Showing papers on "Counterexample published in 1981"
TL;DR: The aim of this paper is to consider both the specific problem posed by counterexamples of Substitution of Equivalent Antecedents (SEA) and the more general problem of what criteria should be satisfied by good explanations for countereXamples.
Abstract: Such counterexamples seem to show that Hypothetical Syllogism (HS) is invalid. Fifteen years ago our first reaction might have been to dig in our heels and seek to explain away the counterexample. Indeed, perhaps some suspicion is warranted in view of the fact that the plausibility of the counterexample diminishes somewhat if we simply reverse the order of the premises. But Robert Stalnaker and David Lewis have shown us that we do not need to explain away such arguments.1 It is possible to give a formal characterization of English counterfactuals according to which HS and other classical inferences such as Contraposition (CT) are not counted valid.2 Hence, we can deal with the counterexample by adopting a formal theory of conditionals that is just slightly weaker than what traditional sentential logic has led us to expect. But there is a residual problem. Donald Nute argued in [13] that the Stalnaker-Lewis characterization of the conditional is still too strong, for there are counterexamples of Substitution of Equivalent Antecedents (SEA) which they assume to be valid. Though Nute has recently modified his position in [16], the general problem of explaining away counterexamples to inferences we think are valid remains with us. Any formal analysis of conditionals (short of one that is extraordinarily weak) must at some point deal with this problem. The aim of this paper is to consider both the specific problem posed by counterexamples to SEA and the more general problem of what criteria should be satisfied by good explanations for counterexamples. It quickly becomes apparent that ad hoc explanations will not do. Good explanations must be developed in formal terms and should be theoretically motivated. We will see that the inferences that bothered Nute can indeed be explained
69 citations
Journal Article•
TL;DR: In this article, the existence of subspaces of L which are isomorphic to an L 1(03BC)-space and are not complemented by complements is shown.
Abstract: The existence is shown of subspaces of L which are isomorphic to an L1(03BC)-space and are not complemented. A more precise local statement is also given.
57 citations
51 citations
TL;DR: For each integer p ≥5 of the form p =2 m +2 n with 0≤ n m , there exist six related pairs of counterexamples to the digraph reconstruction conjecture, including a pair of tournaments.
34 citations
TL;DR: In this paper, a computable necessary and sufficient condition for the spectral controllability of a class of linear hereditary systems is proved and a counterexample is given to a previously published result.
Abstract: A computable necessary and sufficient condition is proved for the spectral controllability of a class of linear hereditary systems and a counterexample is given to a previously published result.
32 citations
TL;DR: Some classes of digraphs are reconstructed from the point-deleted subdigraphs for each of which the degree pair of the deleted point is also known, and none of the known counterexample pairs to the DRC is a countereXample pair to this new conjecture.
27 citations
TL;DR: In this paper, it was shown that the equivalence of the existence of a lower bound to the cost is equivalent to a negative semidefinite, antistabilizing solution to the algebraic Riccati equation.
Abstract: In the problem of infimizing a not necessarily positive semidefinite quadratic cost subject to a linear dynamical constraint, it is usually expected that the existence of a lower bound to the cost is equivalent to the existence of a negative semidefinite, antistabilizing solution to the algebraic Riccati equation. By a counterexample, it is shown that this equivalence breaks down in the discrete-time case. This phenomenon, as well as the whole question of the existence of the appropriate solution to the algebraic Riccati equation, are investigated in detail.
22 citations
IBM1
TL;DR: In this article, a simple deterministic dynamic programming model is used as a general framework for the analysis of stochastic versions of three classical optimization problems: knapsack, traveling salesperson, and assembly line balancing problems.
16 citations
TL;DR: In this article, a counterexample to the conjecture of F.K. Chung and F.R. Hwang is presented, showing that if all assignments of players to starting positions are equally likely, then stronger players (as defined by the preference scheme) are more likely to win the tournament.
Abstract: We present a counterexample to the following conjecture of F.R.K. Chung and F.K. Hwang (1978): for any knockout tournament plan and any preference scheme satisfying strong stochastic transitivity, if all assignments of players to starting positions are equally likely, then stronger players (as defined by the preference scheme) are more likely to win the tournament.
16 citations
1 Jan 1981
TL;DR: In this article, the authors give a series of examples of simple finitely generated dimension groups which cannot be obtained as the inductive limit of a system A. A, A-A-
Abstract: We give a series of examples of simple finitely generated dimension groups which cannot be obtained as the inductive limit of a system A. A, A-
15 citations
TL;DR: In this article, the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind.
Abstract: First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them.
TL;DR: An example is given of an arrangement of eight pseudoplanes, i.e., topological planes, in P 3 , and three points which do not lie in any pseudoplane compatible with the arrangement, which provides a counterexample to the "Levi enlargement lemma" in dimension > 2.
TL;DR: In this paper, it is shown that for simple oriented matroids, the union matroid can be obtained by concatenation of two set systems. But the concatenated set is not a union matroid.
1 Jan 1981
TL;DR: In this article, an infinite family of counterexamples to the maximum number of edges in a diameter critical graph was constructed, which is a conjecture of Caccetta and Haggkvist.
Abstract: A graph is diameter critical if removal of any edge increases the diameter of the graph. In this note we construct an infinite family of counterexamples to a recent conjecture by Caccetta and Haggkvist [2] on the maximum number of edges in a diameter critical graph.
TL;DR: Through a counterexample, Redinbo's upper bound on the ratio of the probability of a group and that of its coset is shown to be incorrect and a new upper bound is provided.
Abstract: Through a counterexample, Redinbo's upper bound on the ratio of the probability of a group and that of its coset is shown to be incorrect. The given example is also shown to provide a new upper bound.
TL;DR: The refinement properties and extensions of filters in Boolean algebras are studied. The paper identifies the necessary and sufficient conditions for a family to have a disjoint refinement, and shows that these conditions are the only possible ones.
Abstract: We consider the question, under what conditions a given family $A$ in a Boolean algebra $\mathcal {B}$ has a disjoint refinement. Of course, $A$ cannot have a disjoint refinement if $A$ is a dense subset of an atomless $\mathcal {B}$, or if $\mathcal {B}$ is complete and $A$ generates an ultrafilter on $\mathcal {B}$. We show in the first two sections that these two counterexamples can be the only possible ones. The third section is concerned with the question, how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base.
TL;DR: In this paper, a continuous surjection f : X → Y → Y is defined so that each point-inverse f -1 (y ) is an AR and each non-AR Y is not an AR.
TL;DR: In this article, the authors provide a counterexample to the bounded orbit conjecture by constructing a fixed point free orientation reversing homeomorphism which satisfies the hypothesis of the conjecture.
Abstract: A long outstanding problem in the topology of Euclidean spaces is the Bounded Orbit Conjecture, which states that every homeomorphism of E2 onto itself, with the property that the orbit of every point is bounded, must have a fixed point. It is well known that the conjecture is true for orientation preserving homeomorphisms. We provide a counterexample to the conjecture by constructing a fixed point free orientation reversing homeomorphism which satisfies the hypothesis of the conjecture.
TL;DR: In this paper, a special class of second-order, bilinear systems and a performance index of the Mayer type were considered and it was shown that the necessary conditions for optimality at a junction joining a singular and non-singular subare were correct.
Abstract: A certain theorem in the literature concerns necessary conditions for optimality at a junction joining a singular and non-singular subare. In a previous investigation it was shown that the theorem is correct fur all first-order systems and any performance index of given typo under a chosen set of parameter values. The present paper considers a special class of second-order, bilinear systems and a performance index of Mayer type. It is demonstrated that, contrary to expectations, the earlier analysis carries over to this more general system and the theorem is again found to be correct. The authors conjecture that a similar result is also true for the most general form of second-order system.
TL;DR: In this paper, Blatt considers the case of an individual who must "evaluate the prospect" of an illegal venture or game which is potentially very profitable (with a possible payoff of M dollars) but for which the death penalty is imposed with probability p.
Abstract: In a recent issue of JPKE (Winter 1979-80), John M. Blatt considers the case of an individual who must "evaluate the prospect" of an illegal venture or game which is potentially very profitable (with a possible payoff of M dollars) but for which the death penalty is imposed with probability p. Blatt draws on the work of Karl Borch and concludes from his analysis of this case that he has found a counterexample demonstrating that the preference orderings of practical businessmen do not necessarily correspond to those derived from expected utility theory. This counterexample is one which he believes to be particularly relevant to situations involving risk of disaster (e.g., bankruptcy). In this comment we will argue
1 Jan 1981
TL;DR: In this article, the authors established a theorem concerning the common polynomials of the cospectral classes of signed graphs on a given graph in which all the cycles are of the same length and pass through a single point.
Abstract: In this note, we establish a theorem concerning the common polynomials of the cospectral classes of signed graphs on a given graph in which all the cycles are of the same length and pass through a single point. This theorem is observed to give a doubly infinite class of graphs serving as counterexamples to a recent conjecture on a certain number associated with a cospectral class of unbalanced signed graphs on a given graph.
TL;DR: In this article, the authors show that the results do not necessarily follow from the assumptions, relaxing certain assumptions leads to different results, the underlying structure of the model should be replaced by another structure to capture the characteristics of actual information use, and the results will hold in an average sense with additional assumptions under these other model structures.
Abstract: Examples can be used in research for several purposes. A counterexample may be used to disprove a theory. Examples may also prove to be useful in clarifying a complex analytical approach. And finally an example may be used as a vehicle to gain insight on a problem that is not analytically tractable. I call this latter approach "armchair theorizing" and this is the approach taken by the Hakansson paper. "Armchair theorizing" is a useful tool in generating new ideas, but care must be taken in generalizing from a specific example. The Hakansson paper makes numerous claims about generalizability, but offers no analytics about how the results would obtain if the assumptions were relaxed. In this discussion I would like to do some "armchair theorizing" and demonstrate that (1) the results do not necessarily follow from the assumptions, (2) relaxing certain assumptions leads to different results, (3) the underlying structure of the model should be replaced by another structure to capture the characteristics of actual information use, and (4) the results will hold in an "average" sense with additional assumptions under these other model structures.
TL;DR: This comment gives a slight generalization of the new criteria for the simplex algorithm which guarantee that a nonbasic variable of a basic feasible solution will remain nonbasic in an optimal basic solution.
Abstract: In a recent paper M.C. Cheng proposed new criteria for the simplex algorithm which guarantee that (i) a nonbasic variable of a basic feasible solution will remain nonbasic in an optimal basic solution, (ii) a basic variable of a basic solution will remain basic in an optimal basic solution. This comment gives (i) a slight generalization of the first result and (ii) a counterexample to the second proposition.
TL;DR: In this article, pointwise bounds for the solution of time-independent linear transport problems with surface sources in convex spatial domains are derived for the case where the source is a convex manifold.
Abstract: Pointwise bounds are derived for the solution of time-independent linear transport problems with surface sources in convex spatial domains. Under specified conditions, upper bounds are derived which, as a function of position, decrease with distance from the boundary. Under other conditions, lower bounds are derived which increase with distance from the boundary. Also, sufficient conditions are obtained for the existence of maximum and minimum principles, and a counterexample is given which shows that such principles do not always exist.
1 Jan 1981
TL;DR: The method of counterexample is a proof procedure, combined with a semantic approach, which may be used for deciding whether an argument is valid or not, in sentence logic and in predicate logic.
Abstract: The method of counterexample is a proof procedure, combined with a semantic approach, which may be used for deciding whether an argument is valid or not. The method can be applied in sentence logic and in predicate logic, though in the latter it must sometimes fail, as predicate logic does not possess the property of decidability.
TL;DR: In this article, the prediction operator from a uniformly convex real Orlicz space to a subset of σ-lattice measurable functions is investigated and a sufficient condition for monotonicity to hold is given.
Abstract: In this paper, we investigate the prediction (or best approximation) operator from a uniformly convex real Orlicz space to a subset of σ-lattice measurable functions. In particular, a counterexample to the monotoncity property, which holds in Lp spaces, is given. Also, a sufficient condition for monotonicity to hold is given. Finally, nested σ-lattices, as occur in isotonic regression, are considered.
TL;DR: In this paper, a counterexample was given to the following conjecture: if f * is a linear functional, and e an exposed point of a convex body K such that f * does not attain its maximum on K at e, then there is no f *-strictly increasing path in the one-skeleton of K emanating from e.
Abstract: A counterexample, in E 3, is given to the following conjecture Suppose f * is a linear functional, and e an exposed point of a convex body K such that f * does not attain its maximum on K at e; then there is an f *-strictly increasing path in the one-skeleton of K emanating from e The counterexample shows that a certain generalized ‘simplex algorithm’ fails Furthermore for a different linear functional f, there are no three disjoint f-strictly increasing paths in the one-skeleton of K leading to e
TL;DR: In this paper, it was shown that the ideal of all relations between zeros of the Galois equation has a basis containing root polynomials only, not the equation itself.
Abstract: A conjecture ofH. Kleiman says that over certain fields a Galois equation of degree ≥3 is uniquely determined by its root polynomials. We prove this conjecture for prime degrees ≥3 and a somewhat smaller class of fields than Kleiman's. In this situation, the ideal of all relations between zeros of the equation has a basis containing root polynomials only, not the equation itself. Giving a large class of counterexamples of degree 4, we disprove Kleiman's conjecture in general.