Online research in philosophy

This volume presents ten new essays on the work of Wilfrid Sellars and its implications for contemporary philosophy.

The central concern of McKeever & Ridge’s paper is with whether or not the moral particularist can formulate a defensible distinction between default and non-default reasons. [McKeever & Ridge 2004] But that issue is only of concern to the particularist, they argue, because it allows him or her to avoid a deeper problem, an unacceptable “flattening of the normative landscape”. The particularist ought, McKeever & Ridge claim, to view this corollary of his or her position as a serious embarrassment. Unpacking (...) the metaphor somewhat, the putative problem is that certain moral reasons seem, at their face value, directly to exhibit their relevance to moral decision and others, equally clearly, do not. Examples of the former class are, for example, the fact that an action 2 would inflict pain seems directly to indicate that this is a reason against carrying out the action. McKeever & Ridge cite as an example of an implausible candidate for direct moral relevance the fact that a person’s shoelace is a certain colour. [Little 2000, p. 291] They explain direct relevance as dependent on the content of a moral reason. It is the triviality of the content of this reason, namely, the fact that the colour of a person’s shoelaces is a certain way that makes it seem utterly implausible as a moral reason. McKeever & Ridge further argue that, “there surely is an important difference between considerations of shoelace colour and considerations of pain, pleasure, promising and the like.” [McKeever & Ridge 2004 p.2] They approvingly cite Lance & Little’s observation that an aspect of moral wisdom is that a morally wise person understands “that there is a deep difference in moral status between infliction of pain and shoelace colour”. [Lance & Little, forthcoming]. (shrink)

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x), f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov function, which (...) assigns to each string x its Kolmogorov complexity: • For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n. • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem. • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A. The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity. Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g: • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f. • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f. Furthermore, we deal with the resource-bounded case and give characterizations for the class ${\rm S}_{2}^{{\rm P}}$ introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP. • ${\rm S}_{2}^{{\rm P}}$ is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g. • PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity. • EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACEf for all strong 2-enumerators f for g. Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P = NP. (shrink)