This chapter contains sections titled: Categorical Prescriptiveness Uniformity as a Moral Matter Uniformity Contrasted with Neutrality The Overridingness of Moral Principles.

A uniform theory of conditionals is one which compositionally captures the behavior of both indicative and subjunctive conditionals without positing ambiguities. This paper raises new problems for the closest thing to a uniform analysis in the literature (Stalnaker, Philosophia, 5, 269–286 (1975)) and develops a new theory which solves them. I also show that this new analysis provides an improved treatment of three phenomena (the import-export equivalence, reverse Sobel-sequences and disjunctive antecedents). While these results concern central issues in (...) the study of conditionals, broader themes in the philosophy of language and formal semantics are also engaged here. This new analysis exploits a dynamic conception of meaning where the meaning of a symbol is its potential to change an agent’s mental state (or the state of a conversation) rather than being the symbol’s content (e.g. the proposition it expresses). The analysis of conditionals is also built on the idea that the contrast between subjunctive and indicative conditionals parallels a contrast between revising and consistently extending some body of information. (shrink)

The pessimistic induction is built upon the uniformity principle that the future resembles the past. In daily scientific activities, however, scientists sometimes rely on what I call the disuniformity principle that the future differs from the past. They do not give up their research projects despite the repeated failures. They believe that they will succeed although they failed repeatedly, and as a result they achieve what they intended to achieve. Given that the disuniformity principle is useful in certain cases in (...) science, we might reasonably use it to infer that present theories are true unlike past theories. Hence, pessimists have the burden to show that our prediction about the fate of present theories is more likely to be true if we use the uniformity principle than if we use the disuniformity principle. (shrink)

This paper defines the form of prior knowledge that is required for sound inferences by analogy and single-instance generalizations, in both logical and probabilistic reasoning. In the logical case, the first order determination rule defined in Davies (1985) is shown to solve both the justification and non-redundancy problems for analogical inference. The statistical analogue of determination that is put forward is termed 'uniformity'. Based on the semantics of determination and uniformity, a third notion of "relevance" is defined, both logically and (...) probabilistically. The statistical relevance of one function in determining another is put forward as a way of defining the value of information: The statistical relevance of a function F to a function G is the absolute value of the change in one's information about the value of G afforded by specifying the value of F. This theory provides normative justifications for conclusions projected by analogy from one case to another, and for generalization from an instance to a rule. The soundness of such conclusions, in either the logical or the probabilistic case, can be identified with the extent to which the corresponding criteria (determination and uniformity) actually hold for the features being related. (shrink)

Uniformism about the epistemology of modality is the view that there is only one basic route to modal knowledge; non-uniformism is the view that there are several. Non-uniformism is becoming an increasingly popular stance, but how can it be defended? I prise apart two ways of understanding the uniformism/non-uniformism conflict that are mixed up in the literature. I argue that once separated, it is evident that they lead up to two different non-uniformist theses that need to be argued for in (...) very different ways. In particular, one construal allows only for a weak thesis about the actual lay of the current theoretical land in modal epistemology, while the other allows for a strong thesis that can be defended without reliance on claims about the alleged goodness or badness of individual accounts of modal justification. I argue that philosophers inclined towards non-uniformism ought to go for the strong, rather than the weak, thesis. (shrink)

Can rational communication proceed when interlocutors are uncertain which contents utterances contribute to discourse? An influential negative answer to this question is embodied in the Stalnakerian principle of uniformity, which requires speakers to produce only utterances that express the same content in every possibility treated as live for the purposes of the conversation. The principle of uniformity enjoys considerable intuitive plausibility and, moreover, seems to follow from platitudes about assertion; nevertheless, it has recently proven controversial. In what follows, I defend (...) the principle by developing two arguments for it based on premises reflecting the central aims and assumptions of possibility-carving frameworks for modeling inquiry—that is, frameworks which describe the evolution of individuals’ attitudinal states in terms of set-theoretic operations defined over a domain of objects representing possibilities. (shrink)

This paper presents a uniform general account of regress problems in the form of a pentalemma—i.e., a set of five mutually inconsistent claims. Specific regress problems can be analyzed as instances of such a general schema, and this Regress Pentalemma Schema can be employed to generate deductively valid arguments from the truth of a subset of four claims to the falsity of the fifth. Thus, a uniform account of the nature of regress problems allows for an improved understanding (...) of specific regress objections or arguments, and, correspondingly, of the general logical geography of the debate about infinite regresses. This uniform approach is illustrated by a treatment of the classical epistemological problem of justification, but it encompasses a whole variety of cases including explanation and ontological grounding. Furthermore, this general account is compared and contrasted with the existing literature discussing argument schemata for regress objections, particularly with the work of Jan Willem Wieland. It is shown how such other schemata can be incorporated and superseded by the general Regress Pentalemma Schema. (shrink)

This special issue of Cognitive Linguistics reexamines the notions of uniformity and variability within morphological systems from a cognitive linguistic standpoint. It challenges traditional perspectives that regard morphological variability as mere deviations from the norm, suggesting instead that such variability is systematic and shaped by external influences including language acquisition and processing constraints. The contributions in this issue promote a shift from isolated analysis to a holistic view of paradigms, classes, and systems, advocating for a framework where morphological structures are (...) seen as integral to communicative and functional aspects of language. By accounting for the broad adaptive dynamics of language systems, the complex interplay between uniformity and variability is revealed as an inherent aspect of language usage. (shrink)

Allen, B., Arithmetizing Uniform NC, Annals of Pure and Applied Logic 53 1–50. We give a characterization of the complexity class Uniform NC as an algebra of functions on the natural numbers which is the closure of several basic functions under composition and a schema of recursion. We then define a fragment of bounded arithmetic, and, using our characterization of Uniform NC, show that this fragment is capable of proving the totality of all of the functions in (...) Uniform NC. Lastly, in the spirit of Buss, we show that any function which is definable by a ∑b1-formula in our theory is a function which is in Uniform NC. (shrink)

Chris Heathwood requires the sentence 'Caesar was conscious when he crossed the Rubicon' to be made true in much the same way as 'Caesar was wet when he crossed the Rubicon'. Yet because the Growing Block theorist is committed to the zombiedom of the past,the former is not made true by past objects, although the latter is. Heathwood demands a uniform account of the grounding of truths and he will be given a uniform account. But we should exercise (...) care in deciding just what sort of uniformity is appropriate. As Russell so famously pointed out a century ago the subject/predicate form of a sentence can be misleading. Likewise although the two sentences 'Caesar is conscious' and 'Caesar is wet' have similar subject/predicate forms they have, I say, different kinds of truth-conditions and hence their past tense transformations also have different kinds of truth-conditions. The uniformity I endorse is that in both cases the grounds for the past tense transformation are the same as the grounds the present tense versions used to have when they were true. (shrink)

In this paper we prove that the preordering $\lesssim $ of provable implication over any recursively enumerable theory $T$ containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function $F$ for $\lesssim $. A recursive function $F$ is a density function if it computes, for $A$ and $B$ with $A\lnsim B$, an element $C$ such that $A\lnsim C\lnsim B$. The function is extensional if it preserves $T$-provable equivalence. Secondly, we prove a (...) general result that implies that, for extensions of elementary arithmetic, the ordering $\lesssim $ restricted to $\Sigma_{n}$-sentences is uniformly dense. In the last section we provide historical notes and background material. (shrink)

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitution core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class of all relational models (...) with infinitely many agents, PAL-K_omega, as well as standard extensions thereof, e.g., PAL-T_omega, PAL-S4_omega, and PAL-S5_omega. We introduce a new Uniform Public Announcement Logic (UPAL), prove completeness of a deductive system with respect to UPAL semantics, and show that this system axiomatizes the substitution core of PAL. (shrink)

Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. (...) The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed. (shrink)

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which (...) admits local definable cell decomposition. (shrink)

Philosophers often make exotic-sounding modal claims, such as: “A timeless world is impossible”, “The laws of physics could have been different from what they are”, “There could have been an additional phenomenal colour”. Otherwise popular empiricist modal epistemologies in the contemporary literature cannot account for whatever epistemic justification we might have for making such modal claims. Those who do not, as a result of this, endorse scepticism with respect to their epistemic status typically suggest that they can be justified but (...) have yet to develop some distinct, workable theory of how. That is, they endorse a form of non-uniformism about the epistemology of modality, according to which claims about philosophically interesting modal matters need to be justified differently from e.g. everyday or scientific modal claims, but they fail to provide any more detail. This article aims to fill this gap by outlining how such a non-uniformist view could be spelled out and what story about philosophically interesting modal justification it could contain. (shrink)

In this paper, a proof-theoretic method to prove uniform Lyndon interpolation for non-normal modal logics is introduced and applied to show that the logics E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {E}$$\end{document}, M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {M}$$\end{document}, MC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MC}$$\end{document}, EN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EN}$$\end{document}, MN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MN}$$\end{document} have (...) that property. In particular, these logics have uniform interpolation. Although for some of them the latter is known, the fact that they have uniform Lyndon interpolation is new. Also, the proof-theoretic proofs of these facts are new, as well as the constructive way to explicitly compute the interpolants that they provide. It is also shown that the non-normal modal logics EC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EC}$$\end{document} and ECN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ECN}$$\end{document} do not have Craig interpolation, and whence no uniform interpolation. (shrink)

We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.

Let ℋ be a finite-dimensional complex Hilbert space and D the set of density matrices on ℋ, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on D that can be regarded as the uniform distribution over D. We propose a measure on D, argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to (...) this measure. (shrink)

In this paper, we discuss uniform versions of some axioms of second order arithmetic in the context of higher order arithmetic. We prove that uniform versions of weak weak König's lemma WWKL and Σ01 separation are equivalent to over a suitable base theory of higher order arithmetic, where is the assertion that there exists Φ2 such that Φf1 = 0 if and only if ∃x0 for all f. We also prove that uniform versions of some well-known theorems (...) are equivalent to or the axiom of the existence of the Suslin operator. (shrink)

The principle of uniformity of nature has sometimes been invoked for the purpose of justifying induction. This principle cannot be established "a priori", And in the absence of a justification of induction, It cannot be established "a posteriori". There is no justification for assuming it as a postulate of science. Use of such a principle is, However, Neither sufficient nor necessary for a justification of induction. In any plausible form, It is too weak for that purpose, And hence, It is (...) insufficient. Since a justification which does not rely upon this principle can be given, It is not necessary. (shrink)

A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic \. New is the result that \ has uniform interpolation. The results imply that for modal logics \ and \, which are known not to have uniform interpolation, certain sequent (...) calculi cannot exist. (shrink)

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform (...) definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

We prove uniformly computable versions of the Implicit Function Theorem in its differentiable and non-differentiable forms. We show that the resulting operators are not computable if information about some of the partial derivatives of the implicitly defining function is omitted. Finally, as a corollary, we obtain a uniformly computable Inverse Function Theorem, first proven by M. Ziegler.

We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles are interpreted while maintaining unmoved the -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.

Barbara Fried takes the view that uniform taxation—that is, a single rate applicable to all income levels—cannot be defended on any grounds of justice. She goes further by saying that, of all possible rate structures, it might be “the hardest one”? to ground in “a”? theory of fairness. Using the contractarian-constitutional perspective advanced by John Rawls and James Buchanan, this article argues that tax uniformity can be seen as a requirement of justice. After modelling how the political world realistically (...) decides to distribute tax shares (self-interested parties act under a majority constraint), I show how the uniformity principle could emerge from the constitutional contract. In other words, rational individuals would choose uniformity as a procedural constraint under a “veil of uncertainty”?; that is, with limited knowledge regarding their positions under the future application of the rule. Moreover, I elucidate how the uniformity requirement integrates generalized criteria of fairness and efficiency into fiscal politics as it precludes fiscal exploitation and constrains majorities, and their most influential subgroups, to opt for policies in the direction of the Pareto frontier, and as such promotes outcomes acceptable to all participants. (shrink)

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.

S There is a long tradition of representing causal relationships by directed acyclic graphs (Wright, 1934 ). Spirtes ( 1994), Spirtes et al. ( 1993) and Pearl & Verma ( 1991) describe procedures for inferring the presence or absence of causal arrows in the graph even if there might be unobserved confounding variables, and/or an unknown time order, and that under weak conditions, for certain combinations of directed acyclic graphs and probability distributions, are asymptotically, in sample size, consistent. These results (...) are surprising since they seem to contradict the standard statistical wisdom that consistent estimators of causal effects do not exist for nonrandomised studies if there are potentially unobserved confounding variables. We resolve the apparent incompatibility of these views by closely examining the asymptotic properties of these causal inference procedures. We show that the asymptotically consistent procedures are ‘pointwise consistent’, but ‘uniformly consistent’ tests do not exist. Thus, no finite sample size can ever be guaranteed to approximate the asymptotic results. We also show the nonexistence of valid, consistent confidence intervals for causal effects and the nonexistence of uniformly consistent point estimators. Our results make no assumption about the form of the tests or estimators. In particular, the tests could be classical independence tests, they could be Bayes tests or they could be tests based on scoring methods such as  or . The implications of our results for observational studies are controversial and are discussed briefly in the last section of the paper. The results hinge on the following fact: it is possible to find, for each sample size n, distributions P and Q such that P and Q are empirically indistinguishable and yet P and Q correspond to different causal effects. (shrink)

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

It is consistent that for many cardinals λ there is a family of at least λ + unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵ ω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.

Our paper addresses the following question: Is there a general characterization, for all predicates P that take both declarative and interrogative complements , of the meaning of the P-interrogative clause construction in terms of the meaning of the P-declarative clause construction? On our account, if P is a responsive predicate and Q a question embedded under P, then the meaning of ‘P + Q’ is, informally, “to be in the relation expressed by P to some potential complete answer to Q”. (...) We show that this rule allows us to derive veridical and non-veridical readings of embedded questions, depending on whether the embedding verb is veridical or not, and provide novel empirical evidence supporting the generalization. We then enrich our basic proposal to account for the presuppositions induced by the embedding verbs, as well as for the generation of intermediate exhaustive readings of embedded questions. (shrink)

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of (...) information necessary to compute the factorization of u. Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation. (shrink)

The aim of this paper is the analysis of uniformity in Feferman's explicit mathematics. The proof-strength of those systems for constructive mathematics is determined by reductions to subsystems of second-order arithmetic: If uniformity is absent, the method of standard structures yields that the strength of the join axiom collapses. Systems with uniformity and join are treated via cut elimination and asymmetrical interpretations in standard structures.

We present an extension of Heyting arithmetic in finite types called Uniform Heyting Arithmetic that allows for the extraction of optimized programs from constructive and classical proofs. The system has two sorts of first-order quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripke-style Friedman/Dragalin translation which is inspired by work of Coquand and Hofmann and (...) a variant of the refined A-translation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical first-order proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program. (shrink)

Spirtes, Glymour and Scheines [Causation, Prediction, and Search Springer] described a pointwise consistent estimator of the Markov equivalence class of any causal structure that can be represented by a directed acyclic graph for any parametric family with a uniformly consistent test of conditional independence, under the Causal Markov and Causal Faithfulness assumptions. Robins et al. [Biometrika 90 491–515], however, proved that there are no uniformly consistent estimators of Markov equivalence classes of causal structures under those assumptions. Subsequently, Kalisch and B¨uhlmann (...) [J. Mach. Learn. Res. 8 613–636] described a uniformly consistent estimator of the Markov equivalence class of a linear Gaussian causal structure under the Causal Markov and Strong Causal Faithfulness assumptions. However, the Strong Faithfulness assumption may be false with high probability in many domains. We describe a uniformly consistent estimator of both the Markov equivalence class of a linear Gaussian causal structure and the identifiable structural coefficients in the Markov equivalence class under the Causal Markov assumption and the considerably weaker k-Triangle-Faithfulness assumption. (shrink)

Au cours du xixe siècle, porter l’uniforme est devenu un élément constitutif de l’identité militaire, particulièrement en France où le service militaire s’est progressivement universalisé à la veille de la Première Guerre mondiale. Objet matériel doté de fonctions symboliques, l’uniforme introduit l’historien au cœur du fonctionnement d’un milieu social qui est alors l’un des laboratoires de la masculinité. Décrit, critiqué, modifié par les médecins militaires soucieux de fonctionnalité et de bien-être, il donne à voir un corps qui demeure cependant largement (...) façonné, redressé par une armature de tissu dissimulant une réalité beaucoup moins idéale. Son port est le fruit d’un long apprentissage qui agrège le jeune soldat à la communauté masculine qui l’entoure. Offert au regard d’autrui, à celui des femmes tout particulièrement, il construit une identité masculine en quête de reconnaissance et se trouve situé, par conséquent, au cœur des processus de construction de la différence sexuée. (shrink)

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.

Non-uniformed combat morally infringes on civilians’ fundamental right to immunity and exacts an impermissible form of unofficial conscription that is morally prohibited even if the civilians knowingly consent to it. It is often argued that revolutionary groups burdened by resource disparities relative to the state or who claim alternative sources of political legitimacy (such as national self-determination or the constitution of a political collective) are justified in using unconventional tactics such as non-uniformed combat. Neither those reasons nor the provision of (...) public goods, however, are sufficient to justify such rights violations and this form of conscription, and it calls into question the suitability of current international legal protections for the non-uniformed. (shrink)

Abstract. A historical perspective allows for a different view on the compatibility of theistic views with a crucial foundation of modern scientific practice: the uniformity of nature, which states that the laws of nature are unbroken through time and space. Uniformity is generally understood to be part of a worldview called “scientific naturalism,” in which there is no room for divine forces or a spiritual realm. This association comes from the Victorian era, but a historical examination of scientists from that (...) period shows that uniformity was an important part of both theistic and naturalistic worldviews. Victorian efforts to maintain the viability of miracles and divine action within a universe ruled by natural laws receives special attention. The methodological practices of theistic and naturalistic scientists in the nineteenth century were effectively indistinguishable despite each group's argument that uniformity was closely dependent on their worldview. This similarity is used to reexamine both the reasons for the decline of the role of religion within the scientific community and claims made by the intelligent design movement about the relationship of science and religion. (shrink)

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this (...) additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations. (shrink)

In this paper, we propose a new concept named the uniform single valued neutrosophic graph. An illustrative example and some properties are examined. Next, we develop an algorithmic approach for computing the complement of the single valued neutrosophic graph. A numerical example is demonstrated for computing the complement of single valued neutrosophic graphs and uniform single valued neutrosophic graph.

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 (...) class='Hi'>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)

We give an optimal lower bound in terms of large cardinal axioms for the logical strength of projective uniformization in conjuction with other regularity properties of projective sets of real numbers, namely Lebesgue measurability and its dual in the sense of category . Our proof uses a projective computation of the real numbers which code inital segments of a core model and answers a question in Hauser.

According to some, the historian must for working purposes assume that nature is uniform, i.e., that miracles do not occur. For otherwise, it is suggested, he may place no confidence in the historical reliability of the records and artifacts on which he relies: such confidence can exist only where it is assumed, for example, that ink marks in the form of words do not sometimes appear spontaneously on old bits of paper. In this article I spell out this methodological (...) thesis in greater detail and determine to what extent it should be accepted. (shrink)

Defining a class of sets to be uniform Δ02 if it is derived from a binary {0,1}{0,1}-valued function f≤TKf≤TK, we show that, for any C⊆DeC⊆De induced by such a class, there exists a high Δ02 degree c which is incomparable with every degree b ϵ Ce \ {0e, 0'e}. We show how this result can be applied to quite general subclasses of the Ershov Hierarchy and we also prove, as a direct corollary, that every nonzero low degree caps with (...) both a high and a nonzero low Δ02 degree. (shrink)

In [U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 89–128], the second author obtained metatheorems for the extraction of effective bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT and normed linear spaces and guarantee the independence of the bounds from parameters ranging over metrically bounded spaces. Recently ]), the authors obtained generalizations of these metatheorems which (...) allow one to prove similar uniformities even for unbounded spaces as long as certain local boundedness conditions are satisfied. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proof. Finally, we illustrate our main metatheorem by an example from metric fixed point theory. (shrink)

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

François Jullien, the leading philosopher and specialist in Chinese thought, has always aimed at building on inter-cultural relations between China and the West. In this new book he focuses on the following questions: Do universal values exist? Is dialogue between cultures possible? To answer these questions, he retraces the history of the concept of the universal from its invention as an aspect of Roman citizenship, through its neutralization in the Christian idea of salvation, to its present day manifestations. This raises (...) the question of whether the search for the universal is a uniquely Western preoccupation: do other cultures, like China, even have a notion of the universal, and if so, how does it differ from ours? Having considered the meaning of the concept in the East and West, Jullien argues that, if communication between cultures is to be meaningful, facile assumptions of universal values and complacent relativism need to be examined. It follows, therefore, that dialogue between cultures should not begin with issues of identity and difference, but rather by considering divergence and profusion. By no longer simply assuming universality, we allow for greater self-reflection. This wide-ranging and engaging study will be of particular interest to students and scholars of philosophy and of Chinese culture and society. It will also appeal to a wider readership interested in contemporary thought and the challenges of communication between East and West. (shrink)