We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K (...) for random α. Following work of Downey, Hirschfeldt and LaForte, we say that αK β iff there is a constant such that for all n, . We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that lim supn K−K=∞ where Ω is Chaitin's random real. One is based upon work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that μ=0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n ≠ m then the initial segment complexities of the natural n- and m-random sets and Ω︀) are different. The techniques introduced in this paper have already found a number of other applications. (shrink)

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...) _\alpha ^0 $ orbit (from the proof). (shrink)

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no (...) set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)

In this book R. G. Peffer tackles the challenges of finding in Marx's work an implicit moral theory, of answering claims that Marxism is incompatible with morality, and of developing the outlines of an adequate Marxist moral and social ...

A subspace V of an infinite dimensional fully effective vector space V ∞ is called decidable if V is r.e. and there exists an r.e. W such that $V \oplus W = V_\infty$ . These subspaces of V ∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V ∞ ) and the set of decidable subspaces forms a lower semilattice S(V ∞ ). We analyse S(V ∞ ) and its relationship with L(V (...) ∞ ). We show: Proposition. Let U, V, W ∈ L(V ∞ ) where U is infinite dimensional and $U \oplus V = W$ . Then there exists a decidable subspace D such that U |oplus D = W. Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces. These results allow us to show: Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V ∞ )), is undecidable. This contrasts sharply with the result for recursive sets. Finally we examine various generalizations of our results. In particular we analyse S * (V ∞ ), that is, S(V ∞ ) modulo finite dimensional subspaces. We show S * (V ∞ ) is not a lattice. (shrink)

TheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that for all (...) degreesa, bwhich are notK-trivial anda > d, b > dthere exists a degreevwhich is notK-trivial anda > v, b > v. This work sheds light to the question of the definability of theK-trivial degrees in the c.e. degrees. (shrink)

In my 1990 work – Marxism, Morality, and Social Justice – I argued for four modifications of Rawls’s principles of social justice and rendered a modified version of his theory in four principles, the first of which is the Basic Rights Principle demanding the protection of people’s security and subsistence rights. In both his Political Liberalism and Justice as Fairness Rawls explicitly refers to my version of his theory, clearly accepting three of my four proposed modifications but rejecting the fourth (...) ‐‐ the demand for social and economic democracy – on grounds that it automatically justifies socialism as opposed to capitalism. I argue, contrary to Rawls, that it is not true that this demand automatically picks socialism as the preferablesocioeconomic/political system and that a Social and Economic Democracy Principle demanding workplace and neighborhood democracy is officially neutral between these two systems … although plausible empirical assumptions may, indeed, favor the former. I then significantly elaborate my second version of Rawls’s theory of social justice which is composed of the following principles arranged in a very strong order of priority : Basic Rights Principle, Equal Basic Liberties Principle, Fair Equality of Opportunity Principle, Modified Difference Principle, and Social and Economic DemocracyPrinciple. I argue that this elaborated version of the theory – which I call “Justice as Fair Rights” – is better than either Rawls’s original theory or my previous versions of it. (shrink)

Although I do not find any of Sterba's responses to my recent criticisms of his work How to Make People Just convincing, I shall not attempt to answer them point by point since this would be a boring, scholastic exercise at best.1 Rather, I shall expand upon what I believe continue to be the three major problems with Sterba's theory and explain why his recent responses to my criticisms along these lines are not adequate.

I take up the "What is equality?" controversy begun by Amartya Sen in 1979 by critically considering utility (J. S. Mill), primary goods (John Rawls), property rights (John Roemer) and basic capabilities in terms of what is to be distributed according to principles and theories of social justice. I then consider the four most general principles designed to answer issues raised by the Equality of Welfare principle, Equality of Opportunity for Welfare principle, Equality of Resources principle and Equality of Opportunity (...) for Resources principle. I consider each with respect to the more general normative principle that whatever theory of social or distributive justice we accept should be as ambition sensitive and endowment insensitive as feasible in real world circumstances. In this context I take up the problems of expensive tastes, expensive disabilities, lowered or manipulated preferences or ‘needs,’ and differential needs versus differential talents and abilities. I argue that the best solution is to adopt a modified version of Rawls’ theory which takes primary social goods as that which is to be distributed but which demands a Basic Rights principle that insures basic subsistent rights (as well as basic security rights) as the most fundamental principle of morality (and social justice), and then demands that Rawls’ Difference Principle be applied lexically to the ‘material’ goods of income, wealth, and leisure time, but done so that the social basis of self-respect is never undermined. (shrink)

For over half a century in more than a dozen books and 600 philosophical articles Kai Nielsen has developed and defended a radically egalitarian theory of social justice as well as a political vision demanding a democratic, humane form of socialism and, on an international level, a federative world socialist government embodying these values. In Globalization and Justice Nielsen applies his acute analytical abilities and his substantive theories and views to the present ongoing reality of corporate, capitalist globalization, arguing that (...) this sort of globalization is unjust in that it further disadvantages the developing world. He then argues for an alternative sort of globalization, a process that he believes could and should have, as its end goal, a democratic and humane socialist government and society on a world-wide scale. Along the way he compares his theory and vision to those of such other major contemporary thinkers as John Rawls, Jürgen Habermas, G.A. Cohen, and Richard Rorty representing, respectively, liberal egalitarianism, critical theory, analytical Marxism , and post-modern eclecticism. (shrink)

Given certain well-known empirical facts–including the Bush II administration’s motivations and its actions initiating the war – the U.S. invasion of Iraq in 2003 (and its continuing war of occupation) is not just (i.e., is not morally justified), on any standard interpretation of Just War Theory criteria for jus ad bellum. Since there was no imminent threat of attack by Iraq against the U.S., the U.S. invasion of Iraq was a Preventative or Merely Precautionary War (which is notrecognized by either (...) Just War Theory or international law as a legitimate basis for initiating a war) rather than a Preemptive War (which may sometimes be justified, if there is a real threat of imminent attack) or a Reactive War (responding to an unjustified attack from an aggressor, which is always justified). Moreover, the neo-conservative program for perpetual U.S. world domination by the weakening of other nations and the invasions of weaker nations for purposes of U.S. economic and geopolitical advantage (behind the facade of “spreading democracy and freedom”) is not morally justified. However, the moral status of the U.S. invasion of Afghanistan is much less clear. Many argue that is was morally justified according to both Just War Theory and international law, given certain well-established empirical facts; particularly, al-Qaeda’s involvement with the events of 9-11 and the Taliban government’s protection of al-Qaeda and its terrorist infrastructure within Afghanistan. On this analysis, the U.S. invasion of Afghanistan was justified as both a Reactive War (responding to an unjustified attack against primarily civilian targets) and a Preemptive War (to try to make sure that al-Qaeda did not have the opportunity to use its infrastructure in Afghanistan to arrange other attacks on civilian targets in the U.S. or other nations). But the cogency of this analysis depends on whether there were any realistic alternatives fordisrupting al-Qaeda and bringing its leaders to justice; and some argue that such alternatives did exist. Moreover, even if U.S. military intervention in Afghanistan was morally justified it is arguable that the amount and type of force used – e.g. intensive, wide-spread bombing campaigns that killed many civilians – were not justified. (shrink)

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$ -c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $ -c.a. as shown in [9]. We also show that if cone avoidance is added then there (...) is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$. (shrink)

I canvass the major contending normative theories /approaches concerning the world hungerabsolute poverty problem by going through a set of questions— some normative, some empirical, and some a mixture of both—in order to elucidate what the germane issues are in this ongoing debate and in order to provide a decision procedure for progressively weeding out the less plausible theories from the more plausible ones until we arrive at what I believe to be the most plausible and well-supported theory and solution (...) to this momentous problem. Theories are eliminated if they are empirically unsupportable in terms of their analysis of the problem (or their recommendations aimed at solving the problems) and/or they are morally unsupportable in terms of not showing sufficient concern and respect for people in the design of the principles, rights, duties, and/or obligations they propose or, alternatively, not respecting the root values of autonomy and fairness. My main conclusion is that although Amartya Sen's capability ethic has made important contributions to moral theory, a Rawlsian theory (such as mine) that specifically accepts a Basic Rights principle is preferable to it since, for one reason, it is less vague in what policies are to be recommended on such issues as world hunger (given the same set of empirical assumptions). I also conclude that although Sen's empirical (economic and socialtheoretical) work on famines, hunger, and absolute poverty in general is to be much commended, it contains claims that are highly suspect. (shrink)

I canvass the major contending normative theories /approaches concerning the world hungerabsolute poverty problem by going through a set of questions— some normative, some empirical, and some a mixture of both—in order to elucidate what the germane issues are in this ongoing debate and in order to provide a decision procedure for progressively weeding out the less plausible theories from the more plausible ones until we arrive at what I believe to be the most plausible and well-supported theory and solution (...) to this momentous problem. Theories are eliminated if they are empirically unsupportable in terms of their analysis of the problem (or their recommendations aimed at solving the problems) and/or they are morally unsupportable in terms of not showing sufficient concern and respect for people in the design of the principles, rights, duties, and/or obligations they propose or, alternatively, not respecting the root values of autonomy and fairness. My main conclusion is that although Amartya Sen's capability ethic has made important contributions to moral theory, a Rawlsian theory (such as mine) that specifically accepts a Basic Rights principle is preferable to it since, for one reason, it is less vague in what policies are to be recommended on such issues as world hunger (given the same set of empirical assumptions). I also conclude that although Sen's empirical (economic and socialtheoretical) work on famines, hunger, and absolute poverty in general is to be much commended, it contains claims that are highly suspect. (shrink)

I take up the "What is equality?" controversy begun by Amartya Sen in 1979 by critically considering utility, primary goods, property rights and basic capabilities in terms of what is to be distributed according to principles and theories of social justice. I then consider the four most general principles designed to answer issues raised by the Equality of Welfare principle, Equality of Opportunity for Welfare principle, Equality of Resources principle and Equality of Opportunity for Resources principle. I consider each with (...) respect to the more general normative principle that whatever theory of social or distributive justice we accept should be as ambition sensitive and endowment insensitive as feasible in real world circumstances. In this context I take up the problems of expensive tastes, expensive disabilities, lowered or manipulated preferences or ‘needs,’ and differential needs versus differential talents and abilities. I argue that the best solution is to adopt a modified version of Rawls’ theory which takes primary social goods as that which is to be distributed but which demands a Basic Rights principle that insures basic subsistent rights as the most fundamental principle of morality, and then demands that Rawls’ Difference Principle be applied lexically to the ‘material’ goods of income, wealth, and leisure time, but done so that the social basis of self-respect is never undermined. (shrink)

In this article we show that it is possible to completely classify the degrees of r.e. bases of r.e. vector spaces in terms of weak truth table degrees. The ideas extend to classify the degrees of complements and splittings. Several ramifications of the classification are discussed, together with an analysis of the structure of the degrees of pairs of r.e. summands of r.e. spaces.

We solve a longstanding question of Soare by showing that if ${\mathbf d}$ is a non-low $_2$ computably enumerable degree then ${\mathbf d}$ contains a c.e. set with no r-maximal c.e. superset.