The main result of the present article is the following: Let N be an infinite subset of,, and let be a matrix with infinitely many rows of completely Ramsey subsets of such that for every n,. Then there exist, a sequence of nonempty finite subsets of N, and an infinite subset T of such that for every infinite subset I of. We also give an application of this result to partitions of an uncountable analytic subset of a Polish space (...) X into sets belonging to the σ‐algebra generated by the analytic subsets of X. (shrink)

Local sentences and the formal languages they define were introduced by Ressayre in. We prove that locally finite ω‐languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω‐languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non‐null recursive ordinal there exist some ‐complete and some ‐complete locally finite ω‐languages, and the supremum of the (...) set of Borel ranks of locally finite ω‐languages is the ordinal, which is strictly greater than the first non‐recursive ordinal. This gives an answer to the question of the topological complexity of locally finite ω‐languages, which was asked by Simonnet and also by Duparc, Finkel, and Ressayre in. Moreover we show that the topological complexity of a locally finite ω‐language defined by a local sentence φ may depend on the models of the Zermelo‐Fraenkel axiomatic system. Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω‐languages are ‐complete, hence highly undecidable. (shrink)

The aim of the paper is to prove tha analytic completeness theorem for a logic LAs with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.

Hoover [2] proved a completeness theorem for the logic L[MATHEMATICAL SCRIPT CAPITAL A]. The aim of this paper is to prove a similar completeness theorem with respect to product measurable biprobability models for a logic Lmath image with two integral operators. We prove: If T is a ∑1 definable theory on [MATHEMATICAL SCRIPT CAPITAL A] and consistent with the axioms of Lmath image, then there is an analytic absolutely continuous biprobability model in which every sentence in T is satified.

The set of complete groups is a complete co-analytic subset of the standard Borel space of countably infinite groups.

We show that every analytic filter is generated by a Π 0 2 prefilter, every Σ 0 2 filter is generated by a Π 0 1 prefilter, and if $P \subseteq \mathscr{P}(\omega)$ is a Σ 0 2 prefilter then the filter generated by it is also Σ 0 2 . The last result is unique for the Borel classes, as there is a Π 0 2 -complete prefilter P such that the filter generated by it is Σ 1 (...) 1 -complete. Also, no complete coanalytic filter is generated by an analytic prefilter. The proofs use König's infinity lemma, a normal form theorem for monotone analytic sets, and Wadge reductions. (shrink)

According to the Genuine Modal Realist, there is a plurality of possible worlds, each world nothing more than a maximally inter-related spatiotemporal sum. One advantage claimed for this position is that it offers us the resources to analyse, in a noncircular manner, the modal operators. In this paper, we argue that the prospects for such an analysis are poor. For the analysis of necessity as truth in all worlds to succeed it is not enough that no modal concepts be used (...) in the realist's account of a possible world (a fact we grant); rather, such an analysis will succeed only if the set of worlds that is postulated is complete. By appealing to plausible truths about the number of possible alien natural properties, we show that there are serious difficulties in guaranteeing that such a set exists without taking some modal concept as primitive. Accordingly, at least in its current form, Genuine Modal Realism must curtail its analytic ambitions. (shrink)

In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t, ⊧ f, ⊧ i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first (...) three entailment relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six. (shrink)

The ability to analyze arguments is critical for higher-level reasoning, yet previous research suggests that standard university education provides at best modest improvements in students’ analytical reasoning abilities. What techniques are most effective for cultivating these skills? Here we investigate the effectiveness of a 12-week undergraduate seminar in which students practice a software-based technique for visualizing the logical structures implicit in argumen- tative texts. Seminar students met weekly to analyze excerpts from contemporary analytic philosophy papers, completed argument visualization problem (...) sets, and received individualized feedback on a weekly basis. We found that Seminar students improved substantially more on LSAT Logical Reasoning test forms than Control students (d = .71, p < .001), suggesting that learning how to visualize arguments in the seminar led to large generalized improvements in students’ analytical reasoning skills. Moreover, blind scoring of final essays from Seminar students and Control students, drawn from a parallel lecture course, revealed large differences in favor of seminar students (d = 0.87, p = .005). Seminar students understood the arguments better, and their essays were more accurate and effectively structured. Taken together, these findings deepen our understanding of how visualizations support logical reasoning, and provide a model for improving analytical reasoning pedagogy. (shrink)

We prove that the zero-set of a C ∞ function belonging to a noetherian differential ring M can be written as a finite union of C ∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ∞ functions over the reals, but more generally for definable C (...) ∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ∞ functions over the reals, containing non-analytic functions. (shrink)

First we prove that the set of countable linear orders of the form I + I form a complete analytic set. As a consequence of this we improve a result of Humke and Laczkovich, who showed in [HL] that the set of functions of the form f ⚬ f form a true analytic set in C[0, 1]. We show that these functions form a complete analytic set, solving a problem mentioned on p. 215 of [K1] (...) and on p. 4 of [B]. (shrink)

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝn, ℂn, Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group, the group of measure preserving transformations of the unit interval, etc.In this theory sets are (...) classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy. After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy.There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets. (shrink)

Analytical philosophy begins with the first mathematical and philosophical works of Bolzano published between 1804 and 1817. There, Bolzano set out a project for the global reform of mathematics by means of the axiomatic method. Having completed the Wissenschaftslehre, Bolzano wrote a summary of his logic for the Größenlehre, which he sent to Exner in 1833. The correspondence between Bolzano and Exner covered some of the main subjects treated by analytical philosophy: the status of abstract objects (propositions and objective ideas), (...) intuitions, objectless ideas, the concept of object and many others. While Bolzano argued in favor of abstract entities independent of mind and of language, Exner considered them as abstractions obtained from the subjective judgments and representations. During the XlXth century, Bolzano's philosophy spread over Bohemia and Austria through manuscripts and through the first edition of Zimmermann's textbook of philosophy. The most important Brentanians, Kerry, Twardowski, Meinong and Husserl, discussed his doctrines which may also have influenced Wittgenstein and the Polish school. (shrink)

Analytical philosophy begins with the first mathematical and philosophical works of Bolzano published between 1804 and 1817. There, Bolzano set out a project for the global reform of mathematics by means of the axiomatic method. Having completed the Wissenschaftslehre, Bolzano wrote a summary of his logic for the Größenlehre, which he sent to Exner in 1833. The correspondence between Bolzano and Exner covered some of the main subjects treated by analytical philosophy: the status of abstract objects (propositions and objective ideas), (...) intuitions, objectless ideas, the concept of object and many others. While Bolzano argued in favor of abstract entities independent of mind and of language, Exner considered them as abstractions obtained from the subjective judgments and representations. During the XlXth century, Bolzano's philosophy spread over Bohemia and Austria through manuscripts and through the first edition of Zimmermann's textbook of philosophy. The most important Brentanians, Kerry, Twardowski, Meinong and Husserl, discussed his doctrines which may also have influenced Wittgenstein and the Polish school. (shrink)

Philosophy is, and has been, many things to many people, and that is fine. Some of those persons who do, or have done, philosophy have engaged in the business of creating categoreal schemes. Were one to ask why these persons set about to construct categoreal schemes, the answer would have to be complex—the conscious motivations, purposes, and goals of system-builders are undoubtedly various. And that is fine. So when I suggest, as I am about to, an account of what it (...) is that categoreal schemes are really trying to do, it must be understood that I do not intend my account to be dogmatically a priori. I have simply thought about some of the great systematizers and the categoreal schemes they have concocted, have admittedly done this reflecting from the perspective of Whitehead’s process metaphysics, and have arrived at certain conclusions about what might be viewed as a diagnostic tool for understanding many of the interesting and historically important categoreal schemes. (shrink)

The singularity space consists of all germs , with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a complete, separable space for the metric given by the order to which jets agree after base change. In the terminology of descriptive set-theory, the classification of singularities up to analytic extensions of scalars is a smooth problem. Over , the following (...) two classification problems up to isomorphism are then also smooth: analytic germs; and polarized schemes. (shrink)

We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is (...) class='Hi'>complete for analytic quasi-orders on View the MathML source. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal [17] and Friedman and Motto Ros [6], and it also partially extends a result from Baumgartner [3] concerning the structure of the embeddability relation on linear orders of size κ. (shrink)

This book offers a clear, analytic, and innovative interpretation of Heidegger's late work. This period of Heidegger's philosophy remains largely unexplored by analytic philosophers, who consider it filled with inconsistencies and paradoxical ideas, particularly concerning the notions of Being and nothingness. This book takes seriously the claim that the late Heidegger endorses dialetheism--namely the position according to which some contradictions are true--and shows that the idea that Being is both an entity and not an entity is neither incoherent (...) nor logically trivial. The author achieves this by presenting and defending the idea that reality has an inconsistent structure. In doing so, he takes one of the most discussed topics in current analytic metaphysics, grounding theory, into a completely unexplored area. Additionally, in order to make sense of Heidegger's concept of nothingness, the author introduces an original axiomatic mereological system that, having a paraconsistent logic as a base logic, can tolerate inconsistencies without falling into logical triviality. This is the first book to set forth a complete and detailed discussion of the late Heidegger in the framework of analytic metaphysics. It will be of interest to Heidegger scholars and analytic philosophers working on theories of grounding, mereology, dialetheism, and paraconsistent logic. (shrink)

Although logical settings are typically concerned with tracking alethic considerations, frameworks exist in which topic-theoretic considerations—e.g., tracking subject-matter or topic—are given equal importance. Intuitions about extending topic through a propositional language are generally straightforward for extensional cases. For a number of reasons, arriving at a compelling account of the subject-matter of intensional operators—such as intensional conditionals—is a more difficult task. In particular, the framework of topic-sensitive intentional modals (TSIMs) championed by Francesco Berto and his collaborators leave the topics of intensional (...) formulae undefined, which artificially constricts the expressivity of the theory. This paper proposes an approach to fill in this lacuna, emphasizing an analogous problem in Parry-style containment logics. In this setting, the approach receives a proof-of-concept through the introduction of a natural and general family of subsystems of Parry’s $${\textsf{PAI}}$$ PAI —with sound and complete axiomatizations—that allow a fine degree of control over the topics of intensional conditionals. (shrink)

The reintroduction of Aristotle's Analytics to the Latin West—in particular, the reintroduction of the Posterior Analytics—forever altered the course of medieval epistemological discussions. Although the Analytics fell decidedly from grace in later centuries, the sophisticated account of human cognition developed in the Posterior Analytics appealed so strongly to thirteenth-century European scholars that it became one of the two central theories of knowledge advocated in the later Middle Ages. Robert Grosseteste's 'Commentarius in Posteriorum Analyticorum Libro', written in the 1220s, is most (...) likely the first complete Latin commentary on the Posterior Analytics. As the thirteenth century wore on, Aristotle’s account of human cognition was generally set in opposition to the Augustinian-influenced theory of divine illumination that was de rigueur in the early thirteenth century. Although common consensus holds that this pattern of opposition was set already in Grosseteste’s CPA, I argue that the story of the Posterior Analytics’s early reception is, in fact, quite different. In particular, I maintain that, rather than seeing himself as forced to choose between his earlier theory of divine illumination and the “new” Aristotelian epistemology, Grosseteste is perfectly content to blend and bring together the diverse elements of these systems and to present a consciously synthetic rather than adversarial picture of these differing accounts of human cognition. (shrink)

This book is designed to explain the technical ideas that are taken for granted in much contemporary philosophical writing. Notions like "denumerability," "modal scope distinction," "Bayesian conditionalization," and "logical completeness" are usually only elucidated deep within difficult specialist texts. By offering simple explanations that by-pass much irrelevant and boring detail, Philosophical Devices is able to cover a wealth of material that is normally only available to specialists. The book contains four sections, each of three chapters. The first section is about (...) sets and numbers, starting with the membership relation and ending with the generalized continuum hypothesis. The second is about analyticity, a prioricity, and necessity. The third is about probability, outlining the difference between objective and subjective probability and exploring aspects of conditionalization and correlation. The fourth deals with metalogic, focusing on the contrast between syntax and semantics, and finishing with a sketch of Godel's theorem. Philosophical Devices will be useful for university students who have got past the foothills of philosophy and are starting to read more widely, but it does not assume any prior expertise. All the issues discussed are intrinsically interesting, and often downright fascinating. It can be read with pleasure and profit by anybody who is curious about the technical infrastructure of contemporary philosophy."--p. 4 of cover. (shrink)

BackgroundCommunities’ engagement in priority-setting is a key means for setting research topics and questions of relevance and benefit to them. However, without attention to dynamics of power and diversity, their engagement can be tokenistic. So far, there remains limited ethical guidance on how to share power with communities, particularly those considered disadvantaged and marginalised, in global health research priority-setting. This paper generates a comprehensive, empirically-based “ethical toolkit” to provide such guidance, further strengthening a previously proposed checklist version of the toolkit. (...) The toolkit places community engagement and power-sharing at the heart of priority-setting for global health research projects.MethodsA two part method was used to generate a revised toolkit. Part one was conceptual, consisting of novel analysis of empirical data (previously collected as part of the same overall project) to identify additional concepts relevant to power-sharing between researchers and communities in global health research priority-setting. Part two was empirical, seeking feedback on the initial checklist version of the toolkit in interviews with researchers, ethicists, community engagement practitioners, and community organisation staff.ResultsThe conceptual process identified two additional components of engagement and six additional features that affect who defines, who participates, and who is heard in research priority-setting. New ethical considerations related to sharing power in global health research priority-setting are articulated in relation to those components and features. Interviewees provided suggestions for revising the toolkit’s content and language. The implications of these suggestions and the analytic process for the toolkit are described.ConclusionsThe resultant toolkit is a reflective project planning aid for researchers and their community partners to employ before priority-setting is undertaken for global health research projects. It consists of three worksheets (to be completed collectively) and a companion document detailing how to use them. It is more comprehensive than the initial toolkit, as worksheet questions for discussion cover all phases of priority-setting. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite (...) states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”. (shrink)

We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator |$\blacktriangle \phi $| which is construed as ‘|$\phi $| has the same value in all accessible states’ or ‘all sources give the same information on the truth value of |$\phi $|’. We equip this logic dubbed |$\textbf {K}^\blacktriangle _{\textbf {FDE}}$| with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the |$\blacktriangle $| operator modelling search (...) for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that |$\blacktriangle $| is not definable via the necessity modality |$\Box $| of |$\textbf {K}_{\textbf{FDE}}$|⁠. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, |$\textbf {S4}$| and |$\textbf {S5}$| (among others) frames are definable. (shrink)

IntroductionThe National Institute for Health and Care Excellence guidelines acknowledge the importance of the parent–infant relationship for child development but highlight the need for further research to establish reliable tools for assessment, particularly for parents of children under 1 year. This study explores the acceptability and psychometric properties of a co-developed tool, ‘Me and My Baby’.Study designA cross-sectional design was applied. The MaMB was administered universally with mothers during routine 6–8-week Health Visitor contacts. The sample comprised 467 mothers. Dimensionality of (...) instrument responses were evaluated via exploratory and confirmatory ordinal factor analyses. Item response modeling was conducted via a Rasch calibration to evaluate how the tool conformed to principles of ‘fundamental measurement’. Tool acceptability was evaluated via completion rates and comparing ‘completers’ and ‘non-completers’ demographic differences on age, parity, ethnicity, and English as an additional language. Free-text comments were summarized. Data sharing agreements and data management were compliant with the General Data Protection Regulation, and University of York data management policies.ResultsHigh completion rates suggested the MaMB was acceptable. Psychometric analyses showed the response data to be an excellent fit to a unidimensional confirmatory factor analytic model. All items loaded statistically significantly and substantially on a single underlying factor. The item response modeling showed that most MaMB items fitted the Rasch model. item reliability was high yet the test yielded little information on each respondent, as highlighted by the relatively low ‘person separation index’ of 0.1.Conclusion and next stepsMaMB reliably measures a single construct, likely to be infant bonding. However, further validation work is needed, preferably with ‘enriched population samples’ to include higher-need/risk families. The MaMB tool may benefit from reduced response categories and some modest item wording amendments. Following further validation and reliability appraisal the MaMB may ultimately be used with fathers/other primary caregivers and be potentially useful in research, universal health settings as part of a referral pathway, and clinical practice, to identify dyads in need of additional support/interventions. (shrink)

Let AC and AI denote the collections of graphs with vertex set ω and which have, respectively, no infinite independent subgraph, and no infinite complete subgraph. Both AC and AI are coanalytic sets of reals that are not analytic, and they are disjoint by Ramsey’s theorem. We prove that there exists a Borel set separating AC and AI, and we discuss the sense in which this is an infinite analogue of a weak version of.

We prove that for every family I of closed subsets of a Polish space each Σ 1 1 set can be covered by countably many members of I or else contains a nonempty Π 0 2 set which cannot be covered by countably many members of I. We prove an analogous result for κ-Souslin sets and show that if A ♯ exists for any $A \subset \omega^\omega$ , then the above result is true for Σ 1 2 sets. A theorem (...) of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris, Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding "big" closed sets inside of "big" Σ 1 1 and Σ 1 2 sets are consequences of our results. (shrink)

Two hundred years ago, J.M.W. Turner packed up two large leatherbound sketchbooks, pencils, and watercolors and set off for the north of England. When he returned from the tour that he regarded as one of the most important of his career, Turner had completed more than two hundred sketches - works that later became the basis of more than fifty major oil paintings and watercolors. For this illustrated book, David Hill has taken photographs of many of the actual sites Turner (...) sketched on his northern tour. The result is an look at the whole of Turner's creative process, from site to exhibited picture, and at the quality and intensity of the artist's experience. (shrink)

We give a direct mathematical proof of the Mathias-Silver theorem that every analytic set is Ramsey.

This review places this translation and commentary on Book A of Prior Analytics in historical, logical, and philosophical perspective. In particular, it details the author’s positions on current controversies. The author of this translation and commentary is a prolific and respected scholar, a leading figure in a large and still rapidly growing area of scholarship: Prior Analytics studies PAS. PAS treats many aspects of Aristotle’s Prior Analytics: historical context, previous writings that influenced it, preservation and transmission of its manuscripts, editions (...) of its manuscripts, interpretations, commentaries, translations, and its influence on subsequent logic, philosophy, and mathematics. All this attention is warranted because Prior Analytics marks the origin of logic: the field that, among other things, asks of a given proposition whether it follows from a given set of propositions; and, if it follows, how we determine that it follows; and, if it does not follow, how we determine that it does not follow. This translation and commentary is not suitable for use in an undergraduate course. It has too many quirks that the teacher would want to warn against. A copy editor should have dealt with these things and with other matters such as incorrect punctuation and improper end-of-line divisions. The prose is heavily laden with glaring clichés. The one-page preface contains “longer than I care to remember”, “more than I can possibly list here”, “first and foremost”, and “last and by no means least”—a sentence later is devoted to thanking the “incredibly meticulous and helpful copy-editor”. A few pages later the translator reveals the need “to find a path between the Scylla … and the Charybdis …”. Moreover, the index is far from meeting the needs of undergraduate students. The attention to scholarly detail is not what one hoped for from Oxford University Press. At 26b10-15, this translation reads “let swan and white be chosen as white things” for what Smith correctly translates “let swan and snow be selected from among those white things”. At 41b16, “angles AB and CD” should read “angles AC and BD”. Despite this book’s flaws, it will be found useful if not indispensable for those currently engaged in Prior Analytics studies. The alternatives suggested to Robin Smith’s translation choices are often worth consideration. It is to be emphasized, however, that this book is unsuitable for those entering Prior Analytics studies. (shrink)

The main contribution of this paper is a completeness result for an axiomatization of Hansson [13]'s deontic system DSDL2, whose semantics involves a non-necessarily transitive betterness relation. Reference is made to a deductive system put forth by Åqvist [2, 3].

We use games of Kastanas to obtain a new characterization of the classC ℱ of all sets that are completely Ramsey with respect to a given happy family ℱ. We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC ℱ. We also study some cardinals that can be associated with the ideal ℐℱ of nowhere ℱ-Ramsey sets.

The Summa Theologiae ranks among the greatest documents of the Christian Church, and is a landmark of medieval western thought. It is regularly consulted by scholars of theology, philosophy and a range of related academic disciplines. This paperback reissue of the classic Latin/English edition first published by the English Dominicans in the 1960s and 1970s has been undertaken in response to regular requests from around the world. The original text is unchanged, except for the correction of a small number of (...) typographical errors. The parallel English and Latin texts can be used successfully by anybody with a basic knowledge of Latin, while the presence of the Latin text allowed the translators a degree of freedom in adapting their English version for modern readers. Each volume contains a glossary of technical terms and is designed to be complete in itself to serve for private study or as a course text. (shrink)