Uncertainty Read online

  2.​2 Logic Is Not Empirical

  2.​3 Syllogistic Logic

  2.​4 Syllogisms

  2.​5 Informality

  2.​6 Fallacy

  3 Induction and Intellection

  3.​1 Metaphysics

  3.​2 Types of Induction

  3.​3 Grue

  4 What Probability Is

  4.​1 Probability Is Conditional

  4.​2 Relevance

  4.​3 The Proportional Syllogism

  4.​4 Details

  4.​5 Assigning Probability:​ Seeming Paradoxes and Doomsday Arguments

  4.​6 Weight of Probability

  4.​7 Probability Usually Is Not a Number

  4.​8 Probability Can Be a Number

  5 What Probability Is Not

  5.​1 Probability Is Not Physical

  5.​2 Probability and Essence

  5.​3 Probability Is Not Subjective

  5.​4 Probability Is Not Limiting Relative Frequency

  5.​5 Probability Is Not Always a Number Redux

  5.​6 Confirmation and Paradoxes

  6 Chance and Randomness

  6.​1 Randomness

  6.​2 Not a Cause

  6.​3 Experimental Design and Randomization

  6.​4 Nothing Is Distributed

  6.​5 Quantum Mechanics

  6.​6 Simulations

  6.​7 Truly Random and Information Theory

  7 Causality

  7.​1 What Is Cause Like?​

  7.​2 Causal and Deterministic Models

  7.​3 Paths

  7.​4 Once a Cause, Always a Cause

  7.​5 Falsifiability

  7.​6 Explanation

  7.​7 Under-Determination

  8 Probability Models

  8.​1 Model Form

  8.​2 Relevance and Importance

  8.​3 Independence Versus Irrelevance

  8.​4 Bayes

  8.​5 The Problem and Origin of Parameters

  8.​6 Exchangeability and Parameters

  8.​7 Mystery of Parameters

  9 Statistical and Physical Models

  9.​1 The Idea

  9.​2 The Best Model

  9.​3 Second-Best Models

  9.​4 Relevance and Importance

  9.​5 Measurement

  9.​6 Hypothesis Testing

  9.7 Die, p -Value, Die Die Die

  9.​8 Implementing Statistical Models

  9.​9 Model Goodness

  9.​10 Decisions

  10 Modelling Goals, Strategies, and Mistakes

  10.​1 The Goal of Models

  10.​2 Regression

  10.​3 Risk

  10.​4 Epidemiologist Fallacy

  10.​5 Quantifying the Unquantifiable

  10.​6 Time Series

  10.​7 The Future

  References

  Index

  © Springer International Publishing Switzerland 2016

  William BriggsUncertainty10.1007/978-3-319-39756-6_1

  1. Truth, Argument, Realism

  William Briggs1

  (1)New York City, NY, USA

  “Quid est veritas?”

  The answer to the above, perhaps the most infamous of all questions, was so obvious that Pilate’s interlocutor did not bother to state it. Truth was there, in the flesh, as it were, and utterly undeniable. Everyone knows the sequel. Since that occasion, at which the answer was painfully obvious, the question has been re-asked many times, with answers becoming increasingly skeptical, tortured, and incredulous. The reasons for this are many, not the least of which is that denial of truth leads to interesting, intellectually pleasing, unsolvable but publishable puzzles.

  Skepticism about truth is seen as sophistication; works transgressive to truth are rewarded, so much so that finding an audience accepting of truth is increasingly difficult. More than sixty years ago Donald Williams [224], exasperated over the pretended academic puzzlement over the certainty of truth, said the academy in its dread of superstition and dogmatic reaction, has been oriented purposely toward skepticism: that a conclusion is admired in proportion as it is skeptical; that a jejune argument for skepticism will be admitted where a scrupulous defense of knowledge is derided or ignored; that an affirmative theory is a mere annoyance to be stamped down as quickly as possible to a normal level of denial and defeat.

  Yet truth is our goal, the only destination worth seeking. So we must understand it. There are two kinds of truth: ontological and epistemological, comprising existence and our understanding of existence. Tremendous disservice has been done by ignoring this distinction. There are two modes of truth: necessary and local or conditional. From this seemingly trivial observation, everything flows.

  1.1 Truth

  Truth exists, and so does uncertainty. Uncertainty acknowledges the existence of an underlying truth: you cannot be uncertain of nothing: nothing is the complete absence of anything. You are uncertain of something, and if there is some thing, there must be truth. At the very least, it is that this thing exists. Probability, which is the language of uncertainty, therefore aims at truth. Probability presupposes truth; it is a measure or characterization of truth. Probability is not necessarily the quantification of the uncertainty of truth, because not all uncertainty is quantifiable. Probability explains the limitations of our knowledge of truth, it never denies it. Probability is purely epistemological, a matter solely of individual understanding. Probability does not exist in things; it is not a substance. Without truth, there could be no probability.

  Why a discussion of truth in a book devoted to probability? Since probability is the language of uncertainty, before we can learn what it means we need to understand what it is that probability aims at. Hempel understood this, but couldn’t help himself from writing the word without scare quotes, as if “truth” might not exist, [110]. What is the nature of probability’s target? What does it mean to be uncertain? How do we move from uncertainty to certainty? How certain is certain? It will turn out that statements of probability (assuming they are made without error, an assumption we make of all arguments unless otherwise specified) are true. When we say things like “Given such-and-such evidence, the probability of X is p”, we mean to say either that (the proposition) X is true, or that not-X is. So truth must be our foundation. What follows is not a disquisition on the subject of truth, merely an introduction sufficient to launch us into probability. This chapter is also a necessity because the majority of Western readers have grown up in a culture saturated in relativism. There is ample reason Pilate’s question is so well remembered.

  Our eventual goal is to grasp models, and models of all kinds, probabilistic or otherwise, are ways of arguing, of getting at the truth. All arguments, probabilistic or not, have the same form: a list of premises, supposeds, accepteds, evidence, observations, data, facts, presumptions, and the like, and some conclusion or proposition which is thought related to the list. Related how and in what way is a discussion that comes later, but for now it loosely is associated with what causes the proposition to be true. Arguments can be well or badly structured, formally valid or invalid, and sound or unsound. Unlike most logical, mathematical, and moral arguments, which often end in truth, probabilistic arguments do not lead to certainty. Whenever a probabilistic argument is used, it is an attempt to convince someone how certain a proposition is in relation to a given body of evidence, and only that body of evidence.

  Anybody who engages in any argument thus accepts that certainty and truth exist. We should have no patience for philosophical skepticism, which is always self-defeating. If you are certain there is no certainty, you are certain. If it is true that there is no truth, it is false there is no truth. If you are certain that “Every proposition is subject to uncertainty” then you speak with forked tongue. Certainty and truth therefore exist. But we must understand that truth resides in our intellects and not in objects themselves, except in the sense of existence. That being so, probability also does
not exist physically; it also resides in our intellects and not in things themselves.

  All arguments have stated and tacit premises, with those tacit usually about the meaning of the words and grammar used to state the argument, but also about how arguments themselves are to be interpreted, about how we move from premise to conclusion. Confusion usually enters when there are misunderstandings or disagreements formed about the tacit premises. Badly structured arguments are incautious in their use of tacit premises, containing too many or those which are prone to dispute. Ajdukiewicz confirms this in his lost classic Pragmatic Logic, an excellent book for students to understand the nature of arguments, [4]. There is also a burgeoning field called argumentation theory which can be looked up.

  1.2 Realism

  No definition of truth is better or more succinct than Aristotle’s: “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.” St Thomas Aquinas, following Aristotle’s Metaphysics, in his Summa Theologica (First Part, Q. 16, articles 1 and 8) said “the true denotes that towards which the intellect tends.” “Truth, properly speaking, resides only in the intellect, as said before (1); but things are called true in virtue of the truth residing in an intellect.”

  This view encapsulate what is called correspondence and reflect the metaphysics of (moderate) realism ; see [222] and below in the chapter on Causality. When we later say of a proposition “It is necessarily true”, this is never meant to imply that the proposition is true in or because of some theory. The proposition is necessarily true for reasons in the proposition itself and the evidence which supports it; the proposition is not true “in” or because of a theory. It is true because it is true.

  Moderate realism is the common-sense position that there exist real things, that there is an existence independent of our minds, that an external world is “out there” and that we can know it, that we can “know things as they are in themselves”, to coin a phrase. Moderate realism holds that greenness exists apart from or in addition to individual green things; exists as an intellectual idea, that is. Realism says the idea of color exists independent of individual colored things. Mathematicians are realists when they insist all triangles have three straight sides and an interior sum of angles of 180∘. Individual approximations to or implementations of triangles also exist, but given the way the world is, all are imperfect representations of the universal ideal. Try drawing one. Catness exists and so do individual cats. We can tell cats from dogs because we know the nature or essence of both. Knifeness exists as do individual knives, even though it’s not always clear if a given object is a knife or only acts like one.

  These natures (or ideas) are universals. They don’t exist as physical objects in some ethereal realm, à la Plato; instead they exist in the objects which instantiate them—redness exists in red apples, knifeness exists in cleavers—or they exist as idea in intellects, as immaterial concepts. This is scholastic realism, a modified form of Aristotle’s philosophy. Excellent introductions to moderate realism are given by Feser [70, 71].

  Contrary to realism is nominalism, which denies universals exist. Under this view, individual triangles exist but there is no concept of an ideal, perfect triangle. This appears to leave out mathematical definitions and, it would seem to follow, all of mathematics, since this field is founded on universal truths (see below; also see Franklin’s Aristotelian conception of mathematics, [81]). Under nominalism, two drawings of triangles are not two drawings of triangles, just two drawings which might have vague similarities, the similarities bespeaking of no central thing in common. How, then, if nominalism is true, could we even have the word triangle or even similarity? Man is also therefore a meaningless term: there are individual bipedal creatures which might coincidentally look somewhat alike and share some DNA (but is all DNA actually DNA?), just as they are more dissimilar to quadrupedal creatures. The higher concept of man or human being holds no higher meaning. Things do not instantiate natures. Things just are, never mind how. Most working scientists are not nominalists, for obvious reasons.

  Nominalism comes in various forms and subtleties, but no branch holds any interest for probability and statistics. If there were no universals, there would be little point in conducting experiments or grouping data, which admits of universals or essences. The acts of grouping and collating say, do they not?, “All these data represent the same underlying essential thing.” Even those dismal objects p-values admit of universal “null” and “alternate” hypotheses; these surely bespeak of universal essences and do not point to physical substances (p-values, God rot them, are discussed later). And neither is probability, as de Finetti taught us in a loud voice, a tangible physical quantity, something that can be measured with a physical apparatus. Probability, like logic, as we’ll see, assumes universals.

  The opposite of nominalism (if such a thing could have an opposite) is idealism, the concept that reality does not exist, rather that individual physical objects do not exist, but that only universals do. Our thoughts are capital-I It, our thoughts are everything, our thoughts define existence. If so, how do we know when you and I are thinking of the same thing? We cannot. I don’t consider idealism to be on any interest. The best overview and refutation of idealism is found in David Stove’s essay “Idealism: A Victorian horror story”, [208].

  There are many other ways for thought to go wrong, and those which have a bearing on probability will be outlined later. For now, I’ll boldly state all scientists are realists, or ought to be. There’s no use for a scientist who subscribes to some form of idealism. After all, if the universe is only in his mind, there’s no guarantee that the universe which is my mind is in any way the same thing as the universe in his. If idealism is true, why not make up how the universe is? Saves research time. If nominalism is true, what is true here might not be true there, and it is of little to no use to speak of “laws” or causes.

  1.3 Epistemology

  Can we know any truths? Yes. And if you disagree you necessarily agree. In disagreeing you’d at least know that you can’t know anything, which would be a truth, and then you’d realize you bit yourself in the tail. Any attempt to deny there are truths is self-contradictory. Roger Scruton said that the people who tout theories which deny truth or our knowledge of it are inviting us to disbelieve them, an invitation which we eagerly accept, [193].

  That there are truths and we can know them is traditionally called rationalism. A prime example of a known truth is Aristotle’s principal of non-contradiction. The epistemic version states that a proposition cannot be both true and false simultaneously (given the same evidence). It is impossible, and not just unlikely, for somebody to doubt this principle. It is possible, and unfortunately not uncommon, for some to claim to doubt it. But claiming and doing are not identical as everybody knows, and that is why we have the words like deception, mistaken, and lying—words, incidentally, which admit the existence of truth and knowledge. Claiming to doubt the principle of non-contradiction is like the man who boasts of disbelieving the reality of gravity. No matter the degree of his earnestness or the number of his scholarly credentials, if he takes a long walk off a short dock he is going to end up wet.

  A ontological version of the non-contradiction principle is that something cannot be and not-be at the same time, that something cannot exist and not-exist simultaneously. Existence is an ontological truth. You cannot exist and not-exist at the same time; further, it is impossible, and not just unlikely, to believe that you exist and that at the same time don’t exist. This is not the same as saying, for example with respect to certain very small objects in physics, that you do not know if or where a thing exists or not. A thing’s existence and our knowledge of it are different. Indeed, the mixing up of epistemological and ontological claims is a routine problem in probability.

  Everyone, regardless of what they might claim, knows that an external world exists. And all scientists ought to admit it, e
lse they’re in the wrong business. This is another way to state realism. Anybody asking the question of another, “Does an external world exist?” has answered it affirmatively, since to ask it requires a person to ask and another to answer it, hence an external world in which the other person exists to answer it, hence we can know it exists, hence we know there are other people, too (the traditional way to phrase it is that we know there are “other minds”).

  Another truth known to everybody is that solipsism is impossible. Again, if you disagree with me, you agree with me and acknowledge the complete fallaciousness of your position because, of course, to disagree with me implies someone other than yourself exists, hence solipsism is false.

  But what if I were an illusion? What if, that is, you were hallucinating my obstreperousness? From David Stove’s masterful essay “I only am alone escaped to tell thee: Epistemology and the Ishmael Effect” , [207, pp. 61–82]: