We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.

We prove that the set of all recursive functions cannot be inferred using first-order queries in the query language containing extra symbols $\lbrack +, . The proof of this theorem involves a new decidability result about Presburger arithmetic which is of independent interest. Using our machinery, we show that the set of all primitive recursive functions cannot be inferred with a bounded number of mind changes, again using queries in $\lbrack +, . Additionally, we resolve an open question in [7] (...) about passive versus active learning. (shrink)

We prove that the set of all recursive functions cannot be inferred using first-order queries in the query language containing extra symbols $\lbrack +, < \rbrack$. The proof of this theorem involves a new decidability result about Presburger arithmetic which is of independent interest. Using our machinery, we show that the set of all primitive recursive functions cannot be inferred with a bounded number of mind changes, again using queries in $\lbrack +, < \rbrack$. Additionally, we resolve an open question (...) in [7] about passive versus active learning. (shrink)

In prior papers the following question was considered: which classes of computable sets can be learned if queries about those sets can be asked by the learner? The answer depended on the query language chosen. In this paper we develop a framework for studying this question. Essentially, once we have a result for queries to [S,<]2, we can obtain the same result for many different languages. We obtain easier proofs of old results and several new results. An earlier result we (...) have an easier proof of: the set of computable sets cannot be learned with queries to the language [+,<] . A new result: the set of computable sets cannot be learned with queries to the language [+,<,POWa] where POWa is the predicate that tests if a number is a power of a. (shrink)

Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...) T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice. (shrink)

A recursive graph is a graph whose vertex and edge sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: Let A be a set. An A-recursive graph is a recursive graph that also has the following property: one can recursively-in-A determine the neighbors of a vertex. We show that, if (...) A is r.e. and not recursive, then there exists A-recursive graphs that are 2-colorable but not recursively k-colorable for any k. This is false for highly-recursive graphs but true for recursive graphs. Hence A-recursive graphs are closer in spirit to recursive graphs than to highly recursive graphs. (shrink)

Most theories of learning consider inferring a function f from either observations about f or, questions about f. We consider a scenario whereby the learner observes f and asks queries to some set A. If I is a notion of learning then I[A] is the set of concept classes I-learnable by an inductive inference machine with oracle A. A and B are I-equivalent if I[A] = I[B]. The equivalence classes induced are the degrees of inferability. We prove several results about (...) when these degrees are trivial, and when the degrees are omniscient. (shrink)

If and the function is partial recursive, it is easily seen that A is recursive. In this paper, we weaken this hypothesis in various ways (and similarly for ``min'' in place of ``max'') and investigate what effect this has on the complexity of A. We discover a sharp contrast between retraceable and co-retraceable sets, and we characterize sets which are the union of a recursive set and a co-r.e., retraceable set. Most of our proofs are noneffective. Several open questions are (...) raised. (shrink)

We prove a number of structural theorems about the honest polynomial m-degrees contingent on the assumption P = NP . In particular, we show that if P = NP , then the topped finite initial segments of Hm are exactly the topped finite distributive lattices, the topped initial segments of Hm are exactly the direct limits of ascending sequences of finite distributive lattices, and all recursively presentable distributive lattices are initial segments of Hm ∩ RE. Additionally, assuming ¦∑¦ = 1, (...) we show that the theory of the hpm-degrees is undecidable. We also show that index sets cannot be minimal. Lastly, we examine an alternative definition of honest m-reduction under which recursive minimal sets can be constructed. (shrink)

Higman essentially showed that if A is any language then SUBSEQ(A) is regular, where SUBSEQ(A) is the language of all subsequences of strings in A. Let s1, s2, s3, . . . be the standard lexicographic enumeration of all strings over some finite alphabet. We consider the following inductive inference problem: given A(s1), A(s2), A(s3), . . . . learn, in the limit, a DFA for SUBSEQU). We consider this model of learning and the variants of it that are usually (...) studied in Inductive Inference: anomalies, mind-changes, teams, and combinations thereof. This paper is a significant revision and expansion of an earlier conference version [10]. (shrink)

LetA be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible toA based on bounding the number of queries toA that an oracle machine can make. WhenA is the halting problemK our hierarchy of sets interleaves with the difference hierarchy on the r.e. sets in a logarithmic way; this follows from a tradeoff between the number of parallel queries and the number of serial queries needed to compute a function with oracleK.

For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A: $ODD^A_n = \{(x_1, \dots ,x_n): {\tt\#}^A_n(x_1, \dots, x_n) \text{is odd}\}$ and, for m ≥ 2, $\text{MOD}m^A_n = \{(x_1, \dots ,x_n):{\tt\#}^A_n(x_1, \dots ,x_n) \not\equiv 0 (\text{mod} m)\},$ where ${\tt\#}^A_n(x_1, \dots ,x_n) = A(x_1)+\cdots+A(x_n)$ . (We identify A(x) with χ A (x), where χ A is (...) the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODD A n and $\text{MOD}m^A_n: \bullet\text{ODD}^A_n$ can be decided with n parallel queries to A, but not with n - 1. $\bullet \text{ODD}^A_n$ can be decided with $\lceil log(n + 1)\rceil$ sequential queries to A but not with $\lceil log(n + 1)\rceil - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil n/m\rceil + \lfloor n/m\rfloor$ parallel queries to A but not with $\lceil n/m\rceil + \lfloor n/m\rfloor - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil$ sequential queries to A but not with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil - 1$ . The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that ODD A n cannot be decided with n - 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODD B n can be decided with one query to B, a set's query complexity depends more on its structure than on its degree. For a fixed set A, $Q(n,A) = \{S: S \text{can be decided with n sequential queries to} A\},\\Q_\parallel(n, A) = \{S: S \text{can be decided with n parallel queries to} A\}.$ We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies: $\bullet Q(0,A) \subset Q(1,A) \subset Q(2,A) \subset\cdots\\ \bullet Q_\parallel(0, A) \subset Q_\parallel(1, A) \subset Q_\parallel(2,A) \subset\cdots$ The same is true if A is a jump. (shrink)

"The lectures that follow were delivered at the Lowell Institute in Boston in November and December, 1906, and in January, 1907, at Columbia University, in New York."-Preface, pg. 3.

In The Emergent Self, William Hasker joins one of the most heated debates in contemporary analytic philosophy, that over the nature of mind.

Toward theory a homuncular of believing For years and years, philosophers took thoughts and beliefs to be modifications of incorporeal Cartesian egos. ...

The Gifford Lectures were established in 1885 at the universities of St Andrews, Glasgow, Aberdeen and Edinburgh to promote the discussion of 'Natural Theology in the widest sense of the term - in other words, the knowledge of God', and some of the world's most influential thinkers have delivered them. The 1901–2 lectures given in Edinburgh by American philosopher William James are considered by many to be the greatest in the series. The lectures were published in book form in (...) 1902 and have been reprinted many times. James, who was educated in the United States and Europe, and spent much of his career teaching philosophy at Harvard, was very influential in the development of modern psychology, and in these twenty lectures he explores the personal experience of religion. Some of the topics include religion and neurology, 'the sick soul', saintliness, and mysticism. (shrink)

For this 1897 publication, the American philosopher William James brought together ten essays, some of which were originally talks given to Ivy League societies. Accessible to a broader audience, these non-technical essays illustrate the author's pragmatic approach to belief and morality, arguing for faith and action in spite of uncertainty. James thought his audiences suffered 'paralysis of their native capacity for faith' while awaiting scientific grounds for belief. His response consisted in an attitude of 'radical empiricism', which deals practically (...) rather than ideologically with real-world phenomena. When facing a 'momentous' decision about belief, he says, we both can and must choose the best hypothesis. The first four essays apply this principle to religious faith, and the rest explore the pragmatic approach to such topics as determinism, ethics and individual achievement. James developed his ideas further in The Varieties of Religious Experience and Pragmatism, both of which are reissued in this series. (shrink)

Descartes has often been called the 'father of modern philosophy'. His attempts to find foundations for knowledge, and to reconcile the existence of the soul with the emerging science of his time, are among the most influential and widely studied in the history of philosophy. This is a classic and challenging introduction to Descartes by one of the most distinguished modern philosophers. Bernard Williams not only analyzes Descartes' project of founding knowledge on certainty, but uncovers the philosophical motives for his (...) search. With acute insight, he demonstrates how Descartes' _Meditations _are not merely a description but the very enactment of philosophical thought and discovery. Williams covers all of the key areas of Descartes' thought, including God, the will, the possibility of knowledge, and the mind and its place in nature. He also makes profound contributions to the theory of knowledge, metaphysics and philosophy generally. With a new foreword by John Cottingham. (shrink)

In Morality Bernard Williams confronts the problems of writing moral philosophy, and offers a stimulating alternative to more systematic accounts which seem nevertheless to have left all the important issues somewhere off the page.

One of the most influential men of his time, philosopher, psychologist, educator, and author William James (1842-1910) helped lead the transition from a predominantly European-centered nineteenth-century philosophy to a new "pragmatic" American philosophy. Helping to pave the way was his seminal book Pragmatism (1907), in which he included a chapter on "Truth," an essay which provoked severe criticism. In response, he wrote the present work, an attempt to bring together all he had ever written on the theory of knowledge, (...) including an article on the function of cognition, later polemic and expository contributions, and some replies to previous criticism. The result was a full and definitive expression of the pragmatist "epistemology" from James' point of view. In the book, he urges the reader to "turn away from abstractions, verbal solutions, fixed principles, and pretended absolutes and look for concreteness and facts, action and power," arguing that "the ultimate test for us of what a truth means is the conduct it dictates or inspires." For students, scholars--anyone interested in William James or the history of American philosophical thought--this book offers an in-depth exposition of the influential ideas of one of the greatest American thinkers. Unabridged republication of the edition originally published by Longmans, Green and Co., New York and London, 1909. (shrink)

Complexity, agency, and time -- The vicissitudes of experience -- Belief, spirituality, and time -- The human predicament -- Capital flows, sovereign decisions, and world resonance machines -- The theorist and the seer.

" In a book that seeks to shift the ground of debate within theory of knowledge, William P. Alston finds that the century-lo.

This book analyzes the support that should be given to minority views, reconsidering classic debates in Science and Technology Studies and examining numerous case studies.

Two books bound together, from the religious period of one of the most renowned and representative thinkers. Written for laymen, thus easy to understand, it is penetrating and brilliant as well. Illuminations of age-old religious questions from a pragmatic perspective, written in a luminous style.

Lycan (1985, 1988) defended a “Principle of Credulity”: “Accept at the outset each of those things that seem to be true” (1988, p. 165). Though that takes the form of a rule rather than a thesis, it does not seem very different from Huemer’s (2001, 2006, 2007) doctrine of phenomenal conservatism (PC): “If it seems to S that p , then, in the absence of defeaters, S thereby has at least some degree of justification for believing that p ” (2007, (...) p. 30). My Principle was differently motivated and put to uses different from Huemer’s. In this paper I shall explore some of the differences. (shrink)

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

Noted psychologist and philosopher develops his own brand of pragmatism, based on theories of C. S. Peirce. Emphasis on "radical empiricism," versus the transcendental and rationalist tradition. One of the most important books in American philosophy. Note.

In Part One Leiss traces the idea of the domination of nature from the Renaissance to the nineteenth century.

In 2018, William Lane Craig and Erik J. Wielenberg participated in a debate at North Carolina State University, addressing the question: "God and Morality: What is the best account of objective moral values and duties?" Craig argued that theism provides a sound foundation for objective morality whereas atheism does not. Wielenberg countered that morality can be objective even if there is no God. This book includes the full debate, as well as endnotes with extended discussions that were not included (...) in the debate. It also includes five chapters by other philosophers who have written substantive responses to the debate - J. P. Moreland, David Baggett, Mark Linville, Wes Morriston, and Michael Huemer. The book provides crucial resources for better understanding moral realism and its dependence on, or independence from, theistic foundations. Key Features A valuable debate about whether or not God is the best explanation for objective morality, bringing together theists and atheists working on the same subject who normally are not in conversation with each other. Includes clear coverage of ontological and epistemological issues in metaethical theories, focusing on Divine Command Theory and Non-natural Robust Moral Realism. Engaging and accessible throughout, making the book well suited for undergraduate and seminary classrooms. (shrink)

Science should tell us what the world is like. However, realist interpretations of physics face many problems, chief among them the pessimistic meta induction. This book seeks to develop a realist position based on process ontology that avoids the traditional problems of realism. Primarily, the core claim is that in order for a scientific model to be minimally empirically adequate, that model must describe real experimental processes and dynamics. Any additional inferences from processes to things, substances or objects are not (...) warranted, and so these inferences are shown to represent the locus of the problems of realism. The book then examines the history of physics to show that the progress of physical research is one of successive eliminations of thing interpretations of models in favor of more explanatory and experimentally verified process interpretations. This culminates in collections of models that cannot coherently allow for thing interpretations, but still successfully describe processes. (shrink)

This new critical edition is an examination of William James's Essays in Radical Empiricism in light of the scientific naturalism prominent in James's Principles of Psychology (1890) and the subsequent development of Darwinian, functional psychology and functionalism in psychology, the philosophy psychology and the philosophy of mind.

Introduction: the advance of governmentality -- Foucault, power, and governmentality: introduction; what is governmentality?; beyond the microphysics of power?; from theory of the state to genealogy of the state; history of the art of government; pastoral power; raison d'état; liberal governmentality; five propositions on foucault and governmentality -- Governmentality 3.4.7.: introduction; governmentality after Foucault; governmentality and the political sciences; some problems in governmentality -- Foucault effect redux? some notes on international governmentality studies: constellation; a few preliminary observations; problems and debates (...) in international governmentality studies -- Reconnecting governmentality and genealogy: introduction; genealogy and governmentality; pluralizing genealogy; practicing genealogy; lines of descent: the family tree of power?; counter-memory and re-serialization; the retrieval of forgotten struggles and subjugated knowledges -- Conclusion: encountering governmentality: governmentality as a new cartography of power; from mapping to encountering; governmentality and politics. (shrink)

Science can reveal or conceal the breathtaking wonders of creation. On one hand, knowledge of the natural world can open us up to greater love for the Creator, give us the means of more neighborly care, and fill us with ever-deepening astonishment. On the other hand, knowledge feeding an insatiable hunger for epistemic mastery can become a means of idolatry, hubris, and damage. Crucial to world-respecting science is the role of wonder: curiosity, perplexity, and astonishment. In this volume, philosopher (...) class='Hi'>William Desmond explores the relation of the different modes of wonder to modern science. Responding to his thought are twelve thinkers across the domains of science, theology, philosophy, law, poetry, medicine, sociology, and art restoration. (shrink)