We examine how degrees of computably enumerable equivalence relations under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the (...) degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers. (shrink)

We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of L-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of (N, +, \times) .

We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are (...) definable in $\operatorname {\mathrm {\mathbf {ER}}}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$ needn’t be a strong minimal cover of a self-full degree $\mathbf {d}$. Finally, we show that the theory of the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic. (shrink)

We study effectively inseparable prelattices $\wedge, \vee$ are binary computable operations; ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; the equivalence relation ${ \equiv _L}$ originated by ${ \le _L}$ is a congruence on L such that the corresponding quotient structure is a nontrivial bounded lattice; the ${ \equiv _L}$ -equivalence classes of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in we show, that if (...) L is an e.i. prelattice then ${ \le _L}$ is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relation R there exists a computable function f reducing R to ${ \le _L}$, i.e., $xRy$ if and only if $f\left{ \le _L}f\left$, for all $x,y$. In fact ${ \le _L}$ is locally universal, i.e., for every pair $a{ < _L}b$ and every c.e. preordering relation R one can find a reducing function f from R to ${ \le _L}$ such that the range of f is contained in the interval $\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also ${ \le _L}$ is uniformly dense, i.e., there exists a computable function f such that for every $a,b$ if $a{ < _L}b$ then $a{ < _L}f\left{ < _L}b$, and if $a{ \equiv _L}a\prime$ and $b{ \equiv _L}b\prime$ then $f\left{ \equiv _L}f\left$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. preordering relation on ${{\rm{\Sigma }}_n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s system R or Q is locally universal and uniformly dense; and the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

We describe strongly minimal theories Tn with finite languages such that in the chain of countable models of Tn, only the first n models have recursive presentations. Also, we describe a strongly minimal theory with a finite language such that every non-saturated model has a recursive presentation.

Let$ \le _c $be computable the reducibility on computably enumerable equivalence relations. We show that for every ceerRwith infinitely many equivalence classes, the index sets$\left\{ {i:R_i \le _c R} \right\}$,$\left\{ {i:R_i \ge _c R} \right\}$, and$\left\{ {i:R_i \equiv _c R} \right\}$are${\rm{\Sigma }}_3^0$complete, whereas in caseRhas only finitely many equivalence classes, we have that$\left\{ {i:R_i \le _c R} \right\}$is${\rm{\Pi }}_2^0$complete, and$\left\{ {i:R \ge _c R} \right\}$ is${\rm{\Sigma }}_2^0$complete. Next, solving an open problem from [1], we prove that the index set of (...) the effectively inseparable ceers is${\rm{\Pi }}_4^0$complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a${\rm{\Sigma }}_3^0$complete preordering relation, a fact that is used to show that the preordering relation$ \le _c $on ceers is a${\rm{\Sigma }}_3^0$complete preordering relation. (shrink)

We study the degrees of bi-hyperhyperimmune sets. Our main result characterizes these degrees as those that compute a function that is not dominated by any ∆02 function, and equivalently, those that compute a weak 2-generic. These characterizations imply that the collection of bi-hhi Turing degrees is closed upwards.

We employ an infinite-signature Hrushovski amalgamation construction to yield two results in Recursive Model Theory. The first result, that there exists a strongly minimal theory whose only recursively presentable models are the prime and saturated models, adds a new spectrum to the list of known possible spectra. The second result, that there exists a strongly minimal theory in a finite language whose only recursively presentable model is saturated, gives the second non-trivial example of a spectrum produced in a finite language.

Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined.

We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any Borel (...) quotient of the reals reduces to it. (shrink)

We build a new spectrum of recursive models (SRM(T)) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite structure.

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence (...) relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy. (shrink)

It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with infinitely many (...) classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes. (shrink)

The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $, written ${\dotplus }$, recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$. In particular, we show that this (...) jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$. (shrink)

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \-sentence with few separable randomizations has countably many countable models.

We say that a theory [Formula: see text] satisfies arithmetic-is-recursive if any [Formula: see text]-computable model of [Formula: see text] has an [Formula: see text]-computable copy; that is, the models of [Formula: see text] satisfy a sort of jump inversion. We give an example of a theory satisfying arithmetic-is-recursive non-trivially and prove that the theories satisfying arithmetic-is-recursive on a cone are exactly those theories with countably many [Formula: see text]-back-and-forth types.

We say that a theory T satisfies arithmetic-is-recursive if any X′-computable model of T has an X-computable copy; that is, the models of T satisfy a sort of jump inversion. We give an example of a...

This paper is part of a project that is based on the notion of a dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In Amidei et al. (2016, Rev. Symb. Logic, 9, 1–26) and Amidei et al. (2016, Rev. Symb. Logic, 9, 299–324), we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasi-dialectical systems, that enrich Magari’s systems with a natural mechanism (...) of revision. In the present paper we consider a third class of systems, that of p-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about p-dialectical systems and the sets that they represent. Then we focus on the completions of first-order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and q-dialectical systems coincide with respect to the completions that they can represent. Yet, p-dialectical systems are more powerful; we exhibit a p-dialectical system representing a completion of Peano Arithmetic that is neither dialectical nor q-dialectical. (shrink)

This book focuses on Andrew Arato’s democratic theory and its relevance to contemporary issues such as processes of democratization, civil society, constitution-making, and the modern Executive. Andrew Arato is -both globally and disciplinarily- a prominent thinker in the fields of democratic theory, constitutional law, and comparative politics, influencing several generations of scholars. This is the first volume to systematically address his democratic theory. Including contributions from leading scholars such as Dick Howard, Ulrich Preuss, Hubertus Buchstein, Janos Kis, Uri Ram, Leonardo (...) Avritzer, Carlos de la Torre, and Nicolás Lynch, this book is organized around three major areas of Arato´s influence on contemporary political and social thought. The first section offers a comprehensive view of Arato’s scholarship from his early work on critical theory and Western Marxism to his current research on constitution-making and its application. The second section shifts its focus from the previous, comprehensive approach, to a much more specific one: Arato´s widespread influence on the study of civil society in democratization processes in Latin America. The third section includes a previously unpublished work, ‘A conceptual history of dictatorship (and its rivals,)’ one of the few systematic interrogations on the meaning of a political form of fundamental relevance in the contemporary world. Critical Theory and Democracy will be of interest to critical and social theorists, and all Arato scholars. (shrink)

Review by: Alexander G. Melnikov The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 400-401, September 2013.

How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies that have been deployed in both areas. The first—debunking arguments—aims to put pressure on realism by emphasising the (...) seeming redundancy of mathematical or moral entities when it comes to explaining our judgements. In the moral realm this challenge has been made by Gilbert Harman and Sharon Street; in the mathematical realm it is known as the 'Benacerraf-Field' problem. The second strategy—indispensability arguments—aims to provide support for realism by emphasising the seeming intellectual indispensability of mathematical or moral entities, for example when constructing good explanatory theories. This strategy is associated with Quine and Putnam in mathematics and with Nicholas Sturgeon and David Enoch in ethics. Explanation in Ethics and Mathematics addresses these issues through an explicitly comparative methodology which we call the 'companions in illumination' approach. By considering how argumentative strategies in the philosophy of mathematics might apply to the philosophy of ethics, and vice versa, the papers collected here break new ground in both areas. For good measure, two further companions for illumination are also broached: the philosophy of chance and the philosophy of religion. Collectively, these comparisons light up new questions, arguments, and problems of interest to scholars interested in realism in any area. (shrink)

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that (...) have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

What is free will? Can it exist in a determined universe? How can we determine who, if anyone, possesses it? Philosophers have been debating these questions for millennia. In recent decades neuroscientists have joined the fray with questions of their own. Which neural mechanisms could enable conscious control of action? What are intentional actions? Do contemporary developments in neuroscience rule out free will or, instead, illuminate how it works? Over the past few years, neuroscientists and philosophers have increasingly come to (...) understand that both fields can make substantive contributions to the free-will debate, so working together is the best path forward to understanding whether, when, and how our choices might be free. We therefore asked leading philosophers and neuroscientists which questions related to free will they would most like the other field to answer. Those experts then voted on the 15 most important questions for each field to answer. This book is a collection of the answers to those questions along with follow-up questions from world experts in the neuroscience and philosophy of free will. These varied perspectives will fascinate, illuminate, and stimulate students from both fields along with anyone who wants to be brought up to date on these profound issues. (shrink)

This paper examines the idea of objective phenomenology, or a way of understanding the phenomenal character of conscious experiences that doesn’t require one to have had the kinds of experiences under consideration. My central thesis is that structural facts about experience—facts that characterize purely how conscious experiences are structured—are objective phenomenal facts. I begin by precisifying the idea of objective phenomenology and diagnosing what makes any given phenomenal fact subjective. Then I defend the view that structural facts about experience are (...) objective. I also argue that structural facts about experience, despite being objective, nevertheless still give rise to an explanatory gap. (shrink)

Biological phenomena can be investigated at multiple levels, from the molecular to the cellular to the organismic to the ecological. In typical biology instruction, these levels have been segregated. Yet, it is by examining the connections between such levels that many phenomena in biology, and complex systems in general, are best explained. We describe a computation-based approach that enables students to investigate the connections between different biological levels. Using agent-based, embodied modeling tools, students model the microrules underlying a biological phenomenon (...) and observe the resultant aggregate dynamics. We describe 2 cases in which this approach was used. In both cases, students framed hypotheses, constructed multiagent models that incorporate these hypotheses, and tested these by running their models and observing the outcomes. Contrasting these cases against traditionally used, classical equation-based approaches, we argue that the embodied modeling approach connects more directly to students’ experience, enables extended investigations as well as deeper understanding, and enables “advanced” topics to be productively introduced into the high school curriculum. (shrink)

Perception is riddled with cases of reliable misperception. These are cases in which a perceptual state is tokened inaccurately any time it is tokened under normal conditions. On the face of it, this fact causes trouble for theories that provide an analysis of perceptual content in non-semantic, non-intentional, and non-phenomenal terms, such as those found in Millikan (1984), Fodor (1990), Neander (2017), and Schellenberg (2018). I show how such theories can be extended so that they cover such cases without giving (...) up their core commitments. (shrink)

Some accounts of moral responsibility hold that an agent's responsibility is completely determined by some aspect of the agent's mental life at the time of action. For example, some hold that an agent is responsible if and only if there is an appropriate mesh among the agent's particular psychological elements. It is often objected that the particular features of the agent's mental life to which these theorists appeal (such as a particular structure or mesh) are not necessary for responsibility. This (...) is because there appear to be cases in which an agent acts at an earlier time which causes her to lack the appropriate psychological features at some later time and yet, intuitively, she is responsible at that later .. (shrink)

According to Alix Cohen, Kant defines emotions as ‘feelings’. Although I find her account of Kantian feelings compelling, I provide three reasons to doubt that it is an account of emotions: (1) it is unclear why Cohen identifies emotions with Kantian feelings; (2) some Kantian feelings are not emotions; (3) some Kantian desires may be emotions. I propose, however, that with some qualifications Cohen’s account may be upheld, provided its extra-textual assumptions about emotions are explicated. Against her claim that Kantian (...) feelings have a derived intentionality, I argue that the text is compatible with their being intrinsically intentional. (shrink)

Moral philosophers are, among other things, in the business of constructing moral theories. And moral theories are, among other things, supposed to explain moral phenomena. Consequently, one’s views about the nature of moral explanation will influence the kinds of moral theories one is willing to countenance. Many moral philosophers are (explicitly or implicitly) committed to a deductive model of explanation. As I see it, this commitment lies at the heart of the current debate between moral particularists and moral generalists. In (...) this paper I argue that we have good reasons to give up this commitment. In fact, I show that an examination of the literature on scientific explanation reveals that we are used to, and comfortable with, non-deductive explanations in almost all areas of inquiry. As a result, I argue that we have reason to believe that moral explanations need not be grounded in exceptionless moral principles. (shrink)

We are all familiar with the phenomenon of a proposition seeming true. Many think that these seeming states can yield justified beliefs. Very few have seriously explored what these seeming states are. I argue that seeming states are not plausibly analyzed in terms of beliefs, partial beliefs, attractions to believe, or inclinations to believe. Given that the main candidates for analyzing seeming states are unsatisfactory, I argue for a brute view of seemings that treats seeming states as irreducible propositional attitudes.

This is a discussion of the state of discussion on pragmatic reasons for belief.

Commentators provide two different accounts of desires in Kant: “feeling-based” accounts stress their connection with feelings, while “action-based” accounts view them as causes of action. I argue that “feeling-based” accounts blur the feeling-desire distinction, while the “action-based” accounts conflict with Kantian desires that do not cause action. On my alternative, Kantian desires are dispositions to action normally directed at producing future objects, and so they differ from the feelings they are connected to, which refer to the way we are affected (...) by objects. This account preserves the feeling-desire distinction, is compatible with desires that do not cause action, and anticipates holistic theories of desire that combine dispositional, hedonic, and evaluative components. (shrink)