To be or not to be constructive,that is not the question

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists $x$’ intuitively means ‘an algorithm to compute $x$ is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘$x$ is standard’ typical of Nonstandard Analysis can be interpreted as ‘$x$ is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics.     

Functions,equivalence relations,quotient spaces and subsets in fuzzy set theory

In the usual set theory, the power set of a set X is lattice isomorphicto 2^x. The theory of fuzzy sets, or fuzzy set theory, involves generalizing the lattice 2 = {0, 1} to a more general, usually non-Boolean, lattice. A ‘point’ in this lattice need not have Boolean properties. Thus proper generalizations must be based on notions which are defined without explicitly using points. In the light of this we give equivalent definitions of function, relation and quotient set involving the members of a power set but not specifically the singletons.A number of authors [1,2,5] have worked with generalizations of functions and quotient sets which are not strictly point free. It emerges that these are special cases of the ones we produce.     

When is the inverse regression estimator MSE-superior to the standard regression estimator in multivariate controlled calibration situations?

We assume as model a standard multivariate regression of y on x, fitted to a controlled calibration sample and used to estimate unknown x′s from observed y-values. The standard weighted least squares estimator (‘classical’, regress y on x and ‘solve’ for x) and the biased inverse regression estimator (regress x on y) are compared with respect to mean squared error. The regions are derived where the inverse regression estimator yields the smaller MSE. For any particular component of x this region is likely to contain ‘most’ future values in usual practice. For simultaneous estimation this needs not be true, however.     

A characterization of fuzzy subgroups

An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)?T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b ?1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with $\text{T =}\text{‘minimum’}$ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.     

Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

Sequential fuzzy system identification

The problem of deriving the structure of a non-deterministic system from its behaviour is a difficult one even when that behaviour is itself well-defined. When the behaviour can be described only in fuzzy terms structural inference may appear virtually impossible. However, a rigorous formulation and solution of the problem for stochastic automata has recently been given [1] and, in this paper, the results are extended to fuzzy stochastic automata and grammars. The results obtained are of interest on a number of counts. (1) They are a further step towards an integrated ‘theory of uncertainty’; (2) They give new insights into problems of inductive reasoning and processes of ‘precisiation’; (3) They are algorithmic and have been embodied in a computer program that can be applied to the modelling of sequential fuzzy data; (4) They demonstrate that sequential fuzzy data may be modelled naturally in terms of ‘possibility’ vectors.     

Axioms for a fuzzy measure of inequality

It has been argued that the concept of inequality is inherently imprecise. A difficulty with standard inequality measures is that they generally make no allowances for this, and when they do, it is by dropping the ‘completeness’ axiom in ranking social states (e.g. the Lorenz criterion). It is argued here that the erring axiom is not ‘completeness’ but ‘exactness’ which, being implicit, tends to escape notice. A fuzzy measure of inequality, along with a set of necessary and sufficient axioms, is established. The new measure has several attractive properttes: It allows for tentative judgements and doubts. It is easy to interpret and compute, and the Gini ranking turns out to be a nearest exact approximation of it.     

Analyticity and the Deviant Logician: Williamson’s Argument from Disagreement

One way to discredit the suggestion that a statement is true just in virtue of its meaning is to observe that its truth is the subject of genuine disagreement. By appealing to the case of the unorthodox philosopher, Timothy Williamson has recast this response as an argument foreclosing any appeal to analyticity. Reconciling Quine’s epistemological holism with his treatment of the ‘deviant logician’, I show that we may discharge the demands of charitable interpretation even while attributing trivial semantic error to Williamson’s philosophers. Williamson’s effort to generalize the argument from disagreement therefore fails.     

Fuzzy P-measure on the real line

The fuzzy measurable space on the real line, β_ϱ say, is defined by means of fuzzy relation ‘less or equal’ [5]. The main properties of β_ϱ are given. These results are used for investigation of fuzzy P-measure on β_ϱ. The principal emphasis is laid on the connections between a fuzzy P-measure and its cumulative distribution function. A theorem extending a fuzzy P-measure to β_ϱ is shown.     

By doctrines fashioned to the varying hour or the calculus of horrors

A deliberate attempt is made in Business Mathematics oriented text books as well as in some reform calculus oriented text books to interpret the derivative ?′(a) of a function y = ? (x) at the value x= a as the change in the y -value of the function per ‘unit’ of change in the x-value. This note questions the above interpretation and suggests the necessary modification for the correct interpretation.     

Stochastic and convex orders and lattices of probability measures,with a martingale interpretation

Letμ be any probability measure on ? with ∫|x|dμ(x)<∞ and letμ* denote the associated Hardy and Littlewood maximal p.m., the p.m. of the Hardy and Littlewood maximal function obtained fromμ. Dubins and Gilat [6] showed thatμ* is the least upper bound, in the usual stochastic order, of the collection of p.m.’sν on ? for which there is a martingale (X _t )_0≤t≤1 having distributions ofX ₁ and sup_0≤t≤1 X _t given byμ andν respectively. In this paper, a type of ‘dual representation’ is given. Specifically, letν be any p.m. on ? with lim sup _x →∞x ν[x,∞)=0xν[x, ∞)=0 and finitex ₀=inf{z :ν(?∞,z]0}. Then there is a ‘minimal p.m.’ν _Δ which is the greatest lower bound, in the usual convex order, of the collection of p.m.’sμ on ? for which there is a martingale (X _t )_0≤t≤1 having distributions ofX ₁ and sup_0≤t≤1 X _t given byμ andν respectively. To demonstrate existence and to obtain identification of these minimal p.m.’s, we use, in particular, a lattice structure on the set of p.m.’s with the convex order, and an equivalence between a convex order of p.m.’s and the stochastic order of their maximal p.m.’s. Consequences of these order results include sharp expectation-based inequalities for martingales. These martingale inequalities form a new class of ‘prophet inequalities’ in the context of optimal stopping theory.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.     

Local times for two-parameter Lévy processes

In this article we study the problem of existence of jointly continuous local time for two-parameter Lévy processes. Here, ‘local time’ is understood in the sense of occupation density, kand by 2-parameter Lévy process we mean a process X = {X_z: z ? [0, +∞)²} with independent and stationary increments (over rectangles of the type (s, s′] × (t, t′]). We prove that if X is $\text{R}$-valued and its lower index is greater than one, then a jointly continuous (at least outside {(x,s,t): x = 0}) local time can be obtained via Berman's method. Also, we extend to 2-parameter processes a result of Getoor and Kesten for usual Lévy processes. Implications in terms of ‘approximate local growth’ of X are stated.     

Existence as the Possibility of Reference

The mere fact that ontological debates are possible requires us to address the question, what is it to claim that a certain entity or kind of entity exists—in other words, what do we do when we make an existence-claim? I develop and defend one candidate answer to this question, namely that to make an existence-claim with regard to Fs is to claim that we can refer to Fs. I show how this theory can fulfil the most important explanatory desiderata for a theory of existence; I also defend it against the charges of illegitimate ‘semantic ascent’ and of making existence counterfactually dependent on human linguistic ability.     

Problem 25 : Posed by C.D. Godsil

Let x and y be two letters and m a fixed integer; a ‘bayonet’ is a word of the type xⁱyxⁱ, i + j?m ? 1. We compute the number t_n of all the words obtained by the concatenation of n bayonets. A consequence for the ‘triangle conjecture’ is deduced from the result.     

On the continued Erlang loss function

We prove two fundamental results in teletraffic theory. The first is the frequently conjectured convexity of the analytic continuation B(x, a) of the classical Erlang loss function as a function of x, x ⩾ 0. The second is the uniqueness of the solution of the basic set of equations associated with the ‘equivalent random method’.     

Searle freed of every flaw

Strong Al presupposes (1) that Super-Searle (henceforth ‘Searle’) comes to know that the symbols he manipulates are meaningful, and (2) that there cannot be two or more semantical interpretations for the system of symbols that Searle manipulates such that the set of rules constitutes a language comprehension program for each interpretation. In this paper, I show that Strong Al is false and that presupposition #1 is false, on the assumption that presupposition #2 is true. The main argument of the paper constructs a second program, isomorphic to Searle’s, to show that if someone, say Dan, runs this isomorphic program, he cannot possibly come to know what its mentioned symbols mean because they do not mean anything to anybody. Since Dan and Searle do exactly the same thing, except that the symbols they manipulate are different, neither Dan nor Searle can possibly know whether the symbols they manipulate are meaningful (let alone what they mean, if they are meaningful). The remainder of the paper responds to an anticipated Strong Al rejoinder, which, I believe, is a necessary extension of Strong Al.     

The algebra of fuzzy logic

In the following, human thinking based on premises with no complete truth value is reviewed for controlling the algebra of fuzzy sets operations. Assuming a system may be developed in this sphere, it should be considered as the algebra of fuzzy sets, as the same algebra is satisfied by classical logic and sets. As will be proved, this algebra is not a lattice and consequently the Zadeh definitions do not constitute an adequate representation. The binary operations of my algebra are “interactive” types. An axiom system is given that, in my opinion, is the foundation of the conception, adequately and without redundancy. The agreement of the theorems deduced from the axiom system with the intuitive expectations is shown. A special arithmetical structure satisfying this algebra is given, and the relation between this structure and the theory of probability is analyzed.Adapting a process of classical logics, fuzzy quantifiers are defined on the basis of the operations of propositional algebra. A “qualifier” is also defined. The qualifier is functional; applying it to Ax we get the statement “usually Ax” s a middle cource between the statements “at least once Ax” and “always Ax”. The concept of entailment of fuzzy logics is introduced. This concept is an innovative generalization of the classical deduction theory, opposite to the concept of entailment of classical multi-valued logics. An important error of the abbreviated system of notation of the fuzzy theory [e.g. m(x, AvB)] appears: the functional type operations (e.g. quantifiers) cannot be interpreted in propositional calculus. Therefore a new system of symbols is proposed in this paper.     

Belief revision conditionals: basic iterated systems

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints.Recent work has shown that in spite of these negative results, the question ‘how to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved.In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing.