Interrelations and Dissimilarities Between Distinct Approaches to Ontic Vagueness

This paper outlines the often striking parallels of various approaches to ontic vagueness, as well as their even more striking differences. Though circling around the same idea, some of these approaches were developed to solve quite diverse theoretical problems and encounter different challenges. In addition to these difficulties, the frequently disregarded epistemological problems of all theories of ontic vagueness turn out to be even more serious under critical scrutiny. The same holds for the difficulties of deciding, for every case of vagueness, whether the vagueness involved is semantic or ontic.     

Ten questions and one problem on fuzzy logic

Recently, I had a very interesting friendly e-mail discussion with Professor Parikh on vagueness and fuzzy logic. Parikh published several papers concerning the notion of vagueness. They contain critical remarks on fuzzy logic and its ability to formalize reasoning under vagueness [10,11]. On the other hand, for some years I have tried to advocate fuzzy logic (in the narrow sense, as Zadeh says, i.e. as formal logical systems formalizing reasoning under vagueness) and in particular, to show that such systems (of many-valued logic of a certain kind) offer a fully fledged and extremely interesting logic [4, 5]. But this leaves open the question of intuitive adequacy of many-valued logic as a logic of vagueness. Below I shall try to isolate eight questions Parikh asks, add two more and to comment on all of them. Finally, I formulate a problem on truth (in)definability in Łukasiewicz logic which shows, in my opinion, that fuzzy logic is not just “applied logic” but rather belongs to systems commonly called “philosophical logic” like modal logics, etc.     

A Novel Category of Vague Abstracta

Much attention has been given to the question of ontic vagueness, and the issues usually center around whether certain paradigmatically concrete entities - cats, clouds, mountains, etc. - are vague in the sense of having indeterminate spatial boundaries. In this paper, however, I wish to focus on a way in which some abstracta seem to be locationally vague. To begin, I will briefly cover some territory already covered regarding certain types of “traditional” abstracta and the ways they are currently alleged to be vague. I then wish to discuss two types of “nontraditional” abstracta and the sense in which I think some of these objects are locationally vague. I will next reexamine some of the traditional abstracta and discuss whether any of these objects are locationally vague in the novel way suggested for the nontraditional sorts. I’ll finish by discussing objections, and conclude with some remarks about characterizing the abstract/concrete distinction.     

A theorem and a question about epicomplete archimedean lattice-ordered groups

An archimedean ?-group is called epicomplete (or universally σ-complete, or sequentially inextensible) if it is divisible, σ-complete and laterally σ-complete. Various characterizations of such G are known in case the G have weak order units. The “theorem” of the title is a characterization of such G which have no weak order unit; it involves the requirement that G have a certain kind of representation. The “question” of the title is whether every epicomplete G has such a representation.     

The Fundamental Pro-groupoid of an Affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π ₁(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π ₁(spec(R)) in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π ₁ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products.     

Balanced sets in an independence structure induced by a submodular function

A submodular (and non-decreasing) function on a set induces an independence structure; the notion of a “balanced” set in this situation helps us determine whether a given independence structure is induced by any submodular function other than its own rank function, answering a question of U. S. R. Murty and I. Simon. The notion “balanced” also has a natural meaning when one independence structure is induced from another across a bipartite graph.     

Characterizing and recognizing generalized polymatroids

Generalized polymatroids are a family of polyhedra with several nice properties and applications. One property of generalized polymatroids used widely in existing literature is “total dual laminarity;” we make this notion explicit and show that only generalized polymatroids have this property. Using this we give a polynomial-time algorithm to check whether a given linear program defines a generalized polymatroid, and whether it is integral if so. Additionally, whereas it is known that the intersection of two integral generalized polymatroids is integral, we show that no larger class of polyhedra satisfies this property.     

Comments on “A distance-based statistical analysis of fuzzy number-valued data” by the SMIRE research group

This paper is a fine review of various aspects related to the statistical handling of “ontic” random fuzzy sets by the means of appropriate distances. It is quite comprehensive and helpful, as it clarifies the status of fuzzy sets in such methods, explains the advantages of using a distance-based approach, specifies the pitfalls in which one should not fall when dealing with “ontic” random fuzzy sets and provides some illustration of practical computations. Not being a statistician but an occasional user of statistics, my discussion will mainly focus on this more practical aspect.     

Real-time assessment of value-at-risk and volatility accuracy

An important question for corporate finance officers is whether risk assessments, such as Value at Risk (VaR), are currently accurate. In contrast to past research on assessing the accuracy of VaR, volatility, and related density estimates, which has focused on backtesting using large samples of fixed size, we develop a class of sequential testing tools for on-line, real-time assessment, based on time windows that vary adaptively with the data.The VaR is determined by a single point of the estimated distribution of the portfolio “gain” and may be positive (profit) or negative (loss). Previous literature has dichotomically tested the sequence of VaR forecasts or the sequence of estimated distributions. A pure test is obtained by converting each observed gain into a binary value indicating whether it was covered by the corresponding VaR forecast or not. A more powerful test results from using the entire distribution, by transforming the observed gain to a random variable that has a known distribution when the forecast is accurate. This, however, also detects errors unrelated to the accuracy of estimating VaR and other measures of risk.We propose an adjustable, continuous compromise between detection power and purity, where “power” refers to quick detection of systematic bias and “purity” refers to insensitivity to errors not relevant to VaR estimation accuracy. Previous approaches focused on either extreme of this continuum. However, we point out that there are few practical situations for which the choice of either extreme would be optimal. Instead, we suggest a compromise that would be much better and very useful in most practical applications.     

Embodying mathematics and science: Microworlds as representations

This paper examines the category of open-ended exploratory computer environments that have been labeled “microworlds.” The paper reviews the ways in which the term “microworld” has been used in the mathematics and science education communities, and analyzes examples of specific computer microworlds. Two definitions of microworld are proposed: a structural definition that focuses on design elements shared by the environments, and a functional definition that highlights commonalties in how students learn with microworlds. In the final section of the paper, the notion that computer microworlds can be said to “embody” mathematical or scientific ideas is addressed, within the context of a re-evaluation of the general concept of representation.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

Vague Analysis

It might be thought that vagueness precludes the possibility of classical conceptual analysis and, thus, that the classical or definitional view of the nature of complex concepts is incorrect. The present paper argues that classical analysis can be had for concepts expressed by vague language since (1) all of the general theories of vagueness are compatible with the thesis that all complex concepts have classical analyses and also that (2) the meaning of vague expressions can be analyzed by having the degree of vagueness of a given analysandum be “mapped” onto the vagueness of an analysans.     

Largeur et nilpotence

The notion of a large set in an arbitrary group is introduced in analogy to the generic sets in an algebraic or stable group. The question is studied which properties “satisfied largely” by a group hold on the entire group.

ZUSAMMENFASSUNG. Wir definieren den Begriff einer groβen Teilmenge einer be-liebigen Gruppe in Analogie zu den generischen Mengen einer algebraischen oder stabilen Gruppe und untersuchen, welche “grofiteils erfullten” Gruppeneigenschaften fur die ganze Gruppe gelten müssen.     

A New Series: Blacks and Colleges: 1: The Dot and The Elephant

As recently as last year, I thought that “education”—the discipline—was too tiresome a subject to sustain my interest for more than a moment or two. I hastily abandoned this conviction after five minutes in front of a classroom. As a teacher, or as someone trying to teach, education became for me a living thing.

By happy, unplanned circumstance, my teaching and the research for this project began at the same time. The two assignments have dovetailed in a way that no amount of planning could have forecast.

This article is written from a black perspective—something which does not necessarily reside within a black skin. Retaining this perspective, particularly as a writer, requires among other things an agonizing bending of the mind. One is constantly obliged to see and weigh things including black education—from several points of view and finally to ask: “What does this or that event mean for black people at large—for ‘the movement’?“

The necessity for all of this is not readily understood by white people. The fact is that to be black in this country is to be part of “the movement.” Whether one is conscious of this fact or not, whether one rushes out to “join” something or not, matters little. One belongs whether one is male or female, unemployed, skilled in carpentry, or deep in clinical psychology. One “withdraws” from the movement only at great psychic cost. To be in it is painfully expensive as well, but this pain is offset by that particular joy that comes from embracing one's own people.     

The Reflection Structures of Generalized Co-Minkowski Spaces Leading to K-Loops

We extend the notion co- Minkowski plane to “unitary co- Minkowski space” (P, G, ～) and discuss the problem whether there are subsets Q of P which can be turned into a geometric K- loop.     

Δ1 Ultrapowers are totally rigid

Hirschfeld and Wheeler proved in 1975 that ∑₁ ultrapowers (= “simple models”) are rigid; i.e., they admit no non-trivial automorphisms. We later noted, essentially mimicking their technique, that the same is true of Δ₁ ultrapowers (= “Nerode semirings”), a class of models of Π₂ Arithmetic that overlaps, but is mutually non-inclusive with, the class of Σ₁ ultrapowers. Hirschfeld and Wheeler left as open the question whether some Σ₁ ultrapowers might admit proper isomorphic self-injections. We do not answer that question; but we do answer the corresponding question, in the negative, for the Δ₁ case.     

Consensus of opinion

In a decision analysis, it is often necessary to combine a group of individuals' beliefs into a consensus probability distribution. This paper addresses the question whether it is possible to base such consensus distributions only upon the information present within the group or must some arbitrary rule be used to resolve disagreement. Some earlier work on modifying beliefs in the light of another's opinion is developed to apply to groups of n people. Using this as a “benchmark of rationality”, standard methods of forming group consensus probability distributions are found somewhat arbitrary. Furthermore it is argued that the possibility of constructing better procedures is remote.     

Discrete hamstrung squad car games

Isaacs introduced a game of perfect information played on the integer grid in which a pursuer with certain restrictions on his movements seeks to capture a slower moving evader. He showed that in a special case it was possible for the evader to avoid capture, and went on to pose the research question as to whether the same situation arose when the pursuer's “speed” was increased. We show that the answer to his question is “No”.     

Logrolling and budget allocation games

In this paper we approach the concept of logrolling by examining a voting system where choices are made among sets of competing projects as a game in characteristic function form. We translate the question: “Will there be prices for votes on different projects which clear the market?” into a different, but equivalent question: “Is the formal game we have described amarket game?” We show that in general the answer is no, unless all voters have virtually the same preferences.