We show that for no chain C there is a monadic-second order interpretation of the full binary tree in C.

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to Σ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^0_2$$\end{document}-induction over RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {RCA}_0$$\end{document}. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees TT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {T}\,}{\mathsf {T}}^1$$\end{document} with an extra condition on the solution tree. (shrink)

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to \-induction over \. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees \ with an extra condition on the solution tree.

Kristiansen and Murwanashyaka recently proved that Robinson arithmetic, Q, is interpretable in an elementary theory of full binary trees, T. We prove that, conversely, T is interpretable in Q by producing a formal interpretation of T in an elementary concatenation theory QT+, thereby also establishing mutual interpretability of T with several well-known weak essentially undecidable theories of numbers, strings, and sets. We also introduce a “hybrid” elementary theory of strings and trees, WQT*, and establish its mutual (...) interpretability with Robinson’s weak arithmetic R, the weak theory of trees WT of Kristiansen and Murwanashyaka, and the weak concatenation theory WTCε of Higuchi and Horihata. (shrink)

We give a sound and complete axiomatization for the full computation tree logic, CTL*, of R-generable models. This solves a long standing open problem in branching time temporal logic.

It is known that the full computation-tree logic CTL * is an important base logic for model checking. The bisimulation theorem for CTL* is known to be useful for abstraction in model checking. In this paper, the bisimulation theorems for two paraconsistent four-valued extensions 4CTL* and 4LCTL* of CTL* are shown, and a translation from 4CTL* into CTL* is presented. By using 4CTL* and 4LCTL*, inconsistency-tolerant and spatiotemporal reasoning can be expressed as a model checking framework.

Computability, while usually performed within the context of ω, may be extended to larger ordinals by means of α-recursion. In this article, we concentrate on the particular case of ω1-recursion, and study the differences in the behavior of ${\rm{\Pi }}_1^0$-classes between this case and the standard one.Of particular interest are the ${\rm{\Pi }}_1^0$-classes corresponding to computable trees of countable width. Classically, it is well-known that the analog to König’s Lemma—“every tree of countable width and uncountable height has an uncountable (...) branch”—fails; we demonstrate that not only does it fail effectively, but also that the failure is as drastic as possible. This is proven by showing that the ω1-Turing degrees of even isolated paths in computable trees of countable width are cofinal in the ${\rm{\Delta }}_1^1\,{\omega _1}$-Turing degrees.Finally, we consider questions of nonisolated paths, and demonstrate that the degrees realizable as isolated paths and the degrees realizable as nonisolated ones are very distinct; in particular, we show that there exists a computable tree of countable width so that every branch can only be ω1-Turing equivalent to branches of trees with ${\aleph _2}$-many branches. (shrink)

It is known that the full computation-tree logic CTL∗is an important base logic for model checking. The bisimulation theorem for CTL∗is known to be useful for abstraction in model checking. In this paper, thebisimulation theorems for two paraconsistent four-valued extensions 4CTL∗and 4LCTL∗of CTL∗are shown, and a translation from 4CTL∗into CTL∗ispresented. By using 4CTL∗and 4LCTL∗, inconsistency-tolerant and spatiotemporal reasoning can be expressed as a model checking framework.

Let S2S [WS2S] espectively be the storn [weak] monadic second order theory of the binary tree T in the language of two successor functions. An S2S-formula whose free variables are just individual variables defines a relation on T (rather than on the power set of T). We show that S2S and WS2S define the same relations on T, and we give a simple characterization of these relations.

The set of all words in the alphabet {l, r} forms the full binary tree T. If x ∈ T then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of T and over arbitrary subsets of T. We prove that there is no monadic second-order formula φ * (X, y) (...) such that for every nonempty subset X of T there is a unique y ∈ X that satisfies φ * (X, y) in T. (shrink)

Formal topology is today an established topic in the development of constructive mathematics and constructive proofs for many classical results of general topology have been obtained by using this approach. Here we analyze one of the main concepts in formal topology, namely, the notion of formal point. We will contrast two classically equivalent definitions of formal points and we will see that from a constructive point of view they are completely different. Indeed, according to the first definition the formal points (...) of the formal topology of the real numbers can be indexed by a set whereas this is not possible according to the second one. (shrink)

We compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree . Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π 2 0 -conservative over primitive recursive arithmetic PRA . In Kohlenbach 1239–1273) we established such conservation results relative to finite type extensions PRA ω of PRA . In this setting one can consider also a (...) uniform version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA ω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms of extensionality for all finite types. In this paper we show that in the presence of , UWKL is much stronger than WKL: whereas WKL remains to be Π 2 0 -conservative over PRA, PRA ω ++ UWKL contains full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRA ω both with and without the Markov principle. (shrink)

Jones argues that extending Seneca kin terms to second cousins requires a revised version of Optimality Theoretic grammar. I extend Seneca terms using three constraints on expression of markers in nested binary classification trees. Multiple constraint rankings on a nested set coupled with local parity checking determines how a given kin classification grammar marks structural endogamy.

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate simple term (...) algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (shrink)

This article aims to rethink the meaning of “nature” and the human in Plato, more specifically through some examples contained in the _Phaedrus_, a rare dialogue further away from the city. Phaedrus and Socrates leave Athens on a path outside the walls, past the Ilisus stream and the breeze of the woods, and end up sitting in the shadows of trees full of singing cicadas. What is the meaning of this scenario in the construction o the text? Is (...) it possible to learn from oak trees or insects? The philosophizing of this dialogue passes through the coolness of the shadows of the trees in the hot summer noon. This sensuous topography through the text reveals the writer Plato wary of the enveloping meaning of the living drama between Phaedrus and Socrates. The audience and participation of natural elements, trees and cicadas in the dialogue reveals an involvement of the discourse with the surrounding atmosphere, a special concern of Socrates in his palinode and in his final prayer to Pan. In this dialogue, more than distinct and removed from nature, we think the human being (and reason) immersed in it - in an attempt to integrate a mythological and dialogical search between river, rock, plants, animals, humans, gods and demigods. The image of an overly rationalist and humanist Plato finds in this text a perspective of an inspired writer, a hybrid poet and philosopher, with a mission to restore language and the city to animal and cosmic life. (shrink)

We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.

Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to (...) prove that the Dedekind reals form a set. Here we show that the Cauchy reals also form a set. More generally, binary refinement ensures that one remains in the realm of sets when one starts from discrete sets and one applies the operations of exponentiation and binary product a finite number of times. (shrink)

A modal logic is developed to deal with finite ordered binary trees a they are used in linguistics. A modal language is introduced with operators for the ‘mother of’, ‘first daughter of’ and ‘second daughter of’ relations together with their transitive reflexive closures. The relevant class of tree models is defined and three linguistic applications of this language are discussed: context free grammars, command relations, and trees decorated with feature structures. An axiomatic proof system is given for (...) which completeness is shown with respect to the class of finite ordered binary trees. A number of decidability results follow. (shrink)

ABSTRACT It is fruitless to interpret Constant's modern liberty from the binary perspective of either the negative/positive freedom opposition or the liberal/republican freedom opposition. Both oppositional perspectives reduce the relationally complex nature of modern liberty to one or another component of the relation. Such reduction inevitably results in an incomplete and, therefore, inadequate interpretation of Constant's modern liberty. Consequently, either of these binary frames of interpretation obscures rather than illuminates the full nature of Constant's modern liberty. Boxed (...) into their irreconcilably opposed alternatives, both binary perspectives fail to appreciate that Constant's conception of modern liberty is a complex achievement irreducible without loss to either liberal negative liberty as non-interference or republican freedom as non-domination. Nor does combining liberal negative freedom and positive freedom (in the sense of ancient liberty), as Holmes well establishes, adequately tells the whole story of Constant's modern liberty. As a complex achievement, Constant's conception of modern liberty, I shall argue, blends negative freedom as associated with neo-Roman republican freedom as non-subjection to arbitrary power, negative freedom as non-interference, associated with the liberal tradition, positive freedom in the sense of inner self-development, and positive freedom as collective self-government or civic republican freedom. (shrink)

In this paper we show the adequacy of tense logic with unary operators for dealing with finite trees. We prove that models on finite trees can be characterized by tense formulas, and describe an effective method to find an axiomatization of the theory of a given finite tree in tense logic. The strength of the characterization is shown by proving that adding the binary operators "Until" and "Since" to the language does not result in a better description (...) than that given by unary tense logic; although the greater expressive power of "Until" and "Since" can be exploited by using the semantics of e-frames instead of traditional Kripke semantics. (shrink)

The first collection of Leibniz's key writings on the binary system, newly translated, with many previously unpublished in any language. -/- The polymath Gottfried Wilhelm Leibniz (1646–1716) is known for his independent invention of the calculus in 1675. Another major—although less studied—mathematical contribution by Leibniz is his invention of binary arithmetic, the representational basis for today's digital computing. This book offers the first collection of Leibniz's most important writings on the binary system, all newly translated by the (...) authors with many previously unpublished in any language. Taken together, these thirty-two texts tell the story of binary as Leibniz conceived it, from his first youthful writings on the subject to the mature development and publication of the binary system. -/- As befits a scholarly edition, Strickland and Lewis have not only returned to Leibniz's original manuscripts in preparing their translations, but also provided full critical apparatus. In addition to extensive annotations, each text is accompanied by a detailed introductory “headnote” that explains the context and content. Additional mathematical commentaries offer readers deep dives into Leibniz's mathematical thinking. The texts are prefaced by a lengthy and detailed introductory essay, in which Strickland and Lewis trace Leibniz's development of binary, place it in its historical context, and chart its posthumous influence, most notably on shaping our own computer age. (shrink)

Two new modes of thinking have spread through systematics in the twentieth century. Both have deep historical roots, but they have been widely accepted only during this century. Population thinking overtook the field in the early part of the century, culminating in the full development of population systematics in the 1930s and 1940s, and the subsequent growth of the entire field of population biology. Population thinking rejects the idea that each species has a natural type (as the earlier essentialist (...) view had assumed), and instead sees every species as a varying population of interbreeding individuals. Tree thinking has spread through the field since the 1960s with the development of phylogenetic systematics. Tree thinking recognizes that species are not independent replicates within a class (as earlier group thinkers had tended to see them), but are instead interconnected parts of an evolutionary tree. It lays emphasis on the explanation of evolutionary events in the context of a tree, rather than on the states exhibited by collections of species, and it sees evolutionary history as a story of divergence rather than a story of development. Just as population thinking gave rise to the new field of population biology, so tree thinking is giving rise to the new field of phylogenetic biology. (shrink)

We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...) always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. (shrink)

We examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surprising result that on such sets, logic with three variables is as powerful in expression as full first order logic.

While Bayesian methods have become very popular in phylogenetic systematics, the foundations of this approach remain controversial. The star-tree paradox in Bayesian phylogenetics refers to the phenomenon that a particular binary phylogenetic tree sometimes has a very high posterior probability even though a star tree generates the data. I argue that this phenomenon reveals an unattractive feature of the Bayesian approach to scientific inference and discuss two proposals for how to address the star-tree paradox. In particular, I defend the (...) polytomy prior as a solution of the paradox and argue that it is preferable to a data-size dependent branch lengths prior from a methodological perspective. However, while this reply dissolves the star-tree paradox, the general challenge to Bayesian confirmation theory remains unmet. (shrink)

We show, relative to the base theory RCA₀: A nontrivial tree satisfies Ramsey's Theorem only if it is biembeddable with the complete binary tree. There is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA₀. Ramsey's Theorem for singletons for the complete binary tree is stronger than $B\sum_{2}^{0}$ , hence stronger than Ramsey's Theorem for singletons for ω. These results lead to extensions of results, or answers to questions, of Chubb, Hirst, and (...) McNicholl [3]. (shrink)

Pereira’s “The Projective Theory of Consciousness” is an experimental statement, drawing on many diverse sources, exploring how consciousness might be produced by a projective mechanism that results both in private selves and an experienced world. Unfortunately, pulling together so many unrelated sources and methods means none gets full attention. Furthermore, it seems to me that the uncomfortable breadth of this paper unnecessarily complicates his project; in fact it may hide what it seeks to reveal. If this conglomeration of diverse (...) sources and methods were compared to trees, the reader may feel like the explorer who cannot see the forest for the trees. Then again, it may be the author who is so preoccupied with foreground figures that the everpresent background is ultimately obscured. (shrink)

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

In lieu of an abstract, here is a brief excerpt of the content:Women Under the Bo TreeLucinda Joy PeachWomen Under the Bo Tree. By Tessa Bartholomeusz. Cambridge, Great Britain: Cambridge University Press, 1994. xx + 284 pp.Tessa Bartholomeusz has made an important contribution to our understanding of Buddhist women with her carefully researched study of the emergence of “pious lay women” or “lay female renunciant” (upasika) as a new category of Buddhists in nineteenth- and twentieth-century Sri Lanka. Bartholomeusz focuses on (...) the period of the nineteenth and twentieth centuries as this period was not only one of a Buddhist revival in Sri Lanka, partially in reaction against European colonialism and its imposition of Christianity on the island, but also the period during which the upasika role developed.Part One of the book traces the historical background of the tradition of female renunciation within Buddhism and its reemergence in the nineteenth century. Part Two describes twentieth-century developments in this tradition up to the date of publication (1994). The author uses a wide variety of documentary sources to describe the various factors that contributed to the emergence of the category of female lay renunciant in Sri Lanka beginning in the nineteenth century: the exclusion of women from full ordination; the rise of Protestant Buddhism, with its belief in the responsibility of the laity for both the welfare of Buddhism and their own salvation (p. 24); the promotion of Buddhism as an anti-Western and anticolonialist [End Page 218] strategy by nationalistic Ceylonese; and the disestablishment of Buddhism, which created a less close-knit relationship between the monastic institution (sangha) and the state.Bartholomeusz does a fine job of describing the relationship between gender stereotypes of women in Sri Lankan society and Buddhist scriptures on the one hand and the actual roles of the lay renunciant women she studies on the other. She describes how the prevailing sexism of Sri Lankan society (and of the Indian society of early Buddhism prior to that) created obstacles for Buddhist women wishing to renounce the world—they are expected to be wives and mothers, and to support the male sangha. Bartholomeusz persuasively describes how the category of upasika developed to fill the gap between householder and full monastic roles for women. The order of fully ordained Buddhist nuns (bhikkunis)—who take ten monastic precepts (dasa sila) in addition to 311 monastic Vinaya rules—established at the time of the Buddha disappeared in Sri Lanka in the eleventh century (and eventually in all Theravada Buddhist countries). Consequently, Buddhist women in these countries are essentially limited to taking lay vows—or taking ordination from nuns in the Mahayana lineage, which, as Bartholomeusz points out, is also quite controversial, especially because it contradicts the view of Sri Lanka as “the bastion of orthodox Buddhism” (p. 14).Lay renunciant women have carefully negotiated their status in relation to both the traditional roles of women in Sri Lankan Buddhism as well as the nineteenth- and twentieth-century efforts by reformers to modernize those roles. Bartholomeusz observes that while maintaining certain elements of women’s traditional social roles as caregivers and nurturers (e.g., p. 152), lay renunciant women have also broken with the traditions of recent centuries by refusing to be satisfied with lives as householders and insisting on their right to renounce lay life—and most in the absence of full monastic vows. It is unfortunate that Bartholomeusz does not address in any depth the implications of the “liminal” status of women lay renunciants (dasa sila mattas) for the status of women in Sri Lanka more generally. Nor does the author provide her own view of whether this intermediate status has provided Sri Lankan women with more or less opportunities and freedoms than held by their Buddhist sisters in other Buddhist countries, especially in Mahayana cultures where full ordination for women exists.Jan Nattier’s analysis of status dissonance and hierarchy in early Indian Buddhism 1 provides a useful framework for considering the various hierarchies of status that Bartholomeusz describes in connection with lay renunciant women. The material that Bartholomeusz presents complexifies this typology. Besides the categories of gender, caste, and seniority that Nattier finds in early Indian Buddhist sutras, Bartholomeusz points to several other status hierarchies, some... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Women Under the Bo TreeLucinda Joy PeachWomen Under the Bo Tree. By Tessa Bartholomeusz. Cambridge, Great Britain: Cambridge University Press, 1994. xx + 284 pp.Tessa Bartholomeusz has made an important contribution to our understanding of Buddhist women with her carefully researched study of the emergence of “pious lay women” or “lay female renunciant” (upasika) as a new category of Buddhists in nineteenth- and twentieth-century Sri Lanka. Bartholomeusz focuses on (...) the period of the nineteenth and twentieth centuries as this period was not only one of a Buddhist revival in Sri Lanka, partially in reaction against European colonialism and its imposition of Christianity on the island, but also the period during which the upasika role developed.Part One of the book traces the historical background of the tradition of female renunciation within Buddhism and its reemergence in the nineteenth century. Part Two describes twentieth-century developments in this tradition up to the date of publication (1994). The author uses a wide variety of documentary sources to describe the various factors that contributed to the emergence of the category of female lay renunciant in Sri Lanka beginning in the nineteenth century: the exclusion of women from full ordination; the rise of Protestant Buddhism, with its belief in the responsibility of the laity for both the welfare of Buddhism and their own salvation (p. 24); the promotion of Buddhism as an anti-Western and anticolonialist [End Page 218] strategy by nationalistic Ceylonese; and the disestablishment of Buddhism, which created a less close-knit relationship between the monastic institution (sangha) and the state.Bartholomeusz does a fine job of describing the relationship between gender stereotypes of women in Sri Lankan society and Buddhist scriptures on the one hand and the actual roles of the lay renunciant women she studies on the other. She describes how the prevailing sexism of Sri Lankan society (and of the Indian society of early Buddhism prior to that) created obstacles for Buddhist women wishing to renounce the world—they are expected to be wives and mothers, and to support the male sangha. Bartholomeusz persuasively describes how the category of upasika developed to fill the gap between householder and full monastic roles for women. The order of fully ordained Buddhist nuns (bhikkunis)—who take ten monastic precepts (dasa sila) in addition to 311 monastic Vinaya rules—established at the time of the Buddha disappeared in Sri Lanka in the eleventh century (and eventually in all Theravada Buddhist countries). Consequently, Buddhist women in these countries are essentially limited to taking lay vows—or taking ordination from nuns in the Mahayana lineage, which, as Bartholomeusz points out, is also quite controversial, especially because it contradicts the view of Sri Lanka as “the bastion of orthodox Buddhism” (p. 14).Lay renunciant women have carefully negotiated their status in relation to both the traditional roles of women in Sri Lankan Buddhism as well as the nineteenth- and twentieth-century efforts by reformers to modernize those roles. Bartholomeusz observes that while maintaining certain elements of women’s traditional social roles as caregivers and nurturers (e.g., p. 152), lay renunciant women have also broken with the traditions of recent centuries by refusing to be satisfied with lives as householders and insisting on their right to renounce lay life—and most in the absence of full monastic vows. It is unfortunate that Bartholomeusz does not address in any depth the implications of the “liminal” status of women lay renunciants (dasa sila mattas) for the status of women in Sri Lanka more generally. Nor does the author provide her own view of whether this intermediate status has provided Sri Lankan women with more or less opportunities and freedoms than held by their Buddhist sisters in other Buddhist countries, especially in Mahayana cultures where full ordination for women exists.Jan Nattier’s analysis of status dissonance and hierarchy in early Indian Buddhism 1 provides a useful framework for considering the various hierarchies of status that Bartholomeusz describes in connection with lay renunciant women. The material that Bartholomeusz presents complexifies this typology. Besides the categories of gender, caste, and seniority that Nattier finds in early Indian Buddhist sutras, Bartholomeusz points to several other status hierarchies, some... (shrink)

The two‐dimensional version of the classical Mycielski theorem says that for every comeager or conull set there exists a perfect set such that. We consider a strengthening of this theorem by replacing a perfect square with a rectangle, where A and B are bodies of some types of trees with. In particular, we show that for every comeager Gδ set there exist a Miller tree and a uniformly perfect tree such that and that cannot be a Miller tree. In (...) the case of measure we show that for every subset F of of full measure there exists a uniformly perfect tree such that and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions from which we derive nonstandard proofs of Mycielski‐like theorems via the Shoenfield Absoluteness Theorem. (shrink)

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, (...) we seek to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory. (shrink)

Genetic modification of trees has the potential to change our forests forever, yet there has been little publicly available information or debate on this important topic. Ethical analysis of genetic modification of plants to date has been focussed mainly on food and feed crops and pharmaceutical production. The purpose of this paper is to examine one major ethical issue arising in connection with the genetic modification of trees, the necessity to examine the practice in its full scientific, (...) social, economic, political and environmental context. It is argued that if this is not done, important factors that play a role in a through ethical critique will be overlooked. The analysis focuses mainly on the Canadian situation, but there are implications for other timber-producing nations as well. The paper concludes that a thorough ethical critique must take into account the ends for which biotechnology is being pursued. (shrink)

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most ω, the resulting logics (...) are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel-Dummett logics with quantifiers over propositions. (shrink)

In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2×2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students significantly more in Bayesian reasoning problems than probability visualizations do. Because tree diagrams or double-trees (which are largely unknown in school) are node-branch-structures, these two visualizations (compared to the 2×2 table) can even (...) simultaneously display probabilities on branches and frequencies inside the nodes. This is a teaching advantage as it allows the frequency concept to be used to better understand probabilities. However, 2×2 tables and (double-) trees have a decisive disadvantage: While joint probabilities (e.g., P(AB)) are represented in 2×2 tables but no conditional probabilities (e.g., P(A|B)), it is exactly the other way around with (double-) trees. Therefore, a visualization that is equally suitable for the representation of joint probabilities and conditional probabilities is desirable. In this article, we present a new visualization—the frequency net—in which all absolute frequencies and all types of probabilities can be depicted. In addition to a detailed theoretical analysis of the frequency net, we report the results of a study with 249 university students that shows that “net diagrams” can improve reasoning without previous instruction to a similar extent as 2×2 tables and double-trees. Regarding questions about conditional probabilities, frequency visualizations (2×2 table, double-tree, or net diagram with natural frequencies) are consistently superior to probability visualizations. The frequency net performs as well as the frequency double-tree. Only the 2×2 table with frequencies—the only visualization that participants were already familiar with —led to higher performance rates. If, on the other hand, a question about a joint probability had to be answered, all implemented visualizations clearly supported participants’ performance, and no uniform format effect becomes visible. Participants reached the highest performance in the versions with probability 2×2 tables and probability net diagrams. Furthermore, after conducting a detailed error analysis, we report interesting error shifts between the two information formats and the different visualizations and give recommendations for teaching probability. (shrink)

We consider the structure of all labeled trees, called also infinite terms, in the first order language with function symbols in a recursive signature of cardinality at least two and at least a symbol of arity two, with equality and a binary relation symbol which is interpreted to be the subtree relation. The existential theory over of this structure is decidable (see Tulipani [9]), but more complex fragments of the theory are undecidable. We prove that the theory of (...) the structure is in , where formulas are those in prenex form consisting of a string of unbounded existential quantifiers followed by a string of arbitrary quantifiers all bounded with respect to . Since the fragment of the theory was already known to be -hard (see Marongiu and Tulipani [5]), it is now established to be -complete. (shrink)

We consider the structure $IT_S$ of all labeled trees, called also infinite terms, in the first order language ${\cal L}$ with function symbols in a recursive signature $S$ of cardinality at least two and at least a symbol of arity two, with equality and a binary relation symbol $\sqsubseteq$ which is interpreted to be the subtree relation. The existential theory over ${\cal L}$ of this structure is decidable, but more complex fragments of the theory are undecidable. We prove (...) that the $ \exists\Delta$ theory of the structure is in $\Sigma^1_1$, where $ \exists\Delta$ formulas are those in prenex form consisting of a string of unbounded existential quantifiers followed by a string of arbitrary quantifiers all bounded with respect to $\sqsubseteq$. Since the fragment of the theory was already known to be $\Sigma^1_1$-hard, it is now established to be $\Sigma^1_1$-complete. (shrink)

Mental-threshold egalitarianism, well-known examples of which include Jeff McMahan’s two-tiered account of the wrongness of killing and Tom Regan’s theory of animal rights, divides morally considerable beings into equals and unequals on the basis of their individual mental capacities. In this paper, I argue that the line that separates equals from unequals is unavoidably arbitrary and implausibly associates an insignificant difference in empirical reality with a momentous difference in moral status. In response to these objections, McMahan has proposed the introduction (...) of an intermediate moral status. I argue that this move ultimately fails to address the problem. I conclude that, if we are not prepared to give up moral equality, our full and equal moral status must be grounded in a binary property that is not a threshold property. I tentatively suggest that the capacity for phenomenal consciousness is such a property, and a plausible candidate. (shrink)

The recognition that female embodiment and feminine experience are legitimate and specific sites of the revelation of God’s love has been one of the most significant developments in theology in the last hundred years. However, an over-emphasis on feminine experience as supervening on female embodiment risks erasing unusual sex-gender body-stories and perpetuating the idea that only some bodies can mediate the divine. Feminist Theology’s future must involve a re-examination and re-negotiation of what it is to be feminist theologians without fixed (...) gender essences. Does Feminist Theology have space to hear from and nurture the voices of those whose gender experiences challenge a binary, either-or model? Can Feminist Theology, in contrast to much secular feminist theory, give space at the table to those whose sex-gender life stories undermine the notion that there is such a thing as a common or biologically-contingent feminine experience in the first place? (shrink)

In this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation (...) system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t( $\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ , has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for $t(\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ is not fundamental, we can use the failed proof search to build a relational countermodel for $t(\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ and from this, build a RelKripke countermodel for $\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha$ . These results allow us to add FL, the basic substructural logic, to the list of those logies of importance in computer science with a relational proof theory. (shrink)

Excerpt: In the lecture What Pragmatism Means, William James gives us what became one of the most famous examples of strengths of the pragmatic method. Instead of beginning with an argument, he provides a story. In this story, James and several of his friends are on a camping trip when a “ferocious metaphysical dispute” arises concerning the movements of a squirrelii. A squirrel, the story goes, clings the one side of a tree-trunk, and on the other side a man tries (...) to catch a glimpse of it, by rapidly chasing it around the tree. However, the squirrel moves just as fast as the man does, always keeping the tree between itself and his pursuer – and the man never sees the squirrel. So, why does James call it a “ferocious metaphysical dispute”? The campers ask: while we can concede that the man has circled completely around the tree, has he gone around the squirrel? Some of the campers say yes, others no, and an argument ensues over who is correct. (shrink)

A new construction of a certain conceptual space is presented. Elements of this conceptual space correspond to concept elements of reality, which potentially comprise an infinite number of qualities. This construction of a conceptual space solves a problem stated by Dietz and his co-authors in 2013 in the context of Voronoi diagrams. The fractal construction of the conceptual space is that this problem simply does not pose itself. The concept of convexity is discussed in this new conceptual space. Moreover, the (...) meaning of convexity is discussed in full generality, for example when space is deprived of it, its substitutes for concept domains are considered. (shrink)