A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice.In addition, Ershov’s completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in (...) a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic.This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices. (shrink)

The different behaviour of total and partial numberings with respect to the reducibility preorder is investigated. Partial numberings appear quite naturally in computability studies for topological spaces. The degrees of partial numberings form a distributive lattice which in the case of an infinite numbered set is neither complete nor contains a least element. Friedberg numberings are no longer minimal in this situation. Indeed, there is an infinite descending chain of non-equivalent Friedberg numberings (...) below every given numbering, as well as an uncountable antichain. (shrink)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...) (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with the (...) classical tree-embeddability, but in this article we are interested in more general situations. We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega}}$$\end{document}. Somewhat surprisingly, changing a little bit the well-partially ordered addresses, the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑΩΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega^\Omega}}$$\end{document}. Our results contribute to the research program on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings. (shrink)

We replace Shelah's notion of true cofinality by the notion of the bounding number for an arbitrary partial order and begin to develop a theory similar to Shelah's pcf theory, which gives many analog results, including the existence of the so-called generators, for the more general case of products of partial orders. The development can be strictly divided into an ideal theoretical and a combinatorial part. We also show that pcf theory is a special case of this more (...) general theory and conclude with some remarks about Shelah's function pp, which also show that there are some differences between pcf theory and the presented theory of bounding numbers. (shrink)

Is there any number of people you should save from paralysis rather than saving one person from death? Is there any number of people you should save from a migraine rather than saving one person from death? Many people answer ‘yes’ and ‘no’, respectively. The aim of partially aggregative moral views is to capture and justify combinations of intuitions like these. These views contrast with fully aggregative moral views, which imply that the answer to both questions is ‘yes’, and with (...) non-aggregative moral views, which imply that the answer to both questions is ‘no’. In this paper, I review the most natural and influential ways of developing partially aggregative views and explain the main problems they face. (shrink)

A number of philosophers have recently argued for acknowledging non-augmented partial grounds, partial grounds that are not parts of full grounds. This paper shows how non-augmented partial grounds can be straightforwardly modelled within the framework of generalised identity. I argue that my proposal answers questions concerning the connections between partial grounding, full grounding, and nothing-over-and-aboveness in a motivated way. In this context, I propose and discuss a way to spell out nothing-over-and-aboveness in terms of generalised identity (...) that does justice to the claim that the fully grounded is nothing over and above its full grounds. It will turn out that the related question whether partial grounding and nothing-over-and-aboveness yield full grounding pertains to considerations concerning how finely factual content is individuated. The relationship between non-augmented partial grounding and nothing-over-and-aboveness crucially depends on whether one works with a representational or a worldly notion of grounding and how coarsely grained one takes worldly grounds to be. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each (...) individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

Popper’s original definition of truthlikeness relied on a central insight: that truthlikeness combines truth and information, in the sense that a proposition is closer to the truth the more true consequences and the less false consequences it entails. As intuitively compelling as this definition may be, it is untenable, as proved long ago; still, one can arguably rely on Popper’s intuition to provide an adequate account of truthlikeness. To this aim, we mobilize some classical work on partial entailment in (...) defining a new measure of truthlikeness which satisfies a number of desiderata. The resulting account has some interesting and surprising connections with other accounts on the market, thus shedding new light on current attempts of systematizing different approaches to verisimilitude. (shrink)

We propose partial measurements as a conceptual tool to understand how to operate with counterfactual claims in quantum physics. Indeed, unlike standard von Neumann measurements, partial measurements can be reversed probabilistically. We first analyze the consequences of this rather unusual feature for the principle of superposition, for the complementarity principle, and for the issue of hidden variables. Then we move on to exploring non-local contexts, by reformulating the EPR paradox, the quantum teleportation experiment, and the entanglement-swapping protocol for (...) the situation in which one uses partial measurements followed by their stochastic reversal. This leads to a number of counter-intuitive results, which are shown to be resolved if we give up the idea of attributing reality to the wavefunction of a single quantum system. (shrink)

The author contests the claim made independently by F.M. Kamm and Thomas Hurka that combatants ought to assign greater weight to collateral harm done to their compatriot noncombatants then they assign to collateral harm done to enemy non-combatants. Two arguments by analogy offered in support of such partiality, one of which appeals to permissible self/other asymmetry in cases of harming the few to save the many, and the second of which appeals to parents' justifiable partiality to their children, are found (...) wanting. The author also rebuts Kamm's argument that combatants should assign greater weight to collateral harm done to neutrals than to compatriot non-combatants. However, he contends that in some cases a neutral state's right to sovereignty may entail that a combatant ought to choose the act that will collaterally kill a greater number of compatriot non-combatants over one that involves collaterally killing a lesser number of neutrals. (shrink)

Parents typically favour their own children over others’. For example, most parents invest more time and money in their own children than in other children. This parental partiality is usually regarded as morally permissible, or even obligatory, but it can have undesirable distributive effects. For example, it may create unfair or otherwise undesirable advantages for the favoured child. A number of authors have found it necessary to justify parental partiality in the face of these distributive concerns, and they have typically (...) done so by appealing to features of the parent–child relationship. Parental partiality is said to be justified, despite its undesirable distributive effects, in part because the parent enjoys a special kind of relationship with her child. In this paper, I raise a problem for such relational defences of parental partiality. I report empirical findings suggesting that parental partiality will frequently create advantages—sometimes undesirable—not only for one’s children, but also for one’s more distant descendants; I argue that the creation of these latter advantages stands as much in need of justification as does the creation of advantages for one’s own children; and I claim that existing relational defences do not clearly contain the resources necessary to deliver such a justification. I then examine three possible responses to this problem. (shrink)

Popper’s original definition of truthlikeness relied on a central insight: that truthlikeness combines truth and information, in the sense that a proposition is closer to the truth the more true consequences and the less false consequences it entails. As intuitively compelling as this definition may be, it is untenable, as proved long ago; still, one can arguably rely on Popper’s intuition to provide an adequate account of truthlikeness. To this aim, we mobilize some classical work on partial entailment in (...) defining a new measure of truthlikeness which satisfies a number of desiderata. The resulting account has some interesting and surprising connections with other accounts on the market, thus shedding new light on current attempts of systematizing different approaches to verisimilitude. (shrink)

The unique problem of lost territory poses one of the most important and complex challenges of compensating for loss and damage due to anthropogenic climate change. Anthropogenic climate change will cause a significant increase in the sea level for centuries to come. A rising sea level endangers many low‐lying coastal areas but also entire states. However, the inundation of an entire state will remain a rare event. Partial loss of territory will be far more pervasive. As measured by the (...) number of states, the area of land, and, in particular, the number of people affected, the phenomenon of partial territorial loss is much more significant. I address this from the perspective of compensatory justice. In particular, I offer (i) background on basic concepts, (ii) an overview and critique of existing accounts, and propose (iii) a novel interpretation of territory as well as (iv) a corresponding criterion of a duty to compensate before (v) I flesh out the overall theory, (vi) take into consideration implications and details of application, and (vii) respond to potential objections. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each (...) individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula φ a formula f(φ) such that for every formula ψ, φ → []ψ is provable in L if and only if f(φ) → ψ is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

African ethical theories tend to hold that moral agents ought to be partial, in the sense that they should favour members of their family or close community. This is considered an advantage over the impartiality of many Western moral theories, which are regarded as having counterintuitive implications, such as the idea that it is unethical to save a family member before a stranger. The partiality of African ethics is thought to be particularly valuable in the context of bioethics. Thaddeus (...) Metz, in particular, argues that his African-derived theory best accounts for a number of plausible intuitions, such as the intuition that health care practitioners ought to be biased towards the patients for whom they are directly responsible. In this article, I claim that the plausible intuitions drawn on in favour of partiality can be satisfactorily explained on the basis of impartial moral theories. Moreover, I argue that blanket acceptance of partiality has problematic consequences for distributive justice in resource allocation in biomedical contexts. By contrast, impartial theories can justify plausible instances of partiality whilst avoiding the pitfalls of fully partial moral theories. Although this provides grounds for abandoning partiality in principle, I claim that this finding should not be seen as damaging to African medical ethics. Instead, it prompts investigation of underexplored possibilities of impartial African moral theories. To demonstrate the value of this direction, I sketch some novel and attractive conceptions that combine impartiality with elements of African ethics. (shrink)

Combining testimonial reports from independent and partially reliable information sources is an important epistemological problem of uncertain reasoning. Within the framework of Dempster–Shafer theory, we propose a general model of partially reliable sources, which includes several previously known results as special cases. The paper reproduces these results on the basis of a comprehensive model taxonomy. This gives a number of new insights and thereby contributes to a better understanding of this important application of reasoning with uncertain and incomplete information.

The debate concerning the role of intuitions in philosophy has been characterized by a fundamental disagreement between two main camps. The first, the autonomists, hold that, due to the use in philosophical investigation of appeals to intuition, most of the central questions of philosophy can in principle be answered by philosophical investigation and argument without relying on the sciences. The second, the naturalists, deny the possibility of a priori knowledge and are skeptical of the role of intuition in providing evidence (...) in philosophical inquiry. In spite of the seemingly stark and deep disagreement between the autonomist and the naturalist, in this paper I will attempt to suggest that there is a way to mount a partial defense of intuition-based philosophical practice on naturalist grounds. As will be seen, doing so will require a number of concessions that will satisfy few autonomists while simultaneously disappointing those naturalists who might have thought that all notions of the a priori had been successfully laid to rest. However, I will argue that the account proposed here will preserve those features of traditional philosophical practice that matter most and better explain the history of successes and failures of that practice, thus answering the worries of the autonomist, and that the rejection of the No A Priori Thesis attendant with the account presented here is indeed demanded by the current theories of our best scientific understanding of the mind, thus pacifying the rabid naturalistic critic of any version of apriorism. (shrink)

Suppose we can save either a larger group of persons or a distinct, smaller group from some harm. Many people think that, all else equal, we ought to save the greater number. This article defends this view (with qualifications). But unlike earlier theories, it does not rely on the idea that several people's interests or claims receive greater aggregate weight. The argument starts from the idea that due to their stakes, the affected people have claims to have a say in (...) the rescue decision. As rescuers, our primary duty is to respect these procedural claims, which we must do by doing what these people would decide, in a process where each is given an equal vote on the matter. So in cases where each votes in their own self-interest, respect for their equal right to decide, or their autonomy, will lead us to save the greater number. The argument is explained in detail, with special attention to the questions of how, exactly, it avoids aggregation, and of why majority rule is superior to lottery procedures. The view has further advantages. Especially, it explains the “partial” relevance of numbers in cases involving unequal harms, and it does so in a way that dissolves the appearance of paradox that besets theories of “partial aggregation.”. (shrink)

Consider the Strong Beneficence Principle (SBP): Persons of affluent means ought to give to those who might fail basic human subsistence until the point at which they must give up something of comparable moral importance. This principle has been the subject of much recent discussion. In this paper, I argue that no coherent interpretation of SBP can be found. SBP faces an interpretive trilemma, each horn of which should be unacceptable to fans of SBP; SBP is either (a) so strong (...) as to be patently absurd; (b) implausible given its acceptance of a form of numbers skepticism; (c) drained of all its demanding force. In the conclusion, I show how the problems with SBP generalize to all similarly demanding principles of beneficence. (shrink)

We provide a detailed history of the concepts of atomic number and isotopy before the discovery of protons and neutrons that draws attention to the role of evolving interplays of multiple aims and criteria in chemical and physical research. Focusing on research by Frederick Soddy and Ernest Rutherford, we show that, in the context of differentiating disciplinary projects, the adoption of a complex and shifting concept of elemental identity and the ordering role of the periodic table led to a relatively (...) coherent notion of atomic number. Subsequent attention to valency, still neglected in the secondary literature, and to nuclear charge led to a decoupling of the concepts of elemental identity and weight and allowed for a coherent concept of isotopy. This concept received motivation from empirical investigations on the decomposition series of radioelements and their unstable chemical identity. A new model of chemical order was the result of an ongoing collaboration between chemical and physical research projects with evolving aims and standards. After key concepts were considered resolved and their territories were clarified, chemistry and physics resumed autonomous projects, yet remained bound by newly accepted explanatory relations. It is an episode of scientific collaboration and partial integration without simple, wholesale gestalt switches or chemical revolutions. (shrink)

Partial matching theory, which maintains that some memory representations of target items in immediate memory are overwritten by others, can predict both a “theoretical” and an “actual” maximum memory span provided no chunking takes place during presentation. The latter is around 4 ± 2 items, the exact number being determined by the degree of similarity between the memory representations of two immediately successive target items.

Clinical experience suggests that adult survivors of childhood trauma arrive at their memories in a number of ways, with varying degrees of associated distress and uncertainty and, in some cases, after memory lapses of varying duration and extent. Among those patients who enter psychotherapy as a result of early abuse, three general patterns of traumatic recall are identified: relatively continuous and complete recall of childhood abuse experiences coupled with changing interpretations of these experiences, partial amnesia for abuse events, accompanied (...) by a mixture of delayed recall and delayed understanding, and delayed recall following a period of profound and pervasive amnesia. These patterns are represented by three composite clinical vignettes. Variations among them suggest that the phenomena underlying traumatic recall are continuous not dichotomous. Future research into the nature of traumatic memory should be informed by clinical observation. (shrink)

In recent years, semantical partiality has emerged as an important explanatory concept in philosophical logic as well as in the study of natural language semantics. Despite the many applications, however, a number of mathematically intriguing questions associated with this concept have received only very limited attention. ;The present dissertation aims to present a systematic study of certain types of partiality in the area of basic model theory. Two types of issues are given special attention: Introducing partially defined models, there are (...) many ways to generalize the classical truth definition for sentences of a simple first order language relative to standard, complete models. Different interpretations of the formal framework motivate conflicting truth definitions between language and partial models: A partial model can be taken to represent a part of the world, or a partial information set. The truth of a sentence can be supported directly by a part of the world, but can also follow indirectly from an information set. These notions are related, and the relation motivates a comparison between various weaker and stronger alternative truth definitions. Results are obtained about the extent to which these truth definitions differ, and a number of characterization results are deduced. ;Among other conditions that are not expressible in the framework of standard, complete model theory, a condition of monotonicity or persistence of truth relative to partial models is argued to follow under both the given interpretations of the formal framework. The final chapter investigates the relation between such conditions and expressibility properties in general. These discussions culminate with a combined Lindstrom and persistence characterization theorem. (shrink)

African ethical theories tend to hold that moral agents ought to be partial, in the sense that they should favour members of their family or close community. This is considered an advantage over the impartiality of many Western moral theories, which are regarded as having counterintuitive implications, such as the idea that it is unethical to save a family member before a stranger. The partiality of African ethics is thought to be particularly valuable in the context of bioethics. Thaddeus (...) Metz, in particular, argues that his African-derived theory best accounts for a number of plausible intuitions, such as the intuition that health care practitioners ought to be biased towards the patients for whom they are directly responsible. In this article, I claim that the plausible intuitions drawn on in favour of partiality can be satisfactorily explained on the basis of impartial moral theories. Moreover, I argue that blanket acceptance of partiality has problematic consequences for distributive justice in resource allocation in biomedical contexts. By contrast, impartial theories can justify plausible instances of partiality whilst avoiding the pitfalls of fully partial moral theories. Although this provides grounds for abandoning partiality in principle, I claim that this finding should not be seen as damaging to African medical ethics. Instead, it prompts investigation of underexplored possibilities of impartial African moral theories. To demonstrate the value of this direction, I sketch some novel and attractive conceptions that combine impartiality with elements of African ethics. (shrink)

Argues that a pluralistic understanding of truth can foster productive conversations about common concerns involving religion, science, ethics, politics, economics, and ecology without falling into relativism. In this book, Donald A. Crosby defends the idea that all claims to truth are at best partial. Recognizing this, he argues, is a necessary safeguard against arrogance, close-mindedness, and potentially violent reactions to differences of outlook and practice. Crosby demonstrates how “partial truths” are inevitably at work in conversations and debates about (...) religion, science, morality, economics, ecology, and social and political progress. He then focuses on the concept in the discipline of philosophy, looking at a number of distinctions that are taken to be strictly binary—those between fact and value, continuity and novelty, rationalism and empiricism, mind and body, and good and evil—and demonstrates how in all of these cases, each on its own can offer only an incomplete picture. Partial Truths and Our Common Future invites ongoing dialogue with others for the sake of mutual enlargements of understanding rather than mere civility, and provides incentive for continuing open-minded and shared inquiries into the important issues of life. “This is a transdisciplinary philosophical work that moves with grace across traditions, time periods, and thinkers. It is a master class in the existential and public relevance of philosophy and a rare example of a book that is both timely and timeless.” — Michael S. Hogue, author of The Promise of Religious Naturalism. (shrink)

In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological T 0 -spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded (...) into a totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets. (shrink)

Partial/particular will (al-irādah al-juz’iyyah) and the creation of human acts are two issues related to the predestination belief. Nowadays, it is unarguably accepted that humans have volition. However, the controversy over the formation steps of human will and act does not seem to be settled. Māturīdīs’ approach, taken with the intent to allow some space for freedom for humans in their actions and based on the partial will and postulation that there is a part in human actions that (...) are not created nor subject to creation, could not be understood at times. When it comes to misunderstandings of this kind, Sadr al-Sharī‘a and Ibn al-Humām, who are among the Hanafī-Māturīdī kalam scholars, are mentioned. Both mutakallims are some of the eminent figures who left their marks on the Hanafī-Māturīdī school. They affected a significant number of scholars coming after them by the authentic ideas they presented. However, even though neither Sadr al-Sharī‘a nor Ibn al-Humām did regard humans as creators of their own actions, they are confronted with an accusation as if they vitiated the fact that God is the Absolute Creator. Therefore, it is of grave importance to reveal Sadr al-Sharī‘a’s ideas, which are represented in his “Four Premises” concerning the formation of an act and Ibn Humām’s thoughts put forward in his “Ījādu al-‘Azm al-Musammamah” and “Takhsīsu Irādah al-Juz’iyyah.”. (shrink)

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book is the first (...) to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time. The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. (shrink)

This paper presents a monotonic system of Post algebras of order +* whose chain of Post constans is isomorphic with 012 ... -3-2-1. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, succesor, and predecessor operations. The successor and predecessor operations are basic for number theory.

We consider so‐called extremal numberings that form the greatest or minimal degrees under the reducibility of all A‐computable numberings of a given family of subsets of, where A is an arbitrary oracle. Such numberings are very common in the literature and they are called universal and minimal A‐computable numberings, respectively. The main question of this paper is when a universal or a minimal A‐computable numbering satisfies the Recursion Theorem (with parameters). First we prove that the Turing (...) degree of a set A is hyperimmune if and only if every universal A‐computable numbering satisfies the Recursion Theorem. Next we prove that any universal A‐computable numbering satisfies the Recursion Theorem with parameters if A computes a non‐computable c.e. set. We also consider the property of precompleteness of universal numberings, which in turn is closely related to the Recursion Theorem. Ershov proved that a numbering is precomplete if and only if it satisfies the Recursion Theorem with parameters for partial computable functions. In this paper, we show that for a given A‐computable numbering, in the general case, the Recursion Theorem with parameters for total computable functions is not equivalent to the precompleteness of the numbering, even if it is universal. Finally we prove that if A is high, then any infinite A‐computable family has a minimal A‐computable numbering that satisfies the Recursion Theorem. (shrink)

Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to (...) a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)

A type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings). By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and (...) the hereditary effective operations (HEO) are easily characterized. (shrink)