. And I go on to spell out this bold, brave and beautiful new understanding of the Universe, and with it the Earth, Spirit and Ourselves and with them a new and true foundation for Civilization itself. (shrink)

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}$ consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus ${\in}$ -induction scheme. ZFC+ “there (...) is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and ${\Pi_1^1}$ -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form ${Loc({\rm ZFC}+\phi)}$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved. (shrink)