Unrestricted Quantification and Reality: Reply to Kim

In my book, Worlds and Individuals, Possible and Otherwise, I use the novel idea of modal tense to respond to a number of arguments against modal realism. Peter van Inwagen’s million-carat-diamond objection is one of them. It targets the version of modal realism by David Lewis and exploits the fact that Lewis accepts absolutely unrestricted quantification. The crux of my response is to use modal tense to neutralize absolutely unrestricted quantification. Seahwa Kim says that even when equipped with modal tense, I am unsuccessful, given my view of reality and the proper use of modal tense in speaking of reality. I counter her attempt at resurrecting van Inwagen’s objection and clarify how we should use modal tense and how we should talk about reality.     

Artistic Expression Goes Green

The paper is a critical discussion of the rich and insightful final chapter of Mitchell Green’s Self-Expression. There, Green seeks to elucidate the compelling, but inchoate intuition that when we’re fully and most expertly expressing ourselves, we can ‘push out’ from within not just our inner representations, but also the ways that we feel. I question, first, whether this type of ‘qualitative expression’ is really distinct from the other expressive forms that Green explores, and also whether it’s genuinely ‘expressive’. I then scrutinize the nature of the ‘qualitative congruences’ that lie at the heart of Green’s theory; and I wonder whether they can play the role Green claims they can in providing a novel account of artistic expression.     

Illusory possibilities and imagining counterparts

Given Kripke’s semantic views, a statement, such as ‘Water is H₂O’, expresses a necessary a posteriori truth. Yet it seems that we can conceive that this statement could have been false; hence, it appears that we can conceive impossible states of affairs as holding. Kripke used a de dicto strategy and a de re strategy to address three illusions that arise with respect to necessary a posteriori truths: (1) the illusion that a statement such as ‘Water is H₂O’ possibly expresses a falsehood, (2) the illusion that conceivability can fail to latch on to a genuine metaphysical possibility, and (3) the illusion that one can access a real metaphysical possibility by conceiving that water is not H₂O. In this paper I argue that while Kripke’s de dicto strategy dispels (1), his strategies do not enable him to dispel (2) and (3).     

‘Seeing is believing’ and ‘believing is seeing’

The principal concern of my paper is a distinction between two ways of appreciating works of art, characterised here in terms of the phrases ‘seeing is believing’ and ‘believing is seeing’. I examine this distinction in the light of an epistemological requirement at times at least grounded in what David Davies, in his Art as Performance, refers to as the ‘common sense theory of art appreciation’ in order to assess exactly what aspect of the philosophical approach generally known as aesthetic empiricism his account commits him to reject. I argue that the ‘experiential requirement’, if only conceived in a slightly broader way than is usual, might very well have an important role to play not only in the appropriate appreciation of works that do not demonstrate the need for such a requirement (primarily works of late modern and conceptual art), but also in the ontological account Davies himself favours.     

Program explanations and causal relevance

Frank Jackson and Philip Pettit have defended a non-reductive account of causal relevance known as the ‘program explanation account’. Allegedly, irreducible mental properties can be causally relevant in virtue of figuring in non-redundant program explanations which convey information not conveyed by explanations in terms of the physical properties that actually do the ‘causal work’. I argue that none of the possible ways to spell out the intuitively plausible idea of a program explanation serves its purpose, viz., defends non-reductive physicalism against Jaegwon Kim’s Causal Exclusion Argument according to which non-reductive physicalism is committed to epiphenomenalism because irreducible mental properties are ‘screened off’ from causal relevance by their physical realizers. Jackson and Pettit’s most promising explication of a program explanation appeals to the idea of invariance of effect under variation of realization, but I show that invariance of effect under variation of realization is neither necessary nor sufficient for causal relevance.     

Baker on the Psychological Account of Personal Identity

Lynne Rudder Baker’s Constitution View of human persons has come under much recent scrutiny. Baker argues that each human person is constituted by, but not identical to, a human animal. Much of the critical discussion of Baker’s Constitution View has focused upon this aspect of her account. Less has been said about the positive diachronic account of personal identity offered by Baker. Baker argues that it is sameness of what she labels ‘first-person perspective’ that is essential to understanding personal identity over time. Baker claims that her account avoids the commitment to indeterminacy of personal identity entailed by the psychological account. Further, the psychological account, but not her account, is plagued by what Baker labels the ‘duplication problem’. In the end, I argue that neither of these considerations forces us to renounce the psychological account and adopt Baker’s favored account.

Is There A Quasi-Mereological Account of Property Incompatibility?

Armstrong’s combinatorial theory of possibility faces the obvious difficulty that not all universals are compatible. In this paper I develop three objections against Armstrong’s attempt to account for property incompatibilities. First, Armstrong’s account cannot handle incompatibilities holding among properties that are either simple, or that are complex but stand to one another in the relation of overlap rather than in the part/ whole relation. Secondly, at the heart of Armstrong’s account lies a notion of structural universals which, building on an objection by David Lewis, is shown to be incoherent. I consider and reject two alternative ways of construing the composition of structural universals in an attempt to meet Lewis’ objection. An important consequence of this is that all putative structural properties are in fact simple. Finally, I argue that the quasi-mereological account presupposes modality in a way that undermines the reductionist aim of the combinatorialist theory of which it is a central part. I conclude that Armstrong’ quasi-mereological account of property incompatibility fails. Without that account, however, Armstrong’s combinatorial theory either fails to get off the ground, or else must give up its goal of reducing the notion of possibility to something non-modal.     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

Isotonic tests for spread and tail

Summary One-sample test problem for ‘stochastically more (or less) spread’ is defined and a family of tests with isotonic power is given. The problem is closely related to that for ‘longer (or shorter) tail’ in the reliability theory and the correspondence between them is shown. To characterize the tests three spread preorders inR ⁿ and corre-sponding tail preorders inR ₊ ⁿ are introduced. Functions which are ‘monotone’ in these orders, and subsets which are ‘centrifugal’ or ‘centripetal’ with respect to these orders are studied. These notions generalize the Schur convexity. The Institute of Statistical Mathematics     

Direct realism: Proximate causation and the missing object

Direct Realists believe that perception involves direct awareness of an object not dependent for its existence on the perceiver. Howard Robinson rejects this doctrine in favour of a Sense-Datum theory of perception. His argument against Direct Realism invokes the principle ‘same proximate cause, same immediate effect’. Since there are cases in which direct awareness has the same proximate cerebral cause as awareness of a sense datum, the Direct Realist is, he thinks, obliged to deny this causal principle. I suggest that although Direct Realism is in more than one respect implausible, it does not succumb to Robinson’s argument. The causal principle is true only if ‘proximate cause’ means ‘proximate sufficient cause’, and the Direct Realist need not concede that there is a sufficient cerebral cause for direct awareness of independent objects.     

S-convexity revisited (fuzzy): long version

In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves on, of the name ‘convexity’ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem to be incompatible with the most basic consequences of having the name ‘convexity’ associated to them. We then believe to have fixed the ‘denominations’ associated with Medar’s (et al.) work, up to a point of having it all matching the existing literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s ₁-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s ₂-convex, so far. This article is a long version of our previous review of Medar’s work, published by FJMS (Pinheiro, M.R.: S-convexity revisited. FJMS, 26/3, 2007).     

Roof duality,complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx _i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.     

Peripheral fillings of relatively hyperbolic groups

In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π₁(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26     

Schmidt’s game on fractals

We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some. In particular, we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of ℝ ^N satisfying the open set condition, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then for any countable collection of non-singular affine transformations, Δ _i : ℝ ^N → ℝ ^N ,

‘Fat’ triangulations,or solving certain nonconvex matrix optimization problems

In an earlier paper, the author proposed the problems of determining ‘optimal’ linear transformations of the triangulationsJ ₁ andK ₁, in the sense of minimizing their average directional density for a given mesh size. These tasks were also formulated as optimization problems where the variable is a matrix. Here we solve these problems, and another one which is analogously related to finding an ‘optimal’ linear transformation of the new triangulationJ′. We show thatJ ₁ andJ′ are themselves optimal, while the (α*β*) ofK ₁ developed by van der Laan and Talman is optimal. The latter theorem extends partial results of van der Laan and Talman and Eaves. The optimality of these linear transformations is quite robust: we may change the objective function to maximizing the volume of each simplex, or the constraints to limiting the sum of squares of edge lengths of each simplex, or both, without changing the optimal solutions. Research partially supported by a fellowship from the Alfred P. Sloan Foundation and by NSF Grant ENG82-15361     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.     

Hardy Spaces on ? n with Pointwise Variable Anisotropy

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝ ⁿ that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝ ⁿ and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.     

Bar-On on Self-Knowledge and Expression

I critically discuss the account of self-knowledge presented in Dorit Bar-On’s Speaking My Mind (OUP 2004), focusing on Bar-On’s understanding of what makes our capacity for self-knowledge puzzling and on her ‘neo-expressivist’ solution to the puzzle. I argue that there is an important aspect of the problem of self-knowledge that Bar-On’s account does not sufficiently address. A satisfying account of self-knowledge must explain not merely how we are able to make accurate avowals about our own present mental states, but how we can reasonably regard ourselves as entitled to claim self-knowledge. Addressing this aspect of the problem of self-knowledge requires confronting questions about the metaphysical nature of mental states, questions that Bar-On’s approach seeks to avoid.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Moore’s Paradox,Truth and Accuracy

G. E. Moore famously observed that to assert ‘I went to the pictures last Tuesday but I do not believe that I did’ would be ‘absurd’. Moore calls it a ‘paradox’ that this absurdity persists despite the fact that what I say about myself might be true. Krista Lawlor and John Perry have proposed an explanation of the absurdity that confines itself to semantic notions while eschewing pragmatic ones. We argue that this explanation faces four objections. We give a better explanation of the absurdity both in assertion and in belief that avoids our four objections.