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4059 results in Logic

Page 137 of 203

  • The Strategy of Cooperation

  • Edited by Christopher W. Morris, Bowling Green State University, Ohio, Arthur Ripstein, University of Toronto
  • Book:
    Practical Rationality and Preference
    Published online:
    11 January 2010
    Print publication:
    16 July 2001, pp 189-208

  • Summary

    Introduction

    Starting with Adam Smith, and running like a bright thread throughout virtually all the subsequent theoretical literature on political economy, one can mark a preoccupation with the conditions under which public structures can emerge that will permit individuals to cooperatively transact with one another to their mutual advantage. The good news that there are in fact conditions under which such structures do emerge has to be tempered, however, by the bad news (1) that historical processes of institutional development tend to be path-dependent in ways that work against adaptive efficiency and (2) that virtually all institutional arrangements are subject to manipulation by special interests, to the short-term advantage of some, but often to the longterm disadvantage of all. Recent work in political economy has in fact identified deep pressures on human interaction that tend to prevent anything like the full realization of the mutual gains that cooperation can make possible. A truly optimal arrangement would seem, then, to be an ideal that is remote from social reality.

    In one sense, there should be nothing surprising about this. After all, the historical record hardly supports any other picture. What is surprising, however, is that many of these conclusions are driven by models of ideally rational beings who have substantial and common knowledge of each other's rationality and preferences and the strategic structure of their interactions. On the accounts offered, Pareto-suboptimality does not flow just from assuming that some are less than fully rational, or possess limited or asymmetrical knowledge. Mutually disadvantageous free-riding and conflict over the distribution of goods are taken to be natural to the way in which even hyperrational and fully informed individuals interact.

  • Bombs and Coconuts, or Rational Irrationality

  • Edited by Christopher W. Morris, Bowling Green State University, Ohio, Arthur Ripstein, University of Toronto
  • Book:
    Practical Rationality and Preference
    Published online:
    11 January 2010
    Print publication:
    16 July 2001, pp 81-97

  • Summary

    In an early article, Gauthier argued that, to act rationally, we must act morally. I tried to refute that argument. Since Gauthier was not convinced, I shall try again.

    Gauthier assumes that, to be rational, we must maximize our own expected utility. Though he distinguishes between ‘utility’ and ‘benefit’, this distinction does not affect his main arguments. We can regard him as appealing to the Self-interest Theory.

    Many writers have argued that, in self-interested terms, it is always rational to act morally. According to most of these writers, morality and self-interest coincide. But that is not Gauthier's line. Gauthier concedes that acting morally may be, and be known to be, worse for us. He claims that, even in such cases, it is rational to act morally.

    If we appeal to the Self-interest Theory, it may seem impossible to defend that claim. How can our acts be rational, in self-interested terms, if we know them to be worse for us? But Gauthier revises the Self-interest Theory. On the standard version of this theory, an act is rational if it will maximize our expected benefit – or be expectably-best for us. On Gauthier's version, we should aim to benefit ourselves not with our acts but only with our dispositions. A disposition is rational if having it will be expectably-best for us. An act is rational if it results from such a disposition.

    Besides revising the Self-interest Theory, Gauthier restricts the scope of morality. To act morally, Gauthier claims, we must honour our agreements.

  • 22 - Inductive Behavior as an Evasion of the Problem of Induction

  • from PROBABILITY APPLIED TO PHILOSOPHY
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 261-268

  • Summary

    The frequentist agrees that no reasons can be given for inductive inferences, but holds that reasons can be given for inductive behavior, using certain procedures based on the idea of confidence intervals.

    The Bayesian is able to attach personal probabilities, or degrees of belief, to individual propositions. The hard-line frequency dogmatist thinks that probabilities can be attached only to a series of events.

    Probability, says this dogmatist, just means the relative frequency of some kind of event produced by a chance setup. Or it refers to the propensity of a chance setup to produce events with a certain stable frequency. Or it refers to certain underlying symmetry properties.

    At any rate, we cannot talk sensibly about the probability of a single event, for that event either happens or does not happen. It has “probability” 0 or 1, and that is that.

    So the frequency dogmatist will never talk about the frequency-type probability that a particular hypothesis is true.

    The hypothesis is either true or false, and there is no frequency about it. At most, we can discuss the relative frequency with which hypotheses of a certain kind are true.

    Thus far the dogmatic frequentist is happy to agree with Hume.

    INDUCTIVE BEHAVIOR

    Nevertheless, continues the frequentist, we may sometimes be able to apply a system for making inferences or drawing conclusions such that the conclusions are usually right.

    We can talk about the relative frequency with which inferences drawn by a certain method are in fact correct.

  • 1 - Logic

  • from LOGIC
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 1-10

  • Summary

    Logic is about good and bad reasoning. In order to talk clearly about reasoning, logicians have given precise meanings to some ordinary words. This chapter is a review of their language.

    ARGUMENTS

    Logicians attach a special sense to the word argument. In ordinary language, it usually takes two to argue. One dictionary defines an argument as:

  • A quarrel.

  • A discussion in which reasons are put forward in support of and against a proposition, proposal, or case.

  • A point or series of reasons presented to support a proposition which is the conclusion of the argument.

    • Definition (3) is what logicians mean by an argument.

    Reasoning is stated or written out in arguments. So logicians study arguments (in sense 3).

    An argument thus divides up into:

    A point or series of reasons which are called premises, and a conclusion.

    Premises and conclusion are propositions, statements that can be either true or false. Propositions are “true-or-false.”

    GOING WRONG

    The premises are supposed to be reasons for the conclusion. Logic tries to understand the idea of a good reason.

    We find arguments convincing when we know that the premises are true, and when we see that they give a good reason for the conclusion.

    So two things can go wrong with an argument:

  • ▪ the premises may be false.

  • ▪ the premises may not provide a good reason for the conclusion.

  • 20 - The Philosophical Problem of Induction

  • from PROBABILITY APPLIED TO PHILOSOPHY
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 247-255

  • Summary

    For philosophers, this is the most important question about induction. It is not a problem within inductive logic. It questions the very possibility of inductive reasoning itself.

    DAVID HUME

    In 1739, David Hume (1711–1776), the Scottish philosopher, published A Treatise of Human Nature, one of the half-dozen most influential books of Western philosophy. He was twenty-eight years old at the time. In 1748, he published An Enquiry Concerning Human Understanding.

    These books, especially the second, include the classic statement of what came to be called the problem of induction.

    Hume's problem about induction is only a small part of a very general theory of knowledge. Here we study just this one aspect of Hume's philosophy.

    SKEPTICISM

    In ordinary English, a skeptic is:

  • ♦ Someone who habitually doubts accepted beliefs.

  • ♦ A person who mistrusts other people or their ideas.

  • ♦ Someone who rejects traditional beliefs, such as religious beliefs.

    • PHILOSOPHICAL SKEPTICISM

    Philosophers attach a far more sweeping sense to the idea of skepticism. A philosophical skeptic is someone who claims to:

  • ♦ Doubt that any real knowledge or sound belief about anything is possible.

    • There are more specialized types of philosophical skepticism, depending upon what kind of knowledge is in doubt. Think of any field of knowledge or belief X, where X may be religious, or scientific, or moral. X may be knowledge about other people, or about the reality of the world around us, or even knowledge about yourself.

An Introduction to Probability and Inductive Logic
  • Ian Hacking
  • Published online:
    05 June 2012
    Print publication:
    02 July 2001
  • This is an introductory 2001 textbook on probability and induction written by one of the world's foremost philosophers of science. The book has been designed to offer maximal accessibility to the widest range of students (not only those majoring in philosophy) and assumes no formal training in elementary symbolic logic. It offers a comprehensive course covering all basic definitions of induction and probability, and considers such topics as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction. The key features of this book are a lively and vigorous prose style; lucid and systematic organization and presentation of ideas; many practical applications; a rich supply of exercises drawing on examples from such fields as psychology, ecology, economics, bioethics, engineering, and political science; numerous brief historical accounts of how fundamental ideas of probability and induction developed; and a full bibliography of further reading.

  • 14 - Coherence

  • from PROBABILITY AS A MEASURE OF BELIEF
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 163-170

  • Summary

    Consistent personal betting rates satisfy the basic rules of probability. Consistency in this sense is called coherence.

    Personal probabilities and betting rates are all very well. But thus far they have no structure, no rules–in fact, not much meaning. Now we give one argument that betting rates ought to satisfy the basic rules for probability. We have already had three thought experiments in Chapter 13. Here are two more.

    FOURTH THOUGHT EXPERIMENT: SETS OF BETTING RATES

    A group of beliefs can be represented by a set of betting rates.

    Imagine yourself advertising a set of betting rates. For each of the propositions A, B, … K in the set, you offer betting rates pa, pb, pc , …, pk .

    In this imaginary game, you are prepared to bet, say,

    on A at rate pa , or

    against A at rate (1–pa ).

    FIFTH THOUGHT EXPERIMENT: SIMPLE INCONSISTENCY

    These are personal betting rates. Couldn't you choose any fractions you like?

    Of course. But you might be inconsistent.

    For example, suppose you are concerned with just two possibilities, B (for “below zero”) and ~B:

    B: On the night of next March 21, the temperature will fall below 0°C at the Toronto International Airport meteorological station.

    ~B: On the night of next March 21, the temperature will not fall below 0°C at the Toronto International Airport meteorological station.

  • 17 - Normal Approximations

  • from PROBABILITY AS FREQUENCY
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 201-208

  • Summary

    We have seen that relative frequencies converge on theoretical probabilities. But how fast? When can we begin to use an observed relative frequency as a reliable estimate of a probability? This chapter gives some answers. They are a little more technical than most of this book. For practical purposes, all you need to know is how to use is the three boxed Normal Facts below.

    EXPERIMENTAL BELL-SHAPED CURVES

    On page 191 we had the result of a coin-tossing experiment. The graph was roughly in the shape of a bell. Many observed distributions have this property.

    Example: Incomes. In modern industrialized countries we have come to expect income distributions to look something like Curve 1 on the next page, with a few incredibly rich people at the right end of the graph. But in feudal times there was no middle class, so we would expect the income distribution in Curve 2. It is “bimodal”–it has two peaks.

    Example: Errors. We can never measure with complete accuracy. Any sequence of “exact” measurements of the same quantity will show some variation. We often average the results. We can think of this as a sample mean. A good measuring device will produce results that cluster about the mean, with a small sample standard deviation. A bad measuring device gives results that vary wildly from one another, giving a large standard deviation.

  • 5 - Conditional Probability

  • from HOW TO CALCULATE PROBABILITIES
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 47-57

  • Summary

    The most important new idea about probability is the probability that something happens, on condition that something else happens. This is called conditional probability.

    CATEGORICAL AND CONDITIONAL

    We express probabilities in numbers. Here is a story I read in the newspaper. The old tennis pro Ivan was discussing the probability that the rising young star Stefan would beat the established player Boris in the semifinals. Ivan was set to play Pete in the other semifinals match. He said,

    The probability that Stefan will beat Boris is 40%.

    Or he could have said,

    The chance of Stefan's winning is 0.4.

    These are categorical statements, no ifs and buts about them. Ivan might also have this opinion:

    Of course I'm going to win my semifinal match, but if I were to lose, then Stefan would not be so scared of meeting me in the finals, and he would play better; there would then be a 50–50 chance that Stefan would beat Boris.

    This is the probability of Stefan's winning in his semifinal match, conditional on Ivan losing the other semifinal. We call it the conditional probability. Here are other examples:

    Categorical: The probability that there will be a bumper grain crop on the prairies next summer.

    Conditional: The probability that there will be a bumper grain crop next summer, given that there has been very heavy snowfall the previous winter. […]

  • 8 - Expected Value

  • from HOW TO COMBINE PROBABILITIES AND UTILITIES
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 79-97

  • Summary

    Inductive logic is risky. We need it when we are uncertain. Not just uncertain about what will happen, or what is true, but also when we are uncertain about what to do. Decisions need more than probability. They are based on the value of possible outcomes of our actions. The technical name for value is utility. This chapter shows how to combine probability and utility. But it ends with a famous paradox.

    ACTS

  • Should you open a small business?

  • Should you take an umbrella?

  • Should you buy a Lotto ticket?

  • Should you move in with someone you love?

    • In each case you settle on an act. Doing nothing at all counts as an act.

    Acts have consequences.

    You go broke (or maybe found a great company).

    You stay dry when everyone else is sopping wet (or you mislay your umbrella).

    You waste a dollar (or perhaps win a fortune).

    You live happily ever after (or split up a week later).

    You do absolutely nothing active at all: that counts as an act, too. Some consequences are desirable. Some are not. Suppose you can represent the cost or benefit of a possible consequence by a number–so many dollars, perhaps. Call that number the utility of the consequence.

    Suppose you can also represent the probability of each possible consequence of an act by a number.

  • 16 - Stability

  • from PROBABILITY AS FREQUENCY
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 189-200

  • Summary

    Some core connections between the basic rules of probability, frequency-type probability, and statistical stability. These facts provide the foundations for frequency reasoning. The chapter ends by stating Bernoulli's Theorem, one of the most fundamental facts about probability.

    Now we move from three chapters using the belief perspective to four chapters using the frequency perspective. Chapters 13 and 14 gave one reason why belief-type probabilities should satisfy the basic rules of probability. Chapter 15 showed how to apply that result to “learning from experience.” Chapters 16–19 do something similar from the frequency perspective.

    THE PROGRAM

  • ▪ This chapter describes some deductive connections between probability rules and our intuitions about stable frequencies.

  • ▪ Chapter 17 extends these connections.

  • ▪ Chapter 18 presents one core idea of frequency-type inductive inference–the significance idea.

  • ▪ Chapter 19 presents a second core idea of frequency-type inductive inference–the confidence idea. This idea explains the way opinion polls are now reported. It also explains how we can think of the use of statistics as inductive behavior.

    • All the results described in this chapter are deductions from the basic rules of probability. The results are only stated, and not proved, because the proofs are more difficult than other material in this book.

    BELIEF AND FREQUENCY COMPARED

    The basic rules are for any type of probability. Belief-type and frequency-type probabilities emphasize two fundamentally different types of consequences of the basic rules.

  • 19 - Confidence and Inductive Behavior

  • from PROBABILITY AS FREQUENCY
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 229-246

  • Summary

    Our final example of inductive logic denies that we make inferences at all. Instead, we behave inductively, in such a way that our system for making decisions has good overall consequences for us. This is the theory of confidence intervals.

    SAMPLES AND POPULATIONS

    In Chapter 2 there was a box of 60 oranges–a population of oranges. We drew 4 oranges at random–a sample. In Chapter 2 we distinguished two forms of argument:

    Statement about a population.

    So,

    Statement about a sample.

    Statement about a sample.

    So,

    Statement about a population.

    Bernoulli's Theorem, applied to sampling with replacement from an urn, makes a statement about a sample on the basis of knowledge about the population of the urn and the sampling method. It is an example of the first type of argument.

    Now we want to go in the other direction. We take a sample. We want to draw a conclusion about a population. A significance test involves one type of reasoning but does not go very far. We often want to use a sample to estimate something about a population. The most familiar type of estimate based on a sample is the opinion poll.

    OPINION POLLS

    Before we do some inductive logic, we should pause to think realistically about survey sampling. Consider a controversial survey topic.

  • 6 - The Basic Rules of Probability

  • from HOW TO CALCULATE PROBABILITIES
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 58-68

  • Summary

    This chapter summarizes the rules you have been using for adding and multiplying probabilities, and for using conditional probability. It also gives a pictorial way to understand the rules.

    The rules that follow are informal versions of standard axioms for elementary probability theory.

    ASSUMPTIONS

    The rules stated here take some things for granted:

  • ▪ The rules are for finite groups of propositions (or events).

  • ▪ If A and B are propositions (or events), then so are AvB, A&B, and ∼A.

  • ▪ Elementary deductive logic (or elementary set theory) is taken for granted.

  • ▪ If A and B are logically equivalent, then Pr(A) = Pr(B). [Or, in set theory, if A and B are events which are provably the same sets of events, Pr(A) = Pr(B).]

    • NORMALITY

    The probability of any proposition or event A lies between 0 and 1.

    1. (1) 0 ≤ Pr(A) ≤ 1

    Why the name “normality”? A measure is said to be normalized if it is put on a scale between 0 and 1.

    CERTAINTY

    An event that is sure to happen has probability 1. A proposition that is certainly true has probability 1.

    (2) Pr(certain proposition) = 1

    Pr(sure event) = 1

    Often the Greek letter Ω is used to represent certainty: Pr(Ω) = 1.

    ADDITIVITY

    If two events or propositions A and B are mutually exclusive (disjoint, incompatible), the probability that one or the other happens (or is true) is the sum of their probabilities.

  • 15 - Learning from Experience

  • from PROBABILITY AS A MEASURE OF BELIEF
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 171-188

  • Summary

    Bayes' Rule is central for applications of personal probability. It offers a way to represent rational change in belief, in the light of new evidence.

    BAYES' RULE

    Bayes' Rule is of very little interest when you are thinking about frequency-type probabilities. It is just a rule. On page 70 we derived it in a few lines from the definition of conditional probability.

    For many problems–shock absorbers, tarantulas, children with toxic metals poisoning, taxicabs–it shortens some calculations. That's all.

    But Bayes' Rule really does matter to personal probability, or to any other belief-type probability.

    Today, belief-type probability approaches are often called “Bayesian.” If you hear a statistician talking about a Bayesian analysis of a problem, it means some version of ideas that we discuss in this chapter. But there are many versions, ranging from personal to logical. An independent-minded Bayesian named I. J. Good (see page 184) figured out that, in theory, there are 46,656 ways to be a Bayesian!

    HYPOTHESES

    “Hypothesis” is a fancy word, but daily life is full of hypotheses. Most decisions depend upon comparing the evidence for different hypotheses.

    Should Albert quit this course? The drop date is tomorrow. He is a marginal student and has done poorly so far. Will the course get harder, as courses often do? (Hypothesis 1) Or will it stay at its present level, where he can pass the course? (Hypothesis 2)

  • 21 - Learning from Experience as an Evasion of the Problem of Induction

  • from PROBABILITY APPLIED TO PHILOSOPHY
  • Ian Hacking, University of Toronto
  • Book:
    An Introduction to Probability and Inductive Logic
    Published online:
    05 June 2012
    Print publication:
    02 July 2001, pp 256-260

  • Summary

    How belief-type probability can be applied to the problem of induction using the idea of learning from experience by Bayes' Rule.

    The idea is already present in Chapters 13–15.

  • ▪ We can represent degrees of belief by numbers between 0 and 1.

  • ▪ Degrees of belief represented by these numbers should satisfy the basic laws of probability, on pain of being “incoherent” if they don't.

  • ▪ If they do satisfy these laws, then Bayes' Rule follows.

  • ▪ Hence we can update earlier degrees of belief by new evidence in a coherent, “rational” way.

    • This evasion of the problem of induction is called Bayesian.

    The Bayesian does not claim to be able to justify any given set of degrees of belief as being uniquely rational. He does claim that he can tell you how it is reasonable to change your beliefs in the light of experience.

    The Bayesian says to Hume:

    Hume, you're right. Given a set of premises, supposed to be all the reasons bearing on a conclusion, you can form any opinion you like.

    But you're not addressing the issue that concerns us!

    At any point in our grown-up lives (let's leave babies out of this), we have a lot of opinions and various degrees of belief about our opinions. The question is not whether these opinions are “rational.” The question is whether we are reasonable in modifying these opinions in the light of new experience, new evidence.

Page 137 of 203