Inductive Argument Research Articles

Psychology, epistemology, and skepticism in Hume’s argument about induction

Since the mid-1970s, scholars have recognized that the skeptical interpretation of Hume’s central argument about induction is problematic. The science of human nature presupposes that inductive inference is justified and there are endorsements of induction throughout Treatise Book I. The recent suggestion that I.iii.6 is confined to the psychology of inductive inference cannot account for the epistemic flavor of its claims that neither a genuine demonstration nor a non-question-begging inductive argument can establish the uniformity principle. For Hume, that inductive inference is justified is part of the data to be explained. Bad argument is therefore excluded as the cause of inductive inference; and there is no good argument to cause it. Does this reinstate the problem of induction, undermining Hume’s own assumption that induction is justified? It does so only if justification must derive from “reason”, from the availability of a cogent argument. Hume rejects this internalist thesis; induction’s favorable epistemic status derives from features of custom, the mechanism that generates inductive beliefs. Hume is attracted to this externalist posture because it provides a direct explanation of the epistemic achievements of children and non-human animals—creatures that must rely on custom unsupplemented by argument.

A link invariant from the symplectic geometry of nilpotent slices

We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspaceSm of the Lie algebra sl2m(C) which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submani

Truth and the Unprovability of Consistency

It might be thought that we could argue for the consistency of a mathematical theory T within T, by giving an inductive argument that all theorems of T are true and inferring consistency. By Godel's second incompleteness theorem any such argument must break down, but just how it breaks down depends on the kind of theory of truth that is built into T. The paper surveys the possibilities, and suggests that some theories of truth give far more intuitive diagnoses of the breakdown than do others. The paper concludes with some morals about the nature of validity and about a possible alternative to the idea that mathematical theories are indefinitely extensible.

The Problem of Induction

The problem of induction is sometimes motivated via a comparison between rules of induction and rules of deduction. Valid deductive rules are necessarily truth preserving, while inductive rules are not. So, for example, one valid deductive rule might be this: (D) From premises of the form “All F are G” and “a is F ,” the corresponding conclusion of the form “a is G” follows. The rule (D) is illustrated in the following depressing argument: (DA) All people are mortal. I am a person. So, I am mortal. The rule here is “valid” in the sense that there is no possible way in which premises satisfying the rule can be true without the corresponding conclusion also being true. A possible inductive rule might be this: (I) From premises of the form “Many many F s are known to be G,” “There are no known cases of F s that are not G,” and “a is F ,” the corresponding conclusion can be inferred of the form “a is G.” The rule (I) might be illustrated in the following “inductive argument.” (IA) Many many people are known to have been moral. There are no known cases of people who are not mortal. I am a person. So, I am mortal.

Conditionals and consequences

We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, non-classical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework.

Inference and Uncertainty in Radiology

This paper seeks to enhance understanding of the philosophical underpinnings of our discipline and the resulting practical implications. Radiology reports exist in order to convey new knowledge about a patient's condition based on empiric observations from anatomic or functional images of the body. The route to explanation and prediction from empiric evidence is mostly through inference based on inductive (and sometimes abductive) arguments. The conclusions of inductive arguments are, by definition, contingent and provisional. Therefore, it is necessary to deal in some way with the uncertainty of inferential conclusions (i.e. interpretations) made in radiology reports. Two paradigms for managing uncertainty in natural sciences exist in dialectic tension with each other. These are the frequentist and Bayesian theories of probability. Tension between them is mirrored during routine interactions among radiologists and clinicians. I will describe these core issues and argue that they are quite relevant to routine image interpretation and reporting.

On Dependency Graphs and the Lattice Gas

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász Local Lemma in probabilistic combinatorics. We show that the conclusion of the Lovász Local Lemma holds for dependency graph . Furthermore, we show that the usual proof of the Lovász Local Lemma – which provides a sufficient condition for this to occur – corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer [28] and explicitly by Dobrushin [12, 13]. We also present a generalization of the Lovász Local Lemma that allows for ‘soft’ dependencies. The paper aims to provide an accessible discussion of these results, which are drawn from a longer paper [26] that has appeared elsewhere.

The Possibility of Inductive Moral Argumets

Is it possible to have moral knowledge? ‘Moral justification skeptics’ hold it is not, because moral beliefs cannot have the sort of epistemic justification necessary for knowledge. This skeptical stance can be summed up in a single, eat argument, which includes the premise that ‘Inductive arguments from non-moral premises to moral conclusions are not possible.’ Other premises in the argument may rejected, but only at some cost. It would be noteworthy, therefore, if ‘inductive inferentialism’ about morals were show to be at least possible. Some philosophers may suppose that inductive moral argumets from non-moral premises cannot get off the ground, but I show that a perfectly legitimate inductive moral argument exists. This argument has on-moral premises and a moral conclusion, its premises are related to its conclusion in the right way, and it avoids some of the problems of other, better-know argumets from ‘Is’ to ‘Ought’.

A technique for multicoloring triangle-free hexagonal graphs

In order to avoid interference in cellular telephone networks, sets of radio frequencies are to be assigned to transmitters such that adjacent transmitters are allotted disjoint sets of frequencies. Often these transmitters are laid out like vertices of a triangular lattice in a plane. This problem corresponds to the problem of multicoloring an induced subgraph of a triangular lattice with integer demands associated with each vertex. We deal with the simpler case of triangle-free subgraphs of the lattice. [Frédéric Havet, Discrete Math. 233 (2001) 1–3] uses inductive arguments to prove that triangle-free hexagonal graphs can be colored with 7 6 ω d + o ( 1 ) colors where ω d is the maximum demand on a clique in the graph. We give a simpler proof and hope that our techniques can be used to prove the conjecture by [McDiarmid and Reed, Networks Suppl. 36 (2000) 114–117] that these graphs are 9 8 ω d + o ( 1 ) -multicolorable.

Argument by Analogy

In this essay I characterize arguments by analogy, which have an important role both in philosophical and everyday reasoning. Arguments by analogy are different from ordinary inductive or deductive arguments and have their own distinct features. I try to characterize the structure and function of these arguments. It is further discussed that some arguments, which are not explicit arguments by analogy, nevertheless should be interpreted as such and not as inductive or deductive arguments. The result is that a presumed outcome of a philosophical dispute will have to be reconsidered.

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovasz local lemma -- which provides a sufficient condition for this to occur -- corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer and explicitly by Dobrushin. We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

Linear Complementarity Systems: Zeno States

A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear time-invariant ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the P-property has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly non-Zeno. While related mode-switch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.

PLAYING GAMES IN LOGIC AND REASONING IN LIBERAL ARTS MATHEMATICS AND GETTING STUDENTS' WORK PUBLISHED

ABSTRACT This paper describes two classroom activities and a project that supplement a Liberal Arts Mathematics course's coverage of logic and reasoning. The first classroom activity introduces the writing of inductive and deductive arguments, and the second activity involves analyzing a guest speaker's arguments. The project consists of using logic and reasoning to write an editorial for a newspaper and includes a large bonus for getting it published. These exercises were not only instructional for both the students and the instructor, but also very rewarding and surprisingly fun.

Degraded Gaussian Multirelay Channel: Capacity and Optimal Power Allocation

We determine the capacity region of a degraded Gaussian relay channel with multiple relay stages. This is done by building an inductive argument based on the single-relay capacity theorem of Cover and El Gamal. For an arbitrary distribution of noise powers, we derive the optimal power distribution strategy among the transmitter and the relays and the best possible improvement in signal-to-noise ratio (SNR) that can be achieved from using a given number of relays. The time-division multiplexing operation of the relay channel in the wideband regime is analyzed and it is shown that time division does not achieve minimum energy per bit.

Life as Narrative

Jerome Bruner Life as Narrative I WOULD LIKE TO TRY OUT AN IDEA THAT MAY NOT BE QUITE READY, indeed may not be quite possible. But I have no doubt it is worth a tiy. It has to do with the nature of thought and with one of its uses. It has been traditional to treat thought, so to speak, as an instrument of reason. Good thought is right reason, and its efficacy is measured against the laws of logic or induction. Indeed, in its most recent computational form, it is a view of thought that has sped some of its enthusiasts to the belief that all thought is reducible to machine computability. But logical thought is not the only or even the most ubiquitous mode of thought. For the last several years, I have been looking at another kind of thought (see, e.g., Bruner, 1986), one that is quite different in form from reasoning: the form of thought that goes into the construction not of logical or inductive arguments but of stories or narratives. What I want to do now is to extend these ideas about narrative to the analysis o f the stories we tell about our lives: our “autobiographies.” Philosophically speaking, the approach I shall take to narrative is a constructivist one—a view that takes as its central premise that “world making” is the principal function of mind, whether in the sciences or in the arts. But the moment one applies a constructivist view of narra­ tive to the self-narrative, to the autobiography, one is faced with dilem­ mas. Take, for example, the constructivist view that “stories” do not “happen” in the real world but, rather, are constructed in people’s heads. Or as Henry James once put it, stories happen to people who know how to tell them. Does that mean that our autobiographies are constructed, that they had better be viewed not as a record of what ORIGINALLY PUBLISHED IN SOCIAL RESEARCH VOL. 5 4 , NO. 1 (SPRING 1987) social research Vol 71 : No 3 : Fall 200 4 691 happened (which is in any case a nonexistent record) but rather as a continuing interpretation and reinterpretation of our experience? Just as the philosopher Nelson Goodman argues that physics or painting or histoiy are “ways of world making” (Goodman, 1978), so autobiog­ raphy (formal or informal) should be viewed as a set of procedures for “life making.” And just as it is worthwhile examining in minute detail how physics or histoiy go about their world making, might we not be well advised to explore in equal detail what we do when we construct ourselves autobiographically? Even if the exercise should produce some obdurate dilemmas, it might nonetheless cast some light on what we might mean by such expressions as “a life.” CULTURE AND AUTOBIOGRAPHY Let me begin by sketching out the general shape of the argument that I wish to explore. The first thesis is this: We seem to have no other way of describing “lived tim e” save in the form of a narrative. Which is not to say that there are not other temporal forms that can be imposed on the experience of time, but none of them succeeds in capturing the sense of lived time: not clock or calendrical time forms, not serial or cycli­ cal orders, not any of these. It is a thesis that will be familiar to many of you, for it has been m ost recently and powerfully argued by Paul Ricoeur (1984). Even if we set down annales in the bare form of events (White, 1984), they will be seen to be events chosen with a view to their place in an implicit narrative. My second thesis is that the mimesis between life so-called and narrative is a two-way affair: that is to say, just as art imitates life in Aristotle’s sense, so, in Oscar W ilde’s, life im itates art. Narrative imitates life, life imitates narrative. “Life” in this sense is the same kind of construction of the human imagination as “a narrative” is. It is constructed by human beings through active ratiocination, by the same kind of ratiocination through...

A Conflict Case Approach to Business Ethics

Departing from frequent use of moral conflict cases in business ethics teaching and research, the paper suggests an elaboration of a moral conflict approach within business ethics, both conceptually and philosophically. The conceptual elaboration borrows from social science conflict research terminology, while the philosophical elaboration presents casuistry as a kind of practical, inductive argumentation with a focus on paradigmatic examples.

A lower bound on the quantum query complexity of read-once functions

We establish a lower bound of Ω( n ) on the bounded-error quantum query complexity of read-once Boolean functions. The result is proved via an inductive argument, together with an extension of a lower bound method of Ambainis. Ambainis’ method involves viewing a quantum computation as a mapping from inputs to quantum states (unit vectors in a complex inner-product space) which changes as the computation proceeds. Initially the mapping is constant (the state is independent of the input). If the computation evalutes the function f then at the end of the computation the two states associated with any f-distinguished pair of inputs (having different f values) are nearly orthogonal. Thus the inner product of their associated states must have changed from 1 to nearly 0. For any set of f-distinguished pairs of inputs, the sum of the inner products of the corresponding pairs of states must decrease significantly during the computation, By deriving an upper bound on the decrease in such a sum, during a single step, for a carefully selected set of input pairs, one can obtain a lower bound on the number of steps. We extend Ambainis’ bound by considering general weighted sums of f-distinguished pairs. We then prove our result for read-once functions by induction on the number of variables, where the induction step involves a careful choice of weights depending on f to optimize the lower bound attained.

Category-based induction: an effect of conclusion typicality.

Category-based induction involves the willingness of a thinker to project some newly learned property of one or more classes of objects to another class on the basis of their shared membership in a common superordinate category. Previous research has established that the perceived strength of arguments of the form "Class A has Property P; therefore, Class B has Property P" is influenced by the similarity of A to B and by the typicality or representativeness of A in a shared category, superordinate to both A and B. (The nature of P is also crucial, but we do not examine it in this study.) There is, however, no prior evidence that the relation between B and the category is influential. Three experiments were designed to test whether the typicality of B in the superordinate category also has an effect on inductive argument strength. By using multiple regression (Experiment 1) and an experimental design (Experiment 3), an effect of conclusion typicality was found, so that people are more willing to project properties to more typical conclusions. Experiment 2 ruled out conclusion familiarity as a potential confounding variable. The results are interpreted in the light of current models of category-based induction.

Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates

Feedback control with limited data rates is an emerging area which incorporates ideas from both control and information theory. A fundamental question it poses is how low the closed-loop data rate can be made before a given dynamical system is impossible to stabilize by any coding and control law. Analogously to source coding, this defines the smallest error-free data rate sufficient to achieve "reliable" control, and explicit expressions for it have been derived for linear time-invariant systems without disturbances. In this paper, the more general case of finite-dimensional linear systems with process and observation noise is considered, the object being mean square state stability. By inductive arguments employing the entropy power inequality of information theory, and a new quantizer error bound, an explicit expression for the infimum stabilizing data rate is derived, under very mild conditions on the initial state and noise probability distributions.

Finite sample identifiability of multiple constant modulus sources

We prove that mixtures of continuous alphabet constant modulus sources can be identified with probability 1 with a finite number of samples (under noise-free conditions). This strengthens earlier results which only considered an infinite number of samples. The proof is based on the linearization technique of the analytical constant modulus algorithm (ACMA), together with a simple inductive argument. We then study the finite-alphabet case. In this case, we provide a subexponentially decaying upper bound on the probability of nonidentifiability for a finite number of samples. We show that under practical assumptions, this upper bound is tighter than the currently known bound. We then provide an improved exponentially decaying upper bound for the case of L-PSK signals (L is even).