This paper is intended to give for a general mathematical audience a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below ε0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly (...) universality and renormalization issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws. (shrink)

ABSTRACT The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of a certain length n (...) to the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with $ k\ge 1 $ k ≥ 1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (2020) as well as enriched with some previous results from Kostrzycka and Zaionc (2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. (shrink)

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and (...) proven to be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students. (shrink)

The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of a certain length n to (...) the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with k≥1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (Citation2020) as well as enriched with some previous results from Kostrzycka and Zaionc (Citation2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (Citation2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. (shrink)

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 725–731] , Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. of (...) the LMS 14 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory .In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.For these independence principles we classify their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles.As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results.This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem. (shrink)

In this paper we prove general exact unprovability results that show how a threshold between provability and unprovability of a finite well-quasi-orderedness assertion of a combinatorial class is transformed by the sequence-construction, multiset-construction, cycle-construction and labeled-tree-construction. Provability proofs use the asymptotic pigeonhole principle, unprovability proofs use Weiermann-style compression techniques and results from analytic combinatorics.

We consider logical expressions built on the single binary connector of implication and a finite number of literals . We prove that asymptotically, when the number of variables becomes large, all tautologies have the following simple structure: either a premise equal to the goal, or two premises which are opposite literals.

Drug design methods have made significant new advances over the last ten years, mainly in the areas of molecular modelling. In more recent times important developments in theory have led to a different type of modelling becoming possible, the so‐called de novo or automated design algorithms. In this new method the programs perform much of the chemist's thinking, in finding appropriately sized chemical groups to fit into a target site. However this is a combinatoric problem which has no general analytical (...) solution; it is ripe for optimization. Other advances, such as combinatorial chemical synthesis and screening, will dramatically influence the search for new lead structures for target sites, which at present are poorly understood. Already these methods are being applied to peptide libraries. Peptides do not make good drug compounds because of their poor bioavailability; further, their flexibility reduces their affinity. In some cases peptide backbones can be removed and replaced with rigid non‐peptide scaffolds. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by explaining in (...) simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

It is conventionally understood that computers play a rather limited role in theoretical mathematics. While computation is indispensable in applied mathematics and the theory of computing and algorithms is rich and thriving, one does not, even today, expect to find computers in theoretical mathematics settings beyond the theory of computing. Where computers are used, by those studying combinatorics , algebra, number theory, or dynamical systems, the computer most often assumes the role of an automated and speedy theoretician, performing manipulations (...) and checking cases in a way assumed to be possible for human theoreticians, if only they had the time, the memory, and the precision. Automated proofs have become standard tools in mathematical logic, and it is often expected that proofs be published in a computer-checkable format. It is not surprising, then, that most philosophical work on computers in theoretical mathematics has been on computers' roles as supplementary mathematicians. Donald MacKenzie's 2001 book Mechanizing Proof demonstrates the rich potential for social and historical studies to complement the substantial analytic debate in this area of philosophy. But what of computers in theoretical mathematics behaving as computers, and not as mere mechanized mathematicians? Very little role is commonly assumed for computers working as supplements to mathematicians, rather than as supplementary mathematicians themselves. Accordingly, very little philosophy has attempted to grapple with theoretical mathematics in which computers play an essential but essentially non-theoretical role. My presentation will draw on work I conducted as a researcher in harmonic analysis on fractals at Cornell University. I will analyze the explicit and implicit conceptual apparatus employed in my and my fellow researchers' use of computers in the theoretical study of second order differential equations, such as those for sound and heat flow, on various fractal analogues of the Sierpinski gasket. Such gaskets are easy to visualize in very crude approximation in a low number of dimensions. As one increases the complexity of the gasket or the refinement of one's analysis, visualization and precise computation become impossible, and soon computers are unable to produce even approximate data to model differential equations in these situations. We thus had to carefully choose analytic approaches and methods to make our theoretical mathematics amenable to computer simulation. In my case, studying the transformation of the gaskets as they are expanded into increasingly high dimensions, computer simulation eventually required that the problem be reimagined entirely in terms of interlinked systems of parameters. This computer-approximation-driven theoretical orientation shaped my mathematical intuitions toward the problem and guided my fellow researchers and me in both theoretical and computational directions. We discovered both that computer approximation could be incredibly powerful as an aid to intuition, and that it can be incredibly difficult to transfer computer-oriented mathematics back into the purely theoretical standards of our area of specialty. I will address the philosophical implications of computer-driven theoretical mathematics, asking how computer experiments can shape both the content and standards of theoretical sciences. (shrink)

One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. (...) All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI). (shrink)

'Mr Bennett, as was to be expected, has written a first-rate book on Kant's Analytic. It is vivid, entertaining, and extremely instructive. It will be found of absorbing interest both by those who already know the Critique and by those - if there are any such - who have a developed interest in philosophy, yet no direct acquaintance with Kant. These last it will surely drive to the text and, as surely, will drive them to approach it in a (...) truly philosophical spirit. Bennett's Kant is not a giant immersed, or frozen, in time. He is a great contemporary - a little out of touch, admittedly, with recent developments in mathematics and physics - but one with whom we can all argue, against him, at his side, or obliquely to him. And so Bennett does argue, continuously, fiercely, and fruitfully; and summons to join in the argument, at appropriate moments, those older contemporaries, Locke, Leibniz, Berkeley, and Hume, and those younger contemporaries, Wittgenstein, Ryle, Ayler, Quine, Quinton, Warnock, and others. This is splendid, and a necessary corrective to that extraordinary isolation in which Kant tends to be islanded, partly indeed, by his own unique qualities, but partly by oceans of the wrong kind of respect. Bennett, continuously engaging his great antagonist, shows the right kind.'. (shrink)

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ (...) logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. OPEN ACCESS LINK. (shrink)

R. Lanier Anderson presents a new account of Kant's distinction between analytic and synthetic judgments, and provides it with a clear basis within traditional logic. He reconstructs compelling claims about the syntheticity of elementary mathematics, and re-animates Kant's arguments against traditional metaphysics in the Critique of Pure Reason.

Dainton and Robinson’s Companion traces lines of descent of analytic philosophy from ancestors. They characterize analytic philosophy as a movement, a tradition, a style, and a commitment to the va...

This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is (...) discussed, which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)

We divide analytic metaphysics into naturalistic and non-naturalistic metaphysics. The latter we define as any philosophical theory that makes some ontological (as opposed to conceptual) claim, where that ontological claim has no observable consequences. We discuss further features of non-naturalistic metaphysics, including its methodology of appealing to intuition, and we explain the way in which we take it to be discontinuous with science. We outline and criticize Ladyman and Ross's 2007 epistemic argument against non-naturalistic metaphysics. We then present our (...) own argument against it. We set out various ways in which intellectual endeavours can be of value, and we argue that, in so far as it claims to be an ontological enterprise, non-naturalistic metaphysics cannot be justified according to the same standards as science or naturalistic metaphysics. The lack of observable consequences explains why non-naturalistic metaphysics has, in general, failed to make progress, beyond increasing the standards of clarity and precision in expressing its theories. We end with a series of objections and replies. (shrink)

Research into consumer responses to corporate social responsibility (CSR) initiatives has expanded in the past four decades, yet the evidence thus far provided does not paint a cohesive picture. Results suggest both positive and negative consumer reactions to CSR, and unless such mixed findings can be reconciled, the outcome might be an amalgamation of disparate empirical results rather than a coherent body of knowledge. The current meta-analysis therefore tests whether the mixed findings might reflect consumers’ distinct, altruistic inferences across various (...) contingency factors. On the basis of 337 effect sizes, involving 584,990 unique respondents, in 162 studies published between 1996 and 2021, this study reveals that altruistic inferences are central to the current CSR paradigm, such that they mediate the effects of CSR initiatives on consumer responses across multiple contingencies. The mediation by altruistic inferences is stronger (weaker) in conditions favorable to dispositional (situational) motive attributions. Furthermore, consumers respond more favorably to cause marketing or philanthropy rather than business-related CSR initiatives, when the initiative is environmental (vs. social), the firm’s offering is utilitarian (vs. hedonic), the CSR initiative takes place in self-expressive (vs. survival) cultures and in earlier (vs. later) periods. These findings offer several ethical implications, and they inform both practical recommendations and an agenda for further research directions. (shrink)

Introduction -- 1. Problem of the New -- 2. Problem of Relations -- 3. Problem of Emergence -- 4. Problem of One and Many -- 5. Plato and the Third Man Argument -- 6. Bradley and the Problem of Relations -- 7. Moore, Russell and the Birth of Analytic Philosophy -- 8. Russell and Deleuze on Leibniz -- 9. On Problematic Fields -- 10. Kant and Problematic Ideas -- 11. Armstrong and Lewis on the Problem of One and Many (...) -- 12. Determinables and Determinates -- 13. The Limits of Representational Thought -- 14. Learning from a Cup of Coffee -- 15. Carnap and the Fate of Metaphysics -- 16. Truth and Relevance -- Conclusion. (shrink)

I am concerned with epistemic closure—the phenomenon in which some knowledge requires other knowledge. In particular, I defend a version of the closure principle in terms of analyticity; if an agent S knows that p is true and that q is an analytic part of p, then S knows that q. After targeting the relevant notion of analyticity, I argue that this principle accommodates intuitive cases and possesses the theoretical resources to avoid the preface paradox.

Using quantitative methods, we investigate the role of logic in analytic philosophy from 1941 to 2010. In particular, a corpus of five journals publishing analytic philosophy is assessed and evaluated against three main criteria: the presence of logic, its role and level of technical sophistication. The analysis reveals that logic is not present at all in nearly three-quarters of the corpus, the instrumental role of logic prevails over the non-instrumental ones, and the level of technical sophistication increases in (...) time, although it remains relatively low. These results are used to challenge the view, widespread among analytic philosophers and labeled here “prevailing view”, that logic is a widely used and highly sophisticated method to analyze philosophical problems. (shrink)

Accused of the murder of two men and the rape and murder of a Tombstone businessman's fiancâee, Matt Donohue must find the girl in the middle of Apache territory in order to clear his name.

lntroduction Imaginary and Images M philosophical imaginary refers to both the capacity to imagine and the stock of images philosophers use. ...

_A Brief History of Analytic Philosophy: From Russell to Rawls_ presents a comprehensive overview of the historical development of all major aspects of analytic philosophy, the dominant Anglo-American philosophical tradition in the twentieth century. Features coverage of all the major subject areas and figures in analytic philosophy - including Wittgenstein, Bertrand Russell, G.E. Moore, Gottlob Frege, Carnap, Quine, Davidson, Kripke, Putnam, and many others Contains explanatory background material to help make clear technical philosophical concepts Includes listings of (...) suggested further readings Written in a clear, direct style that presupposes little previous knowledge of philosophy. (shrink)

Critical comments on Guido Melchior’s book, Knowing and Checking: An Epistemological Investigation (2019). In the second part of his book, Melchior aims to employ his sensitivity account of the epistemic concept of checking to explain well-known puzzle cases about knowing. My comments focus on Melchior’s explanation of knowledge-closure puzzles, as exemplified by Dretske’s zebra case. I raise three critical points about the explanation Melchior proposes for puzzles of this type.

Three Steps on the Ladder of Writing is a poetic, insightful, and ultimately moving exploration of 'the strange science of writing.

I shall argue that the prominent realist methodological approach that is adopted by majority of analytic theologians is inconsistent with the Islamic tradition. I will propose that the realist outlook is constituted of two essential components – metaphysical theological realism and epistemic theological realism – both of which fail to be amenable with the Islamic tradition. The prime reason for this, as I shall demonstrate, is that both metaphysical theological realism and epistemic theological realism divest the Islamic God of (...) absolute transcendence in different ways, impinging on the religion in a manner which ironically defies the purpose of analytic theology. In conclusion, this would establish that the prominent realist position adopted by majority of analytic theologians is inconsistent with the Islamic tradition. (shrink)

By virtue of the originality and depth of its thought, Emmanuel Levinas’s masterpiece, _Totality and Infinity: An Essay on Exteriority, _is destined to endure as one of the great works of philosophy. It is an essential text for understanding Levinas’s discussion of “the Other,” yet it is known as a “difficult” book. Modeled after Norman Kemp Smith’s commentary on _Kant’s Critique of Pure Reason, Levinas’s Existential Analytic _guides both new and experienced readers through Levinas’s text. James R. Mensch explicates (...) Levinas’s arguments and shows their historical referents, particularly with regard to Heidegger, Husserl, and Derrida. Students using this book alongside _Totality and Infinity _will be able to follow its arguments and grasp the subtle phenomenological analyses that fill it. (shrink)

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

Seventeenth-century “chance combinatorics” was a self-contained theory. It had an objective notion of chance derived from physical devices with chance properties, such as casts of dice, combinatorics to count chances and, to interpret their significance, a rule for converting these counts into fair wagers. It lacked a notion of chance as a measure of belief, a precise way to connect chance counts with frequencies and a way to compare chances across different games. These omissions were not needed for (...) the theory’s interpretation of chance counts: determining which are fair wagers. The theory provided a model for how indefinitenesses could be treated with mathematical precision in a special case and stimulated efforts to seek a broader theory. (shrink)

Comparing Buddhist and contemporary analytic views about mereological composition reveals significant dissimilarities about the purposes that constrain successful answers to mereological questions, the kinds of considerations taken to be probative in justifying those answers, and the value of mereological inquiry. I develop these dissimilarities by examining three questions relevant to those who deny the existence of composite wholes. The first is a question of justification: What justifies denying the existence of composite wholes as more reasonable than affirming their existence? (...) The second is a question of ontology: Under what conditions are many partless individuals arranged composite-wise? The third is a question of reasonableness: Why, if there are no composites available to experience, do “the folk” find it reasonable to believe there are? I motivate each question, sketch some analytic answers for each, develop in more detail answers from the Theravādin Buddhist scholar Buddhaghosa, and extract comparative lessons. (shrink)

A Brief History of Analytic Philosophy: From Russell to Rawls presents a comprehensive overview of the historical development of all major aspects of analytic philosophy, the dominant Anglo-American philosophical tradition in the twentieth century. Features coverage of all the major subject areas and figures in analytic philosophy - including Wittgenstein, Bertrand Russell, G.E. Moore, Gottlob Frege, Carnap, Quine, Davidson, Kripke, Putnam, and many others Contains explanatory background material to help make clear technical philosophical concepts Includes listings of (...) suggested further readings Written in a clear, direct style that presupposes little previous knowledge of philosophy. (shrink)

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our (...) main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic. (shrink)