Online research in philosophy

There is a clear tendency in contemporary political/legal thought to limit agency to individual agents, thereby denying the existence and relevance of collective moral agency in general, and corporate agency in particular. This tendency is ultimately rooted in two particular forms of individualism – methodological and fictive (abstract) – which have their source in the Enlightenment. Furthermore, the dominant notion of moral agency owes a lot to Kant whose moral/legal philosophy is grounded exclusively on abstract reason and personal autonomy, to (...) the detriment of a due recognition of the socio-historical grounds of moral social conduct.I shall argue that an adequate theory of responsibility is needed, which does not only take into account individual responsibility, but also collective and corporate responsibility, capable of taking into consideration society and its problems. Furthermore, corporations are consciously and carefully structured organisations with different levels of management and have clearly defined aims and objectives, a central feature upon which I shall be focussing in this paper. (shrink)

p { margin-bottom: 0.21cm; } O objetivo do presente artigo é discutir a concepção de Direito, Justiça e Estado no pensamento estoico greco-romano, demonstrando a atualidade do tema e suas conexões com problemas contemporâneos tratados pela Filosofia do Direito, tais como os da legitimidade do poder e do universalismo da ordem jurídica. Em um primeiro momento são apresentados e problematizados elementos centrais da filosofia estoica, tais como as noções de lei natural, liberdade interior, igualdade formal e universalismo. Em seguida, mediante (...) uma análise histórica e filosófica de caráter crítico-comparativo são apresentados os três principais projetos de república pensados pelos estoicos: 1) a república radical e igualitarista de Zenão (ap. 334 a.C. – 262 a.C.), na qual se nega a juridicidade, eis que o direito é entendido como construção artificial que impede a fruição da liberdade; 2) a república legalista de Cícero (106 a.C. – 43 a.C.), quando o direito natural é positivado para servir enquanto instância racional de auto-reflexão e de auto-fundamentação das normas jurídico positivas e, finalmente, 3) a “república” de Sêneca (4 a.C. – 65 d.C.), na qual o direito inicialmente se deixa absorver pelo poder pessoal do Imperador Romano para, em um segundo momento, mostrar-se como elemento universal, preparando assim as bases para a futura concepção de direitos fundamentais própria da modernidade. (shrink)

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...) n-spheres {P k e } k ∈ ω which are also hypersurfaces, such that P k e is diffeomorphic to the standard n-sphere S n (denoted P k e ≈ diff S n ) iff T e halts on input k, and in this case the connected sum N k e =M ♯ P k e ≈ diff M , so N k e ∈ Met(M), and N k e is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time σ e (k) of T e on inputs y i } ∈ ω of c.e. sets so that for all i the settling time of the associated Turing machine for A i dominates that for A i + 1 , even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a "fractal" like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as "the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization." From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structural complexity of the geometry and topology, not merely undecidability results as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they use nontrivial information about c.e. sets, the Soare sequence {A i } i ∈ ω above, not merely G öodel's c.e. noncomputable set K of the 1930's; and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T 5 (see §). (shrink)

Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating properties. (...) Although these computability results had been announced earlier, their proofs have been deferred until this paper. Computably enumerable sets have long been used to prove undecidability of mathematical problems such as the word problem for groups and Hilbert's Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima. (shrink)

We generalize a result on True Arithmetic (TA) by Lachlan and Soare to certain other completions of Peano Arithmetic (PA). If T is a completion of PA, then Rep(T) denotes the family of sets $X \subseteq \omega$ for which there exists a formula φ(x) such that for all n ∈ ω, if n ∈ X, then $\mathscr{T} \vdash \varphi(S^{(n)})$ (0)) and if $n \not\in X$ , then $\mathscr{T} \vdash \neg\varphi(S^{(n)}(0))$ . We show that if $\mathscr{S,J} \subseteq \mathscr{P}(\omega)$ such that S (...) is a Scott set, J is a jump ideal, $\mathscr{S} \subset \mathscr{J}$ and for all X ∈ J, there exists C ∈ S such that C is a "coding" set for the family of subtrees of 2 $^{ computable in X, and if T is a completion of PA such that Rep(T) = S, then there exists a model A of T such that J is the Scott set of A and no enumeration of Rep(T) is computable in A. The model A of T is obtained via a new notion of forcing. Before proving our main result, we demonstrate the existence of uncountably many different pairs (S,J) satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2 $^{ . In particular, $C \subseteq \omega$ is a coding set for the family of subtrees of 2 $^{ computable in X if and only if for all trees T $\subseteq 2^{ computable in X, if χ C is a path through T, then there exists σ ∈ T such that $\sigma \subset \chi_C$ and every extension of σ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2 $^{ . We show they are identical. (shrink)

Say that (a, d) is an isolation pair if a is a c.e. degree, d is a d.c.e. degree, a < d and a bounds all c.e. degrees below d. We prove that there are an isolation pair (a, d) and a c.e. degree c such that c is incomparable with a, d, and c cups d to o', caps a to o. Thus, {o, c, d, o'} is a diamond embedding, which was first proved by Downey in [9]. Furthermore, (...) combined with Harrington-Soare continuity of capping degrees, our result gives an alternative proof of N5 embedding. (shrink)

We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω in (...) which the functions and relations have degree at most c. We show that every degree d ≥ 0 (ω) is the ωth jump degree of a Boolean algebra, but that for $n no Boolean algebra has nth-jump degree $\mathbf{d} > 0^{(n)}$ . The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties. (shrink)

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, $\varepsilon = (\{W_e\}_{e\in \omega}, \subseteq)$ . We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has (...) a certain "slowness" property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ∈ ε there exists B in the orbit of A such that X ≤ T B under relative Turing computability (≤ T ). We produce B using the Δ 0 3 -automorphism method we introduced earlier. (shrink)

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).