We show that the lattice L 20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (J). We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π 2 necessary and sufficient condition for embedding a finite pseudolattice into J.

De Interpretatione is among Aristotle's most influential and widely read writings; C. W. A. Whitaker presents the first systematic study of this work, and offers a radical new view of its aims, its structure, and its place in Aristotle's system. He shows that De Interpretatione is not a disjointed essay on ill-connected subjects, as traditionally thought, but a highly organized and systematic treatise on logic, argument, and dialectic.

Perceptual experience, that paradigm of subjectivity, constitutes our most immediate and fundamental access to the objective world. At least, this would seem to be so if commonsense realism is correct — if perceptual experience is (in general) an immediate awareness of mind-independent objects, and a source of direct knowledge of what such objects are like. Commonsense realism raises many questions. First, can we be more precise about its commitments? Does it entail any particular conception of the nature of perceptual experience (...) and its relation to perceived objects, or any particular view of the way perception yields knowledge? Second, what explains the apparent intuitive appeal of commonsense realism? Should we think of it as a kind of folk theory held by most human adults or is there a sense in which we are pre-theoretically committed to it — in virtue of the experience we enjoy or in virtue of the concepts we use or in virtue of the explanations we give? Third, is commonsense realism defensible, in the face of formidable challenges from epistemology, metaphysics and cognitive science? The project of the present volume is to advance our understanding of these issues and thus to shed light on the commitments and credentials of commonsense realism. As you may have guessed from the title, the volume also aims to highlight the key role the concept of causation plays in these debates. Central issues to be addressed include the status and nature of causal requirements on perception, the causal role of perceptual experience, and the relation between objective perception and causal thinking — issues that, as many chapters in the volume bring out, are inseparable from concerns with the very nature of causation. (shrink)

I first seriously contemplated writing a book on degree theory in 1976 while I was visiting the University of Illinois at Chicago Circle. There was, at that time, some interest in ann-series book about degree theory, and through the encouragement of Bob Soare, I decided to make a proposal to write such a book. Degree theory had, at that time, matured to the point where the local structure results which had been the mainstay of the earlier papers in the area (...) were finding a steadily increasing number of applications to global degree theory. Michael Yates was the first to realize that the time had come for a systematic study of the interaction between local and global degree theory, and his papers had a considerable influence on the content of this book. During the time that the book was being written and rewritten, there was an explosion in the number of global theorems about the degrees which were proved as applications of local theorems. The global results, in turn, pointed the way to new local theorems which were needed in order to make further progress. I have tried to update the book continuously, in order to be able to present some of the more recent results. It is my hope to introduce the reader to some of the fascinat ing combinatorial methods of Recursion Theory while simultaneously showing how to use these methods to prove some beautiful global theorems about the degrees. (shrink)

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.

According to John McDowell and Bill Brewer, our experiences have the type of content which can be the content of judgements - content which is the result of the actualization of specific conceptual abilities. They defend this view by arguing that our experiences must have such content in order for us to be able to think about our environment. In this paper I show that they do not provide a conclusive argument for this view. Focusing on Brewer’s version of the (...) argument, I show that it rests on a questionable assumption - namely, that if a subject can recognize the normative bearing of a mental content upon what she should think and do, then this content must be the result of the actualization of conceptual capacities (and in this sense conceptual). I argue that considerations regarding the roles played by experience and concepts in our mental lives may require us to reject this assumption. (shrink)

We investigate homomorphisms of degree structures with various relations, functions and constants. Our main emphasis is on pseudolattices, i.e., partially ordered sets with a join operation and relations simulating the meet operation. We show that there are no finite quotients of the pseudolattice of degrees or of the pseudolattice of degrees 0′, but that many finite distributive lattices are pseudolattice quotients of the pseudolattice of computably enumerable degrees.

Both introspection and empirical studies suggest that visual attention can affect the phenomenology of our visual experience. However, the exact character of such effects is far from clear. My aim in this chapter is to spell out the main difficulties involved in attempting to achieve a clearer view of these effects, and to make some suggestions as to how we can make progress with this issue while avoiding tempting mistakes. I do this by discussing the question of whether there is (...) a sense in which attention to a seen object can be said to contribute to the object’s visual phenomenal salience. It is often suggested that focusing visual attention on a seen object renders the object phenomenally salient. I look at potential evidence for this suggestion, and consider what form, if any, of visual phenomenal salience it may support. (shrink)

We define a class of finite partial lattices which admit a notion of rank compatible with embedding constructions, and present a necessary and sufficient condition for the embeddability of a finite ranked partial lattice into the computably enumerable degrees.

We present a necessary and sufficient condition for the embeddability of a principally decomposable finite lattice into the computably enumerable degrees. This improves a previous result which required that, in addition, the lattice be ranked. The same condition is also necessary and sufficient for a finite lattice to be embeddable below every non-zero computably enumerable degree.

This book presents a unifying framework for using priority arguments to prove theorems in computability. Priority arguments provide the most powerful theorem-proving technique in the field, but most of the applications of this technique are ad hoc, masking the unifying principles used in the proofs. The proposed framework presented isolates many of these unifying combinatorial principles and uses them to give shorter and easier-to-follow proofs of computability-theoretic theorems. Standard theorems of priority levels 1, 2, and 3 are chosen to demonstrate (...) the framework's use, with all proofs following the same pattern. The last section features a new example requiring priority at all finite levels. The book will serve as a resource and reference for researchers in logic and computability, helping them to prove theorems in a shorter and more transparent manner"--Provided by publisher. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversedviews. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certainmathematical questions as internal questions appealing to logic and convention. On the contrary, Quine wasopposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of thispaper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlosimulations, limiting mathematical inquiry to the domain of (...) logic is unjustified. Robustness studies implemented inMonte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

John McDowell (in Mind and World) and Bill Brewer (in Perception and Reason) argue that the content of our perceptual experience is conceptual in the following sense. It is of the type of content that could be the content of a judgement – that is, a content which results from the actualization of two (or more) conceptual abilities. Specifically, they suggest that the conceptual abilities actualized in experience are demonstrative abilities, and thus the resulting content is of the type we (...) may express by means of sentences of the form ‘this is thus’. In this paper I argue that we cannot construe experiential contents in this way. I first outline a construal of the ability to think about a thing being thus which is based on Brewer's discussion of conceptual experiential contents, and which I take to be the best construal available to the conceptualist. I then show that on this construal the demonstrative abilities that account for our experience of properties require intentional focused attention to the relevant properties. The conceptualist is thus committed to holding that we experience only the properties we are intentionally attending to, and I argue that this is implausible. The interest in examining Brewer's conceptualist construal of experiential content and pointing out its shortcoming is not limited merely to an interest in whether there is a workable conceptualist account of experiential content. I suggest that certain aspects of Brewer's construal capture important (and often neglected) aspects of our perceptual experience, and that understanding why the account fails can contribute to our understanding of both experiential content and demonstrative thought. (shrink)

A choice set for a computable linear ordering is a set which contains one element from each maximal block of the ordering. We obtain a partial characterization of the computable linear order-types for which each computable model has a computable choice set, and a full characterization in the relativized case; Every model of the linear order-type α of degree ≤ d has a choice set of degree ≤ d iff α can written as a finite sum of order-types, each of (...) which either has finitely many blocks, or has order-type n · η for some integer n. (shrink)

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose many-one degrees are pairwise (...) incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′. The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈, where M is a maximal set, and M is its complement. We showed that / 1 ≡ / 2 if, and only if, 1 = m 2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree. (shrink)

This paper investigates the emotional import of literary devices deployed in fiction. Reflecting on the often-favored approach in the analytic tradition that locates fictional characters, events, and narratives as sources of readers’ emotions, I attempt to broaden the scope of analysis by accounting for how literary devices trigger non-cognitive emotions. I argue that giving more expansive consideration to literary devices by which authors present content facilitates a better understanding of how fiction engages emotion. In doing so, I also explore the (...) somatic dimension of reading fiction. (shrink)

Postcolonial science studies entails ostensibly contradictory critical and empirical commitments. Science studies scholars influenced by Bruno Latour and Isabelle Stengers embrace forms of realist, radical empiricism, while postcolonial studies scholars influenced by Jacques Derrida trace the limits of the knowable. This essay takes their common use of the term cosmopolitics as an unexpected point of departure for reconciling Derrida’s program with Stengers’s and Latour’s. I read Derrida’s critique of hospitality and Stengers’s and Latour’s ontological politics as necessary complements for conceiving (...) a care-oriented subalternist cosmopolitics, a process of composing common worlds that remains attentive to the limits of representation. (shrink)