Expressing and Describing Experiences. A Case of Showing Versus Saying

Experiences are interpreted as conscious mental occurrences that are of phenomenal character. There is already a kind of (weak) intentionality involved with this phenomenal interpretation. A stricter conception of experiences distinguishes between purely phenomenal experiences and intentional experiences in a narrow sense. Wittgenstein’s account of psychological (experiential) verbs is taken over: Usually, expressing mental states verbally is not describing them. According to this, “I believe” can be seen as an expression of one’s own belief, but not as an expression of a belief about one’s belief. Hence, the utterance “I believe it is raining” shows that I believe that it is raining, although it is not said by these words that I believe that it is raining. Thinking thoughts such as “I believe it is raining, but it is not raining” (a variant of Moore’s paradox) is an absurdity between what is already said by silently uttering “It is not raining” and what is shown by silently uttering “I believe it is raining.” The paper agrees with a main result of Wittgenstein’s considerations of Moore’s paradox, namely the view that logical structure, deducibility, and consistency cannot be reduced solely to propositions—besides a logic of propositions, there is, for example, a logic of assertions and of imperatives, respectively.     

What Mary’s Aboutness Is About

The aim of this paper is to reinforce anti-physicalism by extending the “hard problem” to a specific kind of intentional states. For reaching this target, I investigate the mental content of the new intentional states of Jackson’s Mary. I proceed in the following way: I start analyzing the knowledge argument, which highlights the “hard problem” tied to phenomenal consciousness. In a second step, I investigate a powerful physicalist reply to this argument: the phenomenal concept strategy. In a third step, I propose a constitutional account of phenomenal concepts that captures the Mary scenario adequately, but implies anti-physicalist referents. In a last step, I point at the ramifications constitutional phenomenal concepts have on the constitution of Mary’s new intentional states. Therefore, by focusing the attention on phenomenal concepts, the so-called “hard problem” of consciousness will be carried over to the alleged “easy problem” of intentional states as well.     

How to make sense of the common prior assumption under incomplete information

Recent contributions have questioned the meaningfulness of the Common Prior Assumption (CPA) in situations of incomplete information. We characterize the CPA in terms of the primitives (individuals' belief hierarchies) without reference to an ex ante stage. The key is to rule out “agreeing to disagree” about any aspect of beliefs. Our results also yield a generalization of single-person Bayesian updating to situations without perfect recall. The entire analysis is carried out locally at the “true state”, using beliefs only, rather than beliefs-plus-knowledge. We discuss the role of truth assumptions on beliefs for a satisfactory notion of the CPA, and point out an important conceptual discontinuity between the case of two and many individuals.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras H^∞(E), which we call Hardy algebras, and which are noncommutative generalizations of classical H^∞, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which “reproduces” the values, on , of the “functions” coming from H^∞(E). We present results that are natural generalizations of the Poisson integral formula. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the “model space” for the canonical model that can be attached to a point in the disc . We also connect our Poisson kernel to various “point evaluations” and to the idea of curvature. The first named author was supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation. The second named author was supported in part by the U.S.-Israel Binational Science Foundation and by the B. and G. Greenberg Research Fund (Ottawa).     

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

Stochastic programming: Potential hazards when random variables reflect market interaction

There are two types of random phenomena modeled in stochastic programs. One type is what we may term “external” or “natural” random variables, such as temperature or the roll of a dice. But in many other cases, random variables are used to reflect the behavior of other market participants. This is the case for such as price and demand of a product. Using simple game theoretic models, we demonstrate that stochastic programming may not be appropriate in these cases, as there may be no feasible way to replace the decisions of others by a random variable, and arrive at the correct decision. Hence, this simple note is a warning against certain types of stochastic programming models. Stochastic programming is unproblematic in pure forms of monopoly and perfect competition, and also with respect to external random phenomena. But if market power is involved, such as in oligopolies, the modeling may not be appropriate.     

Ordered sets with interval representation and (m,n)-Ferrers relation

Semiorders may form the simplest class of ordered sets with a not necessarily transitive indifference relation. Their generalization has given birth to many other classes of ordered sets, each of them characterized by an interval representation, by the properties of its relations or by forbidden configurations. In this paper, we are interested in preference structures having an interval representation. For this purpose, we propose a general framework which makes use of n-point intervals and allows a systematic analysis of such structures. The case of 3-point intervals shows us that our framework generalizes the classification of Fishburn by defining new structures. Especially we define three classes of ordered sets having a non-transitive indifference relation. A simple generalization of these structures provides three ordered sets that we call “d-weak orders”, “d-interval orders” and “triangle orders”. We prove that these structures have an interval representation. We also establish some links between the relational and the forbidden mode by generalizing the definition of a Ferrers relation.     

Conditional lexicographic orders in constraint satisfaction problems

The lexicographically-ordered CSP (“lexicographic CSP” or “LO-CSP” for short) combines a simple representation of preferences with the feasibility constraints of ordinary CSPs. Preferences are defined by a total ordering across all assignments, such that a change in assignment to a given variable is more important than any change in assignment to any less important variable. In this paper, we show how this representation can be extended to handle conditional preferences in two ways. In the first, for each conditional preference relation, the parents have higher priority than the children in the original lexicographic ordering. In the second, the relation between parents and children need not correspond to the importance ordering of variables. In this case, by obviating the “overwhelming advantage” effect with respect to the original variables and values, the representational capacity is significantly enhanced. For problems of the first type, any of the algorithms originally devised for ordinary LO-CSPs can also be used when some of the domain orderings are dependent on assignments to “parent” variables. For problems of the second type, algorithms based on lexical orders can be used if the representation is augmented by variables and constraints that link preference orders to assignments. In addition, the branch-and-bound algorithm originally devised for ordinary LO-CSPs can be extended to handle CSPs with conditional domain orderings.     

Signed words and permutations, V; a sextuple distribution

We calculate the distribution of the sextuple statistic over the hyperoctahedral group B _n that involves the flag-excedance and flag-descent numbers “fexc” and “fdes,” the flag-major index “fmaj,” the positive and negative fixed point numbers “ ” and “ ” and the negative letter number “neg.” Several specializations are considered. In particular, the joint distribution for the pair is explicitly derived.      

Virtual and effective control for distributed systems and decomposition of everything

For the numerical approximation of the solution of boundary value problems (BVP), decomposition techniques are very important, in particular in view of parallel computations. The same is true, in principle, for optimal control of distributed systems, i.e., systems governed (modelled) by partial differential equations (PDE). Very many techniques have been studied for the approximation of BVP, such as DDM (domain decomposition method), decomposition of operators (splitting up, for instance). In contrast, not so many techniques of decomposition have been used in control problems for distributed systems, as pointed out in the contributions of Benamou [1], Benamou and Després [2], and Lagnese and Leugering [11]. However, it has been observed by Pironneau and Lions [24], [26] that by using so-calledvirtual controls, systematic DDM can be obtained, and that problems of optimal control and analysis of BVP can be considered in the same framework. We then deal with virtual control problems for BVP, virtual and effective control problems for the control of PDE (cf. Pironneau and Lions [25]). Using the idea of virtual control in other guises, Glowinski, Lions and Pironneau [9] have shown how to obtain new decomposition methods for the energy spaces (cf. Section 3), and Pironneau and Lions [27] have shown how to obtain systematically operator decomposition in BVP. In the present paper, we show (without assuming prior knowledge) how to apply the virtual control ideas in several different guises to the “decomposition of everything” for PDE of evolution and for their control. In this way, one can decompose the geometrical domain, the energy space and the operator. This is briefly presented in Sections 2, 3 and 4. We show in Section 5 how one can simultaneously apply two of the decomposition techniques and also indicate briefly how virtual control ideas can be used in case of bilinear control. The content of this paper is presented here for the first time. It is part of a systematic program which is in progress, developed with several colleagues. I wish to thank particularly F. Hecht, R. Glowinski, J. Periaux, O. Pironneau, H. Q. Chen and T. W. Pan. Of course we do not claim by any means that the methods based on “virtual control” are “better” than the many decomposition techniques already available (no attempt has been made to compile a Bibliography on these topics). Numerical works in progress show that the methods are “not bad”, but no serious benchmarking has been made yet. Possible interest lies in the fact thatone technique, with some variants, seems to lead to all possible Decompositions of Everything.     

Nevanlinna theory, diophantine approximation, and numerical analysis

This article treats the problem of the approximation of an analytic function f on the unit disk by rational functions having integral coefficients, with the goodness of each approximation being judged in terms of the maximum of the absolute values of the coefficients of the rational function. This relates to the more usual approximation by a rational function in that it could imply how many decimal places are needed when applying a particularly good rational function approximation having non-integrad coefficients. It is shown how to obtain “good” approximations of this type and it is also shown how under certain circumstances “very good” bounds are not possible. As in diophantine approximation this means that many merely “good” approximations do exist, which may be the preferable case. The existence or nonexistence of “very good” approximations is closely related to the diophantine approximation of the first nonzero power series coefficient of at z=0. Nevanlinna theory methods are used in the proofs.     

A Short and Partial History of Probabilistic Normed Spaces

This contribution aims at presenting a survey of a portion of the theory of Probabilistic Normed spaces. No result will be proved, so that the reader is referred to the original sources for the proofs. The theory of PN spaces has many facets and touches on many branches of mathematics, for instance, geometry, functional analysis, topology, probability. This justifies the adjective “partial” that appears in the title. Therefore, it is perhaps better to declare from the start what one may expect from this survey. It is only natural to investigate which “classical”properties of normed spaces are preserved in the new setting. But it is probably more interesting to look for those properties that pertain to the new theory and which have no corresponding analogue in the classical theory. It must also be added that Probabilistic Normed spaces may be approached from different standpoints: they may be studied for their own sake, as a special subject in functional analysis or in topology that is worth investigating simply because it is there, or because it provides a tool to approach open problems or, again, to shed light on topics that one thought had been thoroughly investigated. In this, one thinks immediately of the possible applications in probability and statistics. This paper was written as a part of the project “Metodi stocastici in finanza matematica” of the Italian M.I.U.R..     

Is Believing at Will ‘Conceptually Impossible’?

In this paper I discuss the claim that believing at will is ‘conceptually impossible’ or, to use a formulation encountered in the debate, “that nothing could be a belief and be willed directly”. I argue that such a claim is only plausible if directed against the claim that believing itself is an action-type. However, in the debate, the claim has been univocally directed against the position that forming a belief is an action-type. I argue that the many arguments offered in favor of the ‘conceptual impossibility’ of performing such actions fail without exception. If we are to argue against doxastic voluntarism we are better off by resorting to more modest means.

Polyhedral surfaces in wedge products

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated “subdirect product” as introduced by McMullen (Discrete Math 14:347–358, 1976); it is dual to the “wreath product” construction of Joswig and Lutz (J Combinatorial Theor A 110:193–216, 2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces “of unusually large genus” that first appeared in works by Coxeter (Proc London Math Soc 43:33–62, 1937), Ringel (Abh Math Seminar Univ Hamburg 20:10–19, 1956), and McMullen et al. (Israel J Math 46:127–144, 1983). Via “projections of deformed wedge products” we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in \mathbb R³{{\mathbb R}^3} . As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (“moduli”) for the surfaces in \mathbb R³{{\mathbb R}^3} . In order to prove that there are many moduli, we introduce the concept of “affine support sets” in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in \mathbb R³{{\mathbb R}^3} via dual 4-polytopes.     

Two Characterizations of Optimality in Dynamic Programming

It holds in great generality that a plan is optimal for a dynamic programming problem, if and only if it is “thrifty” and “equalizing.” An alternative characterization of an optimal plan, that applies in many economic models, is that the plan must satisfy an appropriate Euler equation and a transversality condition. Here we explore the connections between these two characterizations.     

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating convex bodies of constant width to the family of “fat” spindle convex bodies. Also, this leads to the spherical analog of the well-known Blaschke–Lebesgue problem.