Truth: Logic of Probability and Statistics

Doubt truth and end up with a haircut like this.
Here, as promised, is rough, incomplete, outline-only, not-yet-finished, gist-only version of Chapter 1, Truth, for the book tentatively titled The Philosophy of Probability and Statistics (I’m also toying with The Philosophy of Science, Probability, and Statistics and This Is The Book You Were Thinking Of).

Truth is meant as an introduction or guide to truth and not a disquisition. I don’t have the space for a full justification of the realism-coherence description of truth (which is anyway obvious; and if you say it isn’t, it is), nor for a complete survey of all the alternatives and why they are wrong. I only have enough to prepare the ground for probability—a subject which, it will be no secret to regular readers, is vaster than the mathematical quantification ordinarily thought of as “scientific.”

Chapter 2, incidentally, is Logic, and contains some material that complements Truth, such as a proof that logic cannot be empirical and that our knowledge of it must be, in part, built in. So if you see something missing in Truth, it might be in Logic. Same thing for Chapter 3, Induction. I’m still wavering whether to put Causality with Induction or bust it out on its own.

These three beginning chapters (and a Preface) are the necessary foundation for understanding fully probability and statistics. They are therefore the hardest to write, especially since they must perforce be terse. I don’t want people skipping over things, so they can’t be too long; yet if they are too short, I risk giving key elements short shrift.

I’m not happy with the sections on Scientism and Faith: consider these well underdone, mere placeholders.

Statisticians (me included) receive no philosophy in their formal training, except for inconsistent occasional unanchored tidbits. This is why, for example, most repeat the false proposition “All models are wrong”, when nearly the exact opposite is true. Others claim to “use falsification all the time”, which itself is falsified. And so on. Like most people who have no education in an area, the limited knowledge statisticians do possess is thought sufficient and complete. Since this is not so, before work commences on the subject proper, I need these three (four?) chapters whose main job is to prove there is more to be known and to highlight and point to places where complete descriptions might be found.

Anyway, here you go. Unless you have something so secret you don’t want any except the NSA and me to know, please leave comments below and don’t email. That way I won’t misplace them. Don’t point out typos. Way too early for that. Oh, the footnotes, references, and index are far, far from complete.


Update Like I said, Faith was merely a sketch, but upon further reflection, I think I’ll add it to Induction, where it is much better placed (given what I want to say about belief in the unseen).


  1. * “probability is a measure of truth” is inconsistent with probability as the measure of an event.
    * recommend change “probability is the language of uncertainty” to “Probability is in the language of uncertainty”
    * I’d describe a universal as a pattern and tell the reader of how patterns are discovered (rather than existing, as Plato thought) in the chapter on induction.
    * “science” is a polysemic word needing disambiguation in order for valid conclusions to be drawn from arguments about it.
    * I feel that mathematical descriptions of the concepts need to be provided for rigor and to avoid applications of the equivocation fallacy.

  2. Scotian

    “That the speed of light is some number might also be false for the same reason.” The speed of light is not a measured quantity. It is defined as 299,792,458 m/s exactly. What is measured is the length of the second with the length of the metre following from the definition.

    “People before Newton knew apples fell, and would say so.” This is not really the point. The advance in knowledge attributed to Newton is that of universal gravitation which is that the force which caused the apple to fall also kept the Moon in orbit around the Earth. It was a centripetal and not a transverse force as previously thought.

    It also seems to me that a book on the philosophy of probability and statistics should be less overtly religious. People might get the wrong idea as to your intentions.

  3. One other idea that I’d like to pass along is that it would be beneficial for the work to be divided into three parts, each addressed to a different age group. Part I would cover the classical logic and logical abstraction; it would be written for sixth or seventh graders. Part II would cover measure theory and probability theory including Bayes’s theorem; it would be addressed high school seniors or college freshmen. Part III would cover information theory, induction, science, epistemology and the principles of reasoning; it would be written for college seniors or first year graduate students. Organized in this way, the material could be fed to students at that age at which it helped their subsequent intellectual development the most.

  4. Briggs


    Good point re: light speed. But point is still same, as is point about removing contingencies. Both groups right about consequence, but the evidence different.

    My intentions? About truth? Well, well.


    Good grief.

  5. Jim Fedako

    The superfluous “copyright” can be removed; your government granted you that supposed right once you began typing. That so-called intellectual property rights are invalid is another discussion.

  6. Uncle Mike

    Dear Matt,

    1. I love you and value your work

    2. Your self-appointed task (this book) is worthy

    3. Your ideas are sound

    4. You badly need an editor. The paragraphs do not flow, the sentences are unwieldly, the ideas pop up hodgepodge.

    I am also an extemporaneous writer. So I know from experience that inspiration does not suffice. There must also be craft.

    Please take some time and make some effort to refine your craft. I strongly advise (as in don’t make me make you, because I do have a 2×4 and an attitude) that you study a writer’s craft instructional book.

    I suggest *Voice and Vision: A Guide to Writing History and Other Serious Nonfiction* by Stephen J. Pyne. (You can find it at Amazon)

    That book will really help you. Yes, it’s true that Pyne is a friend. As are you. But my advice is not about affinity; it’s about crafting your writing so it works for your purposes.

    Right now it does not. I say this with all good will towards you. You will thank me later.

  7. Briggs

    Thanks, Uncle Mike. I’m working on it. These three first chapters are the hardest to write.

    Jim. Right, but it’s there just as a reminder to others.

  8. Scotian

    Uncle Mike,
    You’ll have to change your moniker to Dutch Uncle.

  9. hi:

    I wrote “the Unix Guide to Defenestration” and discovered that most of the audience either had no idea what the word meant, or (worse) thought they did, but didn’t – and even most of the brighter ones didn’t get the joke.

    Your working title has, I think, a similar problem: the people who think they understand what it means will misunderstand its content as a label for your work – and very few will react to the title by grabbing the book for immediate purchase and later review.

    A quick google for “Truth, Probability, and Statistics” didn’t turn up anything frightening – so may I suggest you consider its virtues as both marketing tool and label?

  10. Ye Olde Statisician

    The speed of light does not exist independently of a method of measurement. Even when presented as merely a defined value, it is a value derived from a privileged method of measurement.

    I have always understood Box’s dictum about models to mean that any model of any reasonably complex system must, per Ockham’s Razor, omit some-to-many of the variables in the model. That is, the Ŷs produced by the model will never exactly match the Ys produced by reality.

  11. Scotian

    “The speed of light does not exist independently of a method of measurement.” Of course it does. This was Einstein’s insight, the speed of light as an invariant quantity independent of reference frame.

    “Even when presented as merely a defined value, it is a value derived from a privileged method of measurement.” I’m not really sure what you mean by this. The value was chosen to be consistent with the earlier definition of the metre but now relates time to space as I said above. The cesium clock is used to measure and define the second and time of flight measurements give the metre. This is an improvement over the platinum iridium bar and krypton wavelength used previously and definitely an improvement on using the mean solar day for the second.

    Any dimensioned parameter or constant can be given any value that we want. It is a measurement system (length, time, mass, charge) that has to be measured. That’s why those pesky theoretical physicists like to set them all equal to one. Dimensionless constants such as pi do not have that luxury.

  12. Ye Olde Statisician

    YOS: “…it is a value derived from a privileged method of measurement.”

    Scotian: I’m not really sure what you mean by this. The value was chosen to be consistent with the earlier definition of the metre but now relates time to space as I said above. The cesium clock is used to measure and define the second and time of flight measurements give the metre. This is an improvement over the platinum iridium bar and krypton wavelength used previously and definitely an improvement on using the mean solar day for the second.

    But you just described (in part) the privileged method of measurement, as well as the method it superseded. Before there were cesium clocks, there were rotating mirrors, Kerr cells, geodimeters, even at one time the moons of Jupiter. Each method produced different results, and did not always produce the same result even when carried out by the same physicist.

    The point is that any measurement is a product, produced by a manufacturing process (“measurement process”), and different methods will in general produce different results. Also any method will have an inherent precision (repeatability) to it, as well as a reproducibility (when different people use it). It may also have an accuracy, provided there is a privileged measurement process that is taken as a standard. In that case, accuracy is defined as the difference between the method used and the standard.

    As an aside: Do you know how the cesium clock is calibrated, and to what standard?

  13. Scotian

    If we are in agreement with this, why the original comment? But that is okay, I always enjoy your contributions. As to your question, as far as I know the cesium clock is the standard. There are some details of operation here:

    Of more interest in many ways is the determination of running time as distinct from time duration. The running time must keep tract of dates and time of day and still has a connection to astronomical events. Wikipedia has a review of this at: and
    It is all very complicated.

    An aside, I notice that IUPAC wants us to spell cesium as caesium now. Will the tyranny never end?

  14. ad

    It reads like you have an axe to grind. And when all you have is an axe, everything looks like wood.

  15. anona


    The discussion of “greenness” in the context of realism is introducing distracting metaphysical disputes. Why should all realists have to agree upon all metaphysical issues?

    The claim about mathematicians being realists because of certain Euclidean theorems about triangles appears to confuse realism with analyticity. Why open this can of worms? [Also, I’ve heard some things about triangles drawn on spherical surfaces.]

    You go on to claim that ‘contrary to realism is X, which denies that universals exist’. I have no familiarity with nominalism, so I have no idea how realistic you are being in presenting the nominalist position. A quotation or two would be highly desirable (even if you push them into footnotes). And same criticism re: idealism. [Next thing, you’ll hear from the positivists and pragmatists saying that you have not adequately considered their position.]

    The excerpt from Stowe on pg 6 which allegedly makes an argument against solipsism makes no sense to me (maybe because I watched too many sci-fi movies involving the great simulations … e.g. Matrix). I can easily imagine a situation where the entire planet is high on LSD. Nothing logically inconsistent there. Honest.

  16. Briggs


    Omit a discussion of realism in a discussion of truth? Can’t be done. But as you suggest, more examples might help. And if you can imagine “a situation where the entire planet is high on LSD” then what you have imagined is not solipsism. Think about it.


    It sounds as if you don’t have a real criticism to offer but just want to carp.


    Thanks, brother, yet again, for the assist.


    Thank you. Your cautionary tale and prescient warning good; the prediction looks to already be coming true, in part.

  17. Joe Alvarez

    I agree with your axe grinding. The title includes the word philosophy so sharpening is necessary.

    I suggest a summary take-away on how truth, faith, conditional, logic, data, etc are bundled for the march toward probability and statistics.

  18. Briggs


    Thanks, but like ad you’ve said, “No good” but not said why. What exactly is wrong, what right?

  19. Wits' End

    “What exactly is wrong, what right?”

    My sense from reading the comments and the material posted (by Briggs) is that what is needed is a tight outline of the structure of the book.

    Painful time spent on getting the structure right will pay huge dividends later.

    When I read “I’m still wavering whether to put Causality with Induction or bust it out on its own. ” I know the thinking about the structure is not anywhere near done.

    Not easy I know. Indeed, painful.

  20. Ye Olde Statisician

    “Truth” comes from West Saxon triewð (or Mercian treowð), meaning “faithfulness.” The corresponding adjective, “true” was triewe (W.Saxon), or treowe (Mercian) “faithful, trustworthy,” from P.Gmc. *trewwjaz, “having or characterized by good faith.” Truth is simply the A/S equivalent to the Latinate faith and has the sense of something reliable or dependable.

    “Truth” is therefore a sort of verb. One must be true to something. A scientific theory is supposed to be true to the facts; a measurement is true to a standard; a novel is true to life. Mathematical theorems are true to a set of axioms. (This latter is sometimes called “coherence” rather than “correspondence,” and applies to any body of statements that are mutually coherent even if, like mathematics, there is nothing physical for them to correspond to.)

    Thus, scientific truths can change when new (or more precise) facts become available; mathematical truths can change when a different set of axioms are employed; and a fable or myth or novel can be true without containing a single fact.

    “The English are always degrading truths into facts. When a truth becomes a fact it loses all its intellectual value.” ~Oscar Wilde

  21. Jim S

    @ Ye Olde
    “Thus, scientific truths can change when new (or more precise) facts become available; mathematical truths can change when a different set of axioms are employed; and a fable or myth or novel can be true without containing a single fact.”

    Very well put.

    The issue is not arriving at UNIVERSAL TRUTH. The issue is, “Is the limited, finite knowledge that I have available to me free of wishful thinking and contradictions – and to what degree can I compute uncertainty (whether in cardinal or ordinal form).” Is your knowledge (truth)objective or subjective?

    Gaining knowledge is a process, and all knowledge is contextual. This is not skepticism.

  22. Ye Olde Statisician

    Up to a point. Take the well-known case of the system of the world. The Ptolemaic model was true to the known facts for a very long time, until overthrown by the phases of Venus. The successor Tychonic system was mathematically equivalent to the Copernican and both systems were true to the facts up to the discovery of stellar aberration in the mid-18th cent.

    However, there really is an actual arrangement of the sun and planets to which these various models were (successively better) approximations. That is, there is an underlying Truth in addition to the truths of the various theories. So even though Kuhn (of paradigm-shift fame) could say that the Copernican system was no more true than the Ptolemaic, Jaki (and Stove) noted that these theories are more true or less true to the actual arrangements.

  23. G. Rodrigues


    This is something of a general critique (and that echoes what others have said), but having learned mathematics from handbooks in the stout and solid German style, the first chapter strikes me as too much of a ramble, without an obvious direction and purpose.

  24. Joe Alvarez

    You answered:

    Thanks, but like ad you’ve said, “No good” but not said why. What exactly is wrong, what right?

    I did not say, “No good.” I may have not been clear, but I meant, “Good points.”

    I then said, “I suggest a summary take-away …”

    I agree with G. Rodriques, “…without an obvious direction and purpose.”

    What is needed is to follow the dictum

    Tell’em what you’re gonna say,
    Tell’em what you said.

    I believe you started well with
    Why a discourse on truth in a book devoted to probability?
    Since probability is the language of uncertainty, before we can
    learn what means we need to assimilate the language of truth.
    Since probability aims at truth, what does the target look like?

    Then you told’em pretty good.

    You did not tell’em what you said.

  25. JohnK


    Good: you have now officially stated that you write this work not merely for yourself but for an audience — statisticians (of good will). Thus, you have a practical purpose.

    Then write the work practically, for your audience. Your proposed introductory chapters bury the lede. Instead, give your professed audience many practical reasons to read on. Invite them with practical demonstrations. Inveigle them with the increased power and refinement of the methods you can deploy on practical problems.

    I would start with one climax of your practical work, with a practical discussion of ‘skill’. From thence, delve deeper — backwards — through smaller practical discussions and demonstrations. Give them reasons for wanting to reconsider everything, from p-values, to probability as a part of logic and philosophy.

    I would then put your present introductory chapters in an Appendix, for the further interested reader.

    I remain in awe of your expertise, but I remain unconvinced that it extends infinitely. You have repeatedly proved yourself impervious to any critique or argumentation of mine on matters philosophical or theological, so I won’t attempt any.

    But if critique and argument are in vain, irony remains. For example, the irony of beginning your discussion of Truth with Truth Himself — and then, in the traditional manner, treating this as a mere decoration to the ‘real’ discussion.

    There is the irony of following the customary treatment of ‘truth’ as at best orthogonal to Time, and at worst, as inevitably corrupted by it. So that, e.g., in school Thomism, the actions of Jesus Himself are (un-ironically) formally merely an ‘example’ of a Time-less Truth thus higher, more ‘true’, than Himself.

    There is the irony of a professional who has long bemoaned the fact that non-statisticians essentially teach almost all the ‘statistics’ learned by non-statistician practitioners, now teaching philosophy to statisticians.

    There is the irony of modern-day Aristotelians not even able to imagine that a Greek like Plato would have thought Aristotle simply wrong, incorrect; or that a Homeric-age Greek would have thought the discussion of the ‘obvious’ foundations of rationality to be literally insane.

    Homeric-age Greeks were not exactly like us, except that they lived a long time ago. That all world views and philosophies are, in the end, commensurable, and particularly, translatable into (lesser) versions of our own, is a commonplace, but it is difficult to make it true historically; and theologically, it is pointless.

    I am all for Realism. But sometimes the realist question is, Who is insane?

  26. Briggs


    Good use of the word “impervious.”


    Now you know I love you guys, and it’s my fault. I should have released the Preface before showing Truth, which, to quote myself, was a “rough, incomplete, outline-only, not-yet-finished, gist-only version”. I knew no number of announcements of its crudeness would save me from criticisms of its crudeness, but I wanted people to see the direction. Which, of course, is stated in the Preface.

    A math book this is not. A science book this is not, or at least not directly and not solely. Not all knowledge is scientific.

    The idea, in very brief (I know: no matter how brief, it will be too short) is that the epistemology of truth has a certain form; we know truths conditional on certain premises; some truths we know completely, some only contingently. Logic is the semi-formal way of talking about truth. The formal way is mathematics, which is a fine subject, but it is only a small part of truth and certainty. Normal Socratic logic, which even mathematicians use when discussing their theories, is of vastly more practical import. Induction is the result of this kind of logic, and the way we move from absolute certainty to discussing uncertainty. Probability comes next. And then statistics.

    Again, I appreciate and am very thankful for everybody’s help.

  27. Briggs


    Writing Preface now. Hardest thing in the world to write. I’d rather pen greeting cards. Ugh.

  28. Uncle Mike

    No. Write the preface last.

    Matt, for whom are you writing this book? Who is your intended audience? How do you personify in your imagination the Reader of your book?

    Because if you’re pissed off writing ME a greeting card, if it’s a big burden, then please don’t.

    I get the feeling reading your chapter that you don’t really like your Reader. You have some bone to pick with your simpleton Reader. And if He/She doesn’t get it, then screw them. Or maybe they are academics who somehow wronged you by teaching you crap that you had to unlearn, and now they get their comeuppance?

    Try imagining your reader as someone you know and care about. He/She could be imaginary, such as your great-great-great-great grandson, but a real existing person is better. Someone you like.

    Because the Reader can feel your vibe and will not bother to read very far without backlashing and/or throwing the book away.

    It won’t be worth the effort. You dig your own hole, not your own mountain. The mountain has to be climbed.

  29. Briggs

    Uncle Mike,

    Pissed off? No, no. It is brutal writing a summary, an outline, which many (like the Mrs) say I should write first, not last. Like Mike Royko used to say, every word is like blood from my fingertips. A natural talent, a way with words, I don’t have, so everything is the result of much effort. The ideas are all up there in my head, and I can see them clearly. I just can’t put them into words which are as clear as my thoughts.

    And, of course, I love my readers! The intended reader is anybody who routinely or in great part uses probability and statistical methods and who wants to really understand what they mean. These in large part are professional statisticians. I’m working on the tone.

    I appreciate the comments!

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