A major debate in the philosophy of biology centers on the question of how we should understand the causal structure of natural selection. This debate is polarized into the causal and statistical positions. The main arguments from the statistical side are that a causal construal of the theory of natural selection's central concept, fitness, either (i) leads to inaccurate predictions about population dynamics, or (ii) leads to an incoherent set of causal commitments. In this essay, I argue that neither the (...) predictive inaccuracy nor the incoherency arguments successfully undermine the causal account of fitness. 1 Introduction2 The Importance of Trait Fitness3 Trait Fitness is not a Silver Bullet4 The Fundamental Incoherency Argument5 Car Racing and Trait Fitness Reversals6 Population Subdivisions and Evolution7 The STP and the Argument for the Incoherency of the Causal Account of Fitness8 Conclusions. (shrink)

Paul Ramsey was one of this century's most important ethicists. From the publication of his classic _Basic Christian Ethics_ in 1950 until his death in 1988, his writings decisively shaped moral discourse and reflection in the areas of theology, law, politics, and medicine. This collection of Ramsey's most important essays on Christian, political, and medical ethics displays the scope and depth of his vision, highlighting both the character of his theological commitments and the continuing significance of his work (...) for the pressing moral problems of our day. Selections deal with such issues as race relations, sexuality and marriage, war, the meaning of Christian love, abortion, and medical care for the sick and dying. A general introduction by William Werpehowski and Stephen Crocco evaluates Ramsey's career and accomplishments and reviews contemporary criticism of his output and legacy. Shorter introductions to each selection point out crucial themes and lines of development in Ramsey's thought. (shrink)

A compilation of all previously published writings on philosophy and the foundations of mathematics from the greatest of the generation of Cambridge scholars that included G.E. Moore, Bertrand Russell, Ludwig Wittgenstein and Maynard Keynes.

The present publication forms part of a projected book that F. P. Ramsey drafted but never completed. It survived among his papers and ultimately came into the possession of the University of Pittsburgh in the circumstances detailed in the Editor's Introduction. Our hope in issuing this work at this stage - some sixty years after Ramsey's premature death at the age of 26 - is both to provide yet another token of his amazing philosophical creativity, and also to (...) make available an important datum for the still to be written history of the development of philosophical analysis. This is a book whose appearance will, we hope and expect, be appreciated both by those interested in linguistic philosophy itself and by those concerned for its historical development in the present century. EDITORS'INTRODUCTION 1. THE RAMSEY COLLECTION Frank Plump ton Ramsey was an extra ordinary scholarly phenomenon. Son of a distinguished mathematician and President of Magdalene College, Cambridge and brother of Arthur Michael, eventual Archbishop of Canterbury, Ramsey was closely connected with Cambridge throughout his life, ultimately becoming lecturer in Mathematics in the University. Notwithstanding his great mathematical talent, it was primarily logic and philosophy that engaged his interests, and he wrote original and important contributions to logic, semantics, epistomology, probability theory, philosophy of science, and economics, in addition to seminal work in the foundations of mathematics. (shrink)

Frank Ramsey was a brilliant Cambridge philosopher, mathematician, and economist who died in 1930 at 26 having made landmark contributions to decision theory, game theory, mathematics, logic, semantics, philosophy of science, and the theory of truth. This rich biography tells the story of his extraordinary life and intellectual achievement.

With a new foreword by noted theologian and ethicist Stanley Hauerwas, this classic text on war and the ethics of modern statecraft written at the height of the Vietnam era in 1968 speaks to a new generation of readers. Characterized by a sophisticated yet back-to-basics approach, The Just War begins with the assumption that force is a fact in political life which must either be reckoned with or succumbed to. It then grapples with modern challenges to traditional moral principles of (...) "just conduct" in war, the "morality of deterrence," and a "just war theory of statecraft.". (shrink)

Frank Ramsey's ‘Truth and Probability’ sketches a proposal for the empirical measurement of credences, along with a corresponding set of axioms for a representation theorem intended to characterize the preference conditions under which this measurement process is applicable. There are several features of Ramsey's formal system which make it attractive and worth developing. However, in specifying his measurement process and his axioms, Ramsey introduces the notion of an ethically neutral proposition, the assumed existence of which plays a (...) key role throughout Ramsey's system. A number of later representation theorems have also appealed to ethically neutral propositions. The notion of ethical neutrality has often been called into question — in fact, there seem to be good reasons to suppose that no ethically neutral propositions exist. In this paper, I present several new, Ramsey-inspired representation theorems that avoid any appeal to ethical neutrality. These theorems preserve the benefits of Ramsey's system, without paying the cost of ethical neutrality. (shrink)

According to the Ramsey Test, conditionals reflect changes of beliefs: α > β is accepted in a belief state iff β is accepted in the minimal revision of it that is necessary to accommodate α. Since Gärdenfors’s seminal paper of 1986, a series of impossibility theorems (“triviality theorems”) has seemed to show that the Ramsey test is not a viable analysis of conditionals if it is combined with AGM-type belief revision models. I argue that it is possible to (...) endorse that Ramsey test for conditionals while staying true to the spirit of AGM. A main focus lies on AGM’s condition of Preservation according to which the original belief set should be fully retained after a revision by information that is consistent with it. I use concrete representations of belief states and (iterated) revisions of belief states as semantic models for (nested) conditionals. Among the four most natural qualitative models for iterated belief change, two are identified that indeed allow us to combine the Ramsey test with Preservation in the language containing only flat conditionals of the form α > β. It is shown, however, that Preservation for this simple language enforces a violation of Preservation for nested conditionals of the form α > (β > γ). In such languages, no two belief sets are ordered by strict subset inclusion. I argue that it has been wrong right from the start to expect that Preservation holds in languages containing nested conditionals. (shrink)

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that (...) Wittgenstein was not moved by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey. (shrink)

Haack, S. Is truth flat or bumpy?--Chihara, C. S. Ramsey 's theory of types.--Loar, B. Ramsey 's theory of belief and truth.--Skorupski, J. Ramsey on Belief.--Hookway, C. Inference, partial belief, and psychological laws.--Skyrms, B. Higher order degrees of belief.--Mellor, D. H. Consciousness and degrees of belief.--Blackburn, S. Opinions and chances.--Grandy, R. E. Ramsey, reliability, and knowledge.--Cohen, L. J. The problem of natural laws.--Giedymin, J. Hamilton's method in geometrical optics and Ramsey 's view of theories.

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). (...) In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems. There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f. (shrink)

This searching critique of the United Methodist Bishops' pastoral letter on war and peace in a nuclear age, by America's foremost Christian ethicist, exposes theological flaws from which flow gaps in moral argument and strangely utopian politics. Never before has In Defense of Creation been more thoroughly analyzed. At the same time Paul Ramsey gives a full-length and detailed comparison of the Methodist document with The Challenge of Peace by the U.S. Catholic Bishops. Issues of nuclear ethics, as seen (...) by the leaders of two major churches, are set fully in view for the first time in a single volume. This "ecumenical consultation" is broadened by drawing extensively on the writings of Mennonite theologian John Howard Yoder. The book's larger purpose is to construe an encounter between Christian just-war tradition and Christian pacifism. This comparative discussion of Christian ethics should be of interest to any reader concerned about the nuclear crisis. Some of the questions confronted in these pages are: What do people mean by "nonviolence"? Should we never kill another human being, or never kill another human being unjustly? Do Christian pacifism and Christian just-war teachings have anything in common in their understanding of the Christian moral life? Do different interpretations of the person and work of Jesus Christ give rise to Christian pacifism and to just-war participation? Are these irreducibly different options equally valid for followers of Christ? Do the tests of discrimination and proportion lead to the same prohibitions on war and limits in war in a nuclear age? With an epilogue by Stanley Hauerwas, this volume offers the unusual event of two Methodist laymen engaged in lively debate over their church and the modern world. (shrink)

Frank Ramsey (1931) wrote: If two people are arguing 'if p will q?' and both are in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q. We can say that they are fixing their degrees of belief in q given p. Let us take the first sentence the way it is often taken, as proposing the following test for the acceptability of an indicative conditional: ‘If p then (...) q’ is acceptable to a subject S iff, were S to accept p and consider q, S would accept q. Now consider an indicative conditional of the form (1) If p, then I believe p. Suppose that you accept p and consider ‘I believe p’. To accept p while rejecting ‘I believe p’ is tantamount to accepting the Moore-paradoxical sentence ‘p and I do not believe p’, and so is irrational. To accept p while suspending judgment about ‘I believe p’ is irrational for similar reasons. So rationality requires that if you accept p and consider ‘I believe p’, you accept ‘I believe p’. (shrink)

John Worrall recently provided an account of epistemic structural realism, which explains the success of science by arguing for the correct mathematical structure of our theories. He accounts for the historical failures of science by pointing to bloated ontological interpretations of theoretical terms. In this paper I argue that Worrall’s account suffers from five serious problems. I also show that Pierre Cruse and David Papineau have developed a rival structural realism that solves all of the problems faced by Worrall. This (...) Ramsey-sentence realism is a significant advance in the debate, but still ultimately fails for its incomplete account of reference. (shrink)

This book attempts to explicate and expand upon Frank Ramsey's notion of the realistic spirit. In so doing, it provides a systematic reading of his work, and demonstrates the extent of Ramsey's genius as evinced by both his responses to the Tractatus Logico-Philosophicus , and the impact he had on Wittgenstein's later philosophical insights.

According to philosophical folklore Ramsey maintained three propositions in his famous 1925 paper “Universals”: (i) there is no subject-predicate distinction; (ii) there is no particular-universal distinction; (iii) there is no particular-universal distinction because there is no subject-predicate distinction. The ‘first generation’ of Ramsey commentators dismissed “Universals” because they held that whereas predicates may be negated, names may not and so there is a subject-predicate distinction after all. The ‘second generation’ of commentators dismissed “Universals because they held that the (...) absence of a merely linguistic distinction between subject and predicate does not provide any kind of reason for doubting that a truly ontological (i.e. non-linguistic) distinction obtains between particulars and universals. But both first and second-generation criticisms miss their marks because Ramsey did not maintain the three identified propositions. The failure of commentators to appreciate the point and purpose of the position Ramsey actually advanced in “Universals” results from (a) failing to consider the range of different arguments advanced there, (b) looking at “Universals” in isolation from Ramsey’s other papers and (c) failing to consider Ramsey’s writings in the context of the views that Russell and Wittgenstein held during the early 1920s. Seen from this wider perspective Ramsey arguments in “Universals” take on an altogether different significance. They not only anticipate important contemporary developments⎯the resurgence of Humeanism and the doctrine that the existence of universals can only be established a posteriori⎯but also point beyond them. (shrink)

Can an agent deliberating about an action A hold a meaningful credence that she will do A? 'No', say some authors, for 'Deliberation Crowds Out Prediction' (DCOP). Others disagree, but we argue here that such disagreements are often terminological. We explain why DCOP holds in a Ramseyian operationalist model of credence, but show that it is trivial to extend this model so that DCOP fails. We then discuss a model due to Joyce, and show that Joyce's rejection of DCOP rests (...) on terminological choices about terms such as 'intention', 'prediction', and 'belief'. Once these choices are in view, they reveal underlying agreement between Joyce and the DCOP-favouring tradition that descends from Ramsey. Joyce's Evidential Autonomy Thesis (EAT) is effectively DCOP, in different terminological clothing. Both principles rest on the so-called 'transparency' of first-person present-tensed reflection on one's own mental states. (shrink)

This article rebuts Ramsey's earlier theory, in 'Universals of Law and of Fact', of how laws of nature differ from other true generalisations. It argues that our laws are rules we use in judging 'if I meet an F I shall regard it as a G'. This temporal asymmetry is derived from that of cause and effect and used to distinguish what's past as what we can know about without knowing our present intentions.

'. . . this is a good book that will well serve both students who are new to Ramsey and those who might not be better acquainted with his work . . .'-THE JOURNAL OF RELIGION.

We aim to devise a Ramsey test analysis of actual causation. Our method is to define a strengthened Ramsey test for causal models. Unlike the accounts of Halpern and Pearl ([2005]) and Halpern ([2015]), the resulting analysis deals satisfactorily with both over- determination and conjunctive scenarios.

In the literature over the Ramsey-sentence approach to structural realism, there is often debate over whether structural realists can legitimately restrict the range of the second-order quantifiers, in order to avoid the Newman problem. In this paper, I argue that even if they are allowed to, it won’t help: even if the Ramsey sentence is interpreted using such restricted quantifiers, it is still an implausible candidate to capture a theory’s structural content. To do so, I use the following (...) observation: if a Ramsey sentence did encode a theory’s structural content, then two theories would be structurally equivalent just in case they have logically equivalent Ramsey sentences. I then argue that this criterion for structural equivalence is implausible, even where frame or Henkin semantics are used. (shrink)

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still (...) consistent with V = L. (shrink)

The Ramsey Test is considered to be the default test for the acceptability of indicative conditionals. I will argue that it is incompatible with some of the recent developments in conceptualizing conditionals, namely the growing empirical evidence for the Relevance Hypothesis. According to the hypothesis, one of the necessary conditions of acceptability for an indicative conditional is its antecedent being positively probabilistically relevant for the consequent. The source of the idea is Evidential Support Theory presented in Douven. I will (...) defend the hypothesis against alleged counterexamples, and show that it is supported by growing empirical evidence. Finally, I will present a version of the Ramsey test which incorporates the relevance condition and therefore is consistent with growing empirical evidence for the relevance hypothesis. (shrink)

This paper defends scientific realism from the pessimistic meta-induction from past reference failure. It allows that a descriptive theory of reference implies that scientific terms characteristically fail of determinate reference. But it argues that a descriptive theory of reference also implies an equivalence between scientific theories and quantificational claims in the style of Ramsey. Since these quantificational claims do not use any of the referentially suspect scientific terms, they can be approximately true even when those terms fail to refer (...) determinately.Keywords: Science; Realism; Ramsey sentences; Approximate truth; Reference. (shrink)

I am not so insular and I hope not so presumptuous as to suppose that there is no contemporary philosophy apart from that empiricism which dominates very much of Great Britain, North America and Scandinavia. So let us notice that contemporary philosophy embraces broadly three points of view, though it will be part of my argument that they largely combine in the lessons they have to teach us, and in many of their implications for theology.

Frank Plumpton Ramsey (1903–30) made seminal contributions to philosophy, mathematics and economics. Whilst he was acknowledged as a genius by his contemporaries, some of his most important ideas were not appreciated until decades later; now better appreciated, they continue to bear an influence upon contemporary philosophy. His historic significance was to usher in a new phase of analytic philosophy, which initially built upon the logical atomist doctrines of Bertrand Russell and Ludwig Wittgenstein, raising their ideas to a new level (...) of sophistication, but ultimately he became their successor rather than remain a mere acolyte. (shrink)

The book introduces Ramsey's main doctrines and assesses their contemporary significance. In particular, Jérôme Dokic and Pascal Engel are interested in Ramsey's thoughts on truth and belief, and his pragmatic thesis that the truth of one's beliefs guarantees the success of one's actions. From this, it is a short step to what may be called "Ramsey's principle": the content of a belief is constituted by the success of one's actions. This principle finds its current expression in the (...) work of philosophers who offer evolutionary conceptions of mental states, according to which the success conditions of a belief are constituted by its biological functions. (shrink)

Ancients and moderns alike have constructed arguments and assessed theories on the basis of common sense and intuitive judgments. Yet, despite the important role intuitions play in philosophy, there has been little reflection on fundamental questions concerning the sort of data intuitions provide, how they are supposed to lead us to the truth, and why we should treat them as important. In addition, recent psychological research seems to pose serious challenges to traditional intuition-driven philosophical inquiry. Rethinking Intuition brings together a (...) distinguished group of philosophers and psychologists to discuss these important issues. Students and scholars in both fields will find this book to be of great value. (shrink)

My subject is the arguments brought by Ramsey in his paper “ Universals ” ’ against the generally held distinction between particulars and universals. This paper is provocative, suggestive, and radical, and it is humbling to reflect that its author was just 22 years old when it was published in Mind. As so often with Ramsey, the paper is superficially very easy to follow and hardly requires any introduction other than the imperative, “Read it through”, but underneath the (...) surface are many assumptions which make the paper difficult to interpret, and its argument structure is quite tortuous. Whereas the debate between nominalists and realists has been about whether there are just particulars or whether there are universals as well, Ramsey wants to step back behind the debate and question the basis on which it is made. His aims are primarily destructive: he wants to argue that there is no good reason to believe there is such a distinction. He does not offer much in the way of a positive theory of his own to replace those he considers he is demolishing. It is characteristic of Ramsey’s Cambridge perspective that he puts the debate in terms of the way such questions were considered in Cambridge in his day. His terms of comparison hardly extend beyond Cambridge, which in those days was quite a reasonable stance, since Cambridge provided sufficiently many great philosophers with differing theories for extramural excursions to be an unnecessary luxury. Ramsey accordingly focusses on the supposedly different roles of words for particulars and words for universals in atomic propositions, on the assumption that there will be some fairly straightforward kind of isomorphism between at least atomic propositions and the atomic facts to which they correspond if true. The alert reader will notice that he frequently switches between calling Socrates and ‘Socrates’ the subject of the same proposition. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n (...) and BΣ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

Frank Plumpton Ramsey est, malgré la brièveté de sa vie et de son œuvre, est l’une des figures les plus importantes de la philosophie du vingtième siècle. Elevé dans le Cambridge des années 1920, il fut très vite considéré par Maynard Keynes, Russell, Moore et Wittgenstein comme l’un de leurs pairs. En quelques années, il écrivit un ensemble d’essais pionniers en logique, en mathématiques, en philosophie et en économie. Sa critique de la théorie des types de Russell et son (...) traitement des paradoxes logiques, sa formulation de la théorie des probabilités subjectives et de la théorie de la décision, son analyse de la croyance, de la causalité et des lois, ainsi que du problème des universaux, font aujourd’hui partie de l’héritage de la philosophie analytique et en inspirent encore les travaux les plus contemporains. On trouvera dans ce recueil, issu d’un travail collectif de traduction, ses principaux essais dans ses domaines, de l’article célèbre « Fondements des mathématiques » à ses articles économiques sur la taxation et l’épargne. (shrink)

This paper reconstructs and evaluates the representation theorem presented by Ramsey in his essay 'Truth and Probability', showing how its proof depends on a novel application of Hölder's theory of measurement. I argue that it must be understood as a solution to the problem of measuring partial belief, a solution that in many ways remains unsurpassed. Finally I show that the method it employs may be interpreted in such a way as to avoid a well known objection to it (...) due to Richard Jeffrey. (shrink)

Frank Ramsey was the greatest of the remarkable generation of Cambridge philosophers and logicians which included G. E. Moore, Bertrand Russell, Ludwig Wittgenstein and Maynard Keynes. Before his tragically early death in 1930 at the age of twenty-six, he had done seminal work in mathematics and economics as well as in logic and philosophy. This volume, with a new and extensive introduction by D. H. Mellor, contains all Ramsey's previously published writings on philosophy and the foundations of mathematics. (...) The latter gives the definitive form and defence of the reduction of mathematics to logic undertaken in Russell and Whitehead's Principia Mathematica; the former includes the most profound and original studies of universals, truth, meaning, probability, knowledge, law and causation, all of which are still constantly referred to, and still essential reading for all serious students of these subjects. (shrink)

Guilt poses a unique evolutionary problem. Unlike other dysphoric emotions, it is not immediately clear what its adaptive significance is. One can imagine thriving despite or even because of a lack of guilt. In this article, we review solutions offered by Scott James, Richard Joyce, and Robert Frank and show that although their solutions have merit, none adequately solves the puzzle. We offer an alternative solution, one that emphasizes the role of empathy and posttransgression behavior in the evolution of guilt. (...) Our solution, we contend, offers a better account of why guilt evolved to play its distinctive social role. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence (...) relations on fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

We investigate a variant of the variable convention proposed at Tractatus 5.53ff for the purpose of eliminating the identity sign from logical notation. The variant in question is what Hintikka has called the strongly exclusive interpretation of the variables, and turns out to be what Ramsey initially (and erroneously) took to be Wittgenstein's intended method. We provide a tableau calculus for this identity-free logic, together with soundness and completeness proofs, as well as a proof of mutual interpretability with first-order (...) logic with identity. (shrink)

Most causal discovery algorithms in the literature exploit an assumption usually referred to as the Causal Faithfulness or Stability Condition. In this paper, we highlight two components of the condition used in constraint-based algorithms, which we call “Adjacency-Faithfulness” and “Orientation- Faithfulness.” We point out that assuming Adjacency-Faithfulness is true, it is possible to test the validity of Orientation- Faithfulness. Motivated by this observation, we explore the consequence of making only the Adjacency-Faithfulness assumption. We show that the familiar PC algorithm has (...) to be modified to be correct under the weaker, Adjacency-Faithfulness assumption. The modified algorithm, called Conservative PC (CPC), checks whether Orientation- Faithfulness holds in the orientation phase, and if not, avoids drawing certain causal conclusions the PC algorithm would draw. Howtion: ever, if the stronger, standard causal Faith-. (shrink)

For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is (...) known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following:1.For every integer \, if \ is true for all integers i with \, then \ is true for all integers i with \.2.If \ are any integers such that for some prime p we have \ and \, then in \: \ and \.3.For \, \\\ implies \, and \ implies neither \ nor \ in \.4.For every integer \, \ implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in \ ). In particular, \ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.5.The Chain-AntiChain Principle implies neither \ nor \ in \, for every integer \. (shrink)

The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are nonnull in a certain effective measure-theoretic sense. We show that the sets that can compute infinite homogeneous sets for nonnull many computable stable colorings and the sets that can compute infinite homogeneous sets for all computable stable colorings agree below $\emptyset'$ but (...) not in general. We also answer the analogs of two well-known questions about the stable Ramsey's theorem by showing that our weaker principle does not imply COH or WKL 0 in the context of reverse mathematics. (shrink)

This book provides a much-needed critical introduction to the main doctrines of Frank Ramsey's work and assesses their contemporary significance.