William M. Briggs

Statistician to the Stars!

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Tautology Tautology: Day Two Teaching

Tautologies may be like this screwy circle.

Tautologies may be like this screwy circle.

Since probability is the completion—we do not say “branch”—of logic, and logic is the science of the relations between propositions, probability is also the study of the relations between propositions.

A (provable) rule of logic is that you cannot prove a contingent proposition starting from a necessary truth. Since it is a rule of logic, it is a rule of probability, too. Contingent propositions are the ones most familiar to us. “The car will start”, “The price of lemons will increase”, “A proton is made from quarks”, and on and on. Propositions which are themselves not necessarily true, which even if they are true contingently (lemon prices have risen in my neighborhood), don’t have to be true. The world would continue were lemon prices to remain flat, and the universe could have been built with quarkless protons.

In shorthand, if we have some proposition P and evidence E, Pr(P|E) = Pr(P|ET), where T is a necessary truth. T does nothing. It also does nothing when P is contingent and there is no other evidence than T. Thus Pr(P|T) is not a unique number, but because a tacit premise is P’s contingency we know that 0 < Pr(P|T) < 1. Note that the inequalities are not strict.

I think reader JH pointed to whathisname who wrote a book defending Objective Bayesianism (names escape from me faster than female interns from the late Ted Kennedy’s office). Williamson, maybe. Anyway, he and even subjective Bayesians insist on precise probabilities and so would pick 1/2 for Pr(P|T), which is screwy, as an example will prove.

The one that came up in class was the proposition P = “The Patriots win next year’s Super Bowl.” The necessary truth was the tautology T = “The Patriots might win”, which only acknowledges the contingency of P. T is equivalent to T = “The Patriots might win or they might lose.”

It is well at this point to remind that tacit premises about word definitions and grammar always accompany all our written evidence.

Now T is true no matter what, if by “lose” we mean not that they entered the final competition and lost, but that they do not win, such as is the natural state of the Detroit Lions. Any team which does not take the trophy (or whatever it is; I don’t follow football) “loses.”

Students have a difficult time wrapping their heads around (logical, not grammatical) tautologies, not being familiar with them. T is true no matter what, even if the Patriots disassemble their team tomorrow, even if the NFL goes bust or the universe ends.

Again, many like to say Pr(P|T) = 1/2. But we could have also said T2 = “The Lions might win” or T2 = “The Cowboys might win” and it cannot be true that the probability of each of these is 1/2.

Williamson (I think that’s the name; I’ll look it up later) wants probability to be a unique number and says so. He wants that unique number to be, in the face of uncertainty, the maximally equivocal. That’s what he forces the 1/2. But I could have also used the tautology T3 = “Tomorrow it will rain or it won’t” and surely “Pr(P|T) = 1/2” makes no kind of sense.

Indeed, let Q = “I wore an orange pocket square”, which is true based on observation yesterday. P is the same. Then Pr(P|Q) is undefined or is still the same open interval if we insist the tacit premise of P’s contingency. Q gives no evidence of P, and so there is no possibility of the probability being 1/2.

And there is no problem in asking “What is Pr(P|Q)?” just as there is no problem asking “Given x + y = 7 and w = 13, solve for x.” This is not a typo. The w is not probative or x or y; it adds no information. We cannot come to a unique single number of x, and there is no reason in the world we should insist on one.

Chapter House

That public house, where we usually spend our first night, was mobbed with USA-Ghana watchers, so it was off to Ruloff’s, an indifferent and non-air-conditioned bar in College Town. Greasiest appetizers (a contradiction in terms if ever there was one) you can find. But convenient. They had a small badly tuned television where we were able to celebrate the victory.

I’ll also note for Luis, if he’s reading this—Luis is from Portugal—that suicide is never an option.


All typos today free of charge. No coupon needed.

I Was Wrong About Axioms: Day One Teaching

Mistakes can happen

I was wrong about how belief in axioms are held. I was not wrong about the beliefs themselves, which is to say, the axioms which we all know and love are still true.

But I used to say, and say wrongly, that we knew axioms were true based on “no evidence” save our intuition, or on faith. This is wrong and bone-headed, as I should have recognized. Actually, we know that axioms are true the same way we know any universal is true: via the evidence of observations.

This dawned on me when I was reading Ed Feser’s new book Scholastic Metaphysics when I was reminded of the Aristotelian and Scholastic dictum “there is nothing in the intellect that was not first in the senses”. If that’s true (and it is) then the sort of a priori knowledge I had in mind, which was the fully formed idea being present to us (somehow), must be wrong.

I was originally convinced of that kind of a priori knowledge by a beauty of an argument by David Stove which appeared in his The Rationality of Induction. The argument is still right if only a few words in it are changed. It purports to prove, and does prove, that our knowledge of logic cannot be had empirically. And that’s so; at least, that is true if we mean all our knowledge. His argument went something like this.

In order to know via experience the validity of (say) the schema A = “For all x, all F, all G, either ‘x is F and all F are G’ is false, or ‘x is G’ is true”, we could make observations like O1 = “David is bald and David is a person now in this room and all persons in this room are bald.” But in order to get from O1 to A; that is, to know A is necessarily true, we have to already know that O2 = “O1 confirms A”, and that is to have non-empirical logical knowledge. Or you could insist at O2 was learnt by experience, but that would require knowing some other logical knowledge, call it O3, which somehow confirms O2. And then there would have to be some O4 which somehow confirms O3, and so on. There cannot be an infinite regress—the series must stop somewhere, at a point where we just know (my guess is O2)—so we must possess some innate logical knowledge.

Perhaps this sketch isn’t fully convincing, but it is in the context of the rest of the chapter, which contains several other proofs that our knowledge of logic cannot be (fully) empirical. (I have a more thorough summary coming in my book—which is still in the works; see more examples below.)

Anyway, the conclusion is still true. Consider the major premise in the old standby “All men are mortal”, which is also unobserved, but still true. Our senses confirm instances, and we extrapolate to the universal. Consider my favorite mathematical axiom that “For all natural numbers x, y, if x = y then y = x”. Again, our senses confirm instances, and we extrapolate to the universal, which we call an “axiom.”

The Scholastic approach solves problems the rationalist a priori raises. If there is rationalistic knowledge we “just know”, then does everybody know all of this knowledge or just a few? And if not everybody knows everything, who or what decides who gets what knowledge and who doesn’t? Too confusing. I should have thought of that.

Much more satisfying is the Scholastic approach. We observe, do we not, that not everybody knows Peano, to pick just one. But they can be brought to its knowledge, and brought pretty quickly, by giving them a few experiences where x = y and y = x. This explains the difficulties over the Axiom of Choice, too (look it up); agreement is harder to come by. Solves the “problem” of induction, too.

Of course I speak of the Aristotelian and not Platonic universals, about which there isn’t space to disagree here. Except to insist that because we are sometimes wrong in our judgement of universals/axioms, it does not imply always wrong. Intuition is a tool that can be misused like any other.

Last of course: The above is not a proof that we do not (all or those non-defective) come pre-wired according to some scheme so that we have the ability to come to universals. Let him that readth understand.

Very little thus has to change in what I wrote before about axioms, but there must still be changes. It is not “no evidence save our intuition or faith” but “the evidence of our senses and our intuition (or faith).” I’ll put this up in the Classic Posts page as a warning. Still unsure whether the language of faith and universals is best. I’m leaning on “yes.” More later.


My plan was to write posts about the class discussions of the day before. There is no day before yet. But I talk about axioms on the first day, so this is a natural place.

More Stove

We can learn from observation the following argument is invalid: “‘All men are mortal and David is mortal’ therefore ‘David is a man” if perchance we see David is not a man (maybe he’s a puppy). And we can learn from observation the invalidity of “‘All men are mortal and Peter is mortal’ therefore `Peter is a man” only if we see Peter is not a man (maybe he’s a cow). But we cannot learn the invalidity of “‘All men are mortal and X is mortal’ therefore ‘X is a man” through observation because we would have to measure every imaginable X, and that’s not possible. If we believe “‘All men are mortal and X is mortal’ therefore `X is a man” is unsound, and it surely is, this belief can be informed by experience but it cannot be solely because of it that we have knowledge of it. Another universal is born, though of a more complicated form.

Stove himself: “If an argument from P to Q is invalid, then its invalidity can be learnt from experience if, but also only if, P is true and Q is false in fact, and the conjunction P-and-not-Q, as well as being true, is observational. This has the consequence, first, that only singular judgments of invalidity can be learnt from experience; and second, that very few even of them can be so learnt.” And here’s the kicker: “If the premise P should happen to be false; or the conclusion Q should be true; or if the conjunction P-and-not-Q is not observational but entails some metaphysical proposition, or some scientific-theoretical one, or even a mere universal contingent like ‘All men are mortal’: then it will not be possible to learn, by experience, the invalidity of even this particular argument” (pp. 155–156). The key is that :scarcely any of the vast fund of knowledge of invalidity which every normal human being possesses can have been acquired from experience.” Can we allow the hilarious pun universal universals?

Examples? The invalidity of the argument “Given ‘The moon is made of cheese’ therefore ‘Cats do not understand French'” cannot be learned from experience. Neither can “Given ‘Men can breathe underwater unaided’ therefore ‘The atmosphere is largely transparent to sunlight'”. In neither can we can ever observe the conjunct P-and-not-Q.

Homework: your turn for examples.

Summary Against Modern Thought: First Steps In The Scientific Proof Of God

This may be proved in three ways. The first...

This may be proved in three ways. The first…

See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles.

Previous post.

We have reached at long last the happiest part of our journey. A proof for the existence of God. Aquinas starts and ends with the best, an argument which, once it is understood, grasped at its full, is fully convincing. It is a scientific proof. It is based on the indisputable evidence of our long observation of Nature. It first saw print, as so much foundational knowledge has does, with Aristotle. So strongly does Aquinas feel about this argument that he offers no other in this work. It is an argument, or rather two arguments which share much in common, which he, and which I think you will agree, is best suited for those of a scientific mind. Read all three (short) paragraphs from Aquinas before reading the footnotes.

Chapter 13: Arguments in proof of God’s existence

1 HAVING shown then that it is not futile to endeavour to prove the existence of God, we may proceed to set forth the reasons whereby both philosophers and Catholic doctors have proved that there is a God.

2 In the first place we shall give the arguments by which Aristotle sets out to prove God’s existence: and he aims at proving this from the point of view of movement, in two ways.i

3 The first way is as follows.[1] Whatever is in motion is moved by another: and it is clear to the sense that something, the sun for instance, is in motion.ii Therefore it is set in motion by something else moving it. Now that which moves it is itself either moved or not. If it be not moved, then the point is proved that we must needs postulate an immovable mover: and this we call God. If, however, it be moved, it is moved by another mover. Either, therefore, we must proceed to infinity, or we must come to an immovable mover. But it is not possible to proceed to infinity. Therefore it is necessary to postulate an immovable mover.iii

4 This argument contains two propositions that need to be proved: namely that whatever is in motion is moved by another, and that it is not possible to proceed to infinity in movers and things moved.iv


iThere are various kinds of movement, change, or evolution that first must be understood. There is movement in space, where an object is first here then there, or a change in orientation. There can be change in mass, where an object excretes or accretes. There can be change in energy, such as when a field offers up a newly created particle from a “quantum fluctuation,” or when mass is converted to energy. There can be change in qualitative characteristics, such as in color. In short, any differentiation whatsoever in an object can be considered change or motion. There need not be, of course, any person to witness or measure this differentiation, nor do we need to understand all of the (secondary efficient) causes that bring this differentiation about. Somebody with no knowledge of internal combustion engines is surely aware that a car moves from one place to another.

iiFor bookkeeping, and for commenting, label the first argument “W1”, for “Way 1”, etc.

We begin with the observation that things move, that they change. There can be no doubt about this. Even those who profess complete philosophical skepticism know that their minds change (about objects moving). (I say “profess” because however much somebody might claim to be a skeptic, nobody except possibly the genuinely insane actually believes it. If you argue with me, you necessarily agree with me. But this point is not in the least necessary for us to continue. Feel free to pretend your computer is not really there.)

And what around us does not change? All of Nature, i.e. the physical, does. Trees rot, even protons decay, babies are born, the universe expands. What remains motionless? Well, truth. Mathematical truths are not subject to movement, neither can logical truths shift from place to place or grow hair. Our knowledge of such truths, individually and collectively, surely evolves, or devolves, as the case may be. But the truths themselves are incorruptible. Again, if you disagree, you agree. If you say, “All truths are inventions”, you have either stated a truth or an invention. If it is a truth, your proposition is self-defeating. If it is an invention, there is no reason to trust it. There is no way to speak coherently except by admitting truth exists and is unchangeable.

Anyway, that much is background. Aquinas is not claiming nothing is unchangeable. His first simple indubitable plain commonsensical premise is only that some things move, and that we all see this to be the case, as for example the relative motion of the sun or your finger on the scroll bar. This is all you must advert to now. Do you?

iiiNow W1 will be proved bit by bit. Aquinas will give examples and clarifications of each of the propositions and premises which comprise the argument and when he is finished no loose ends will remain. However, it is well here to ensure you grasp the intention or meaning of this argument before continuing. Here is a sketch.

Take an analogy—an analogy, I say—from Fulton Sheen. You see a boxcar of a train pass by. It moves. Something caused it to move. What? Well, the boxcar in front of it pulls it. But what caused that boxcar to move? The one in front of it. But this cannot go on forever. We cannot have an infinity of boxcars, each pulled by the one in front. At the start of the line there must exist an engine which pulls all along, or there can be no movement.

The same analogy holds for cogs in a machine. This one moves by the one before it, and so on, a series which must terminate at an axle hooked to a motor. There must be a start.

These are only analogies because we left off short. Something is causing the engine to move, and something is causing the motor to spin. These causes are operating now, in this moment. And this what we’re talking about. Movement or change occurring the here in the right now, at this instant.

The classic example is a stick pushing a stone. Imagine yourself holding the stick and applying steady pressure to it, nudging the stone. The stone moves because of the stick. The stone moves now, in this moment because of the stick’s pressure. The stick is also in motion: it has force now, in this moment applied to it via your arm. Your arm is also in motion: its muscles contract or extend now, in this moment. This is happening all at once, at the same instant. Do not let this slip from your mind.

The muscles are also in motion: individual cells contract or extend now, in this moment. The materials in the cell are also in motion: chemicals are moving or reacting now, in this moment. The chemicals are also in motion: the chemicals subcomponents are moving or reacting now, in this moment; and if the chemicals are simple, the electrons in its shell and the neutrons and protons in its nucleus are moving now, in this moment. And the electrons themselves, and the quarks inside the nucleus are moving now, in this moment.

Again, all this is happening at once. Not in the distant past, but right now.

The quarks are also in motion: the strings, or whatever, inside the quarks now, in this moment are also “vibrating”.

Are there things “bellow” strings, or whatever? I don’t know. It doesn’t matter. What does is that if there is, whatever these objects are, have to be in motion now, in this moment too. The chain of all these objects pushes along, as it were, pushed from “below,” all at the same time. This series cannot, however, continue to infinity. There cannot an infinite number of things smaller, or rather more base, than strings and which must first move before any movement is possible.

There must be a base. There must be a thing which moves, which is itself not moved, and is indeed immovable. There must be a start to everything, a first unmoving mover. If there were not, nothing would ever happen. And we have already agreed things happen.

Now two happy things follow from this. The first (and the proof of this is coming: this is not here a proof, but a claim: do not say it was not proved when it is not claimed as being proved) is that this unmoved mover is and must be the same everywhere and for all changes. That is why the unmoved mover is God.

The second is that to be truly scientific, to honestly understand physics, to speak properly of causation and how the universe is run, one must understand God. Because He is there, at base, in everything. He is the root cause of every single thing that happens. This should be cheering, not the least because of the good news we have received some two thousand years ago, but because our task of contemplating the world is proven finite. There must come a point below which physics ends and God begins, to speak loosely. It is not clear if we will ever figure out the whole of mechanics, though. All we have proved is that the causal movement-chain must be finite. We haven’t any idea, through arguments of this kind, how long it is. Finite does not imply short.

ivCall these P1 and P2, or properly W1P1, etc. Next time we start with these same premises. They are given here for the sole reason of noting that there was only one thing proved in this post. That we see some things move. Everything else was a clarification, a heads up. The argument for God being the unmoved Mover was merely sketched. There is thus no point whatsoever in claiming that the main argument was found wanting because it hasn’t yet been fully given.

Clever readers will also have recognized that at no point was scripture invoked. No divine revelation, other than the ordinary kind, i.e. the revelations of our senses, is assumed. Like I said up top, this proof is purely scientific.

Next installment.

[1] 7 Phys. i.

The Joy Of Unknowns In World Cup; Updates

Sports hipsters like to say f-oo-ou-t ball.

Slaughterhouse Five

Five brave individuals, strong and true, have thus far ponied up to be killed off in one of my award-eligible Mini Plays. Strike that: four deaths and one dismemberment.

Inspired by George RR Martin’s proposal to accept $20,000 from each fan who would like to meet a “grisly death” in his next book, I offered the same service for readers on this blog for only $10, a substantial discount. Full details may be had here.

As promised, here’s what will happen. Tomorrow starts my two-week isolation in Ithaca, where I’ll teach nine to five daily. I won’t have much time to write the play until this ordeal ends. Tentatively, then, the play will run on Saturday, 5 July. This is far speedier than Martin can do. More information will be provided as we move along.

Meanwhile, there is still time to get in on the fun. Sign up for immortality, literarily speaking, today!

Batman &

All I could bring to mind after Spain’s character-building exercise was the old phrase, “If you ain’t Dutch, you ain’t much.”

Goal-line Technology

ESPN invented “Goal-line Technology” for the World Cup. A solid white on solid blue graphic, a cartoon resembling the 1970s game of Pong, that shows the audience that the ball has, as the audience a moment ago saw with its own eyes, indeed crossed the goal line. There is the ball resting on the white line in the blue background, rotating to show you that something has happened. A computer has confirmed the crossing, therefore the ball really did cross.

GK Chesterton: “I entertain a private suspicion that physical sports were much more really effective and beneficent when they were not taken quite so seriously. One of the first essentials of sport being healthy is that it should be delightful; it is rapidly becoming a false religion with austerities and prostrations.”

He meant instant replay and goal-line “technology.”

Keller Calls ‘Em

Kasey Keller was color-man for the Chile-Australia match. After Chile pulled ahead, Keller opined, “Chile can afford to slow it down.” Minutes later, Australia scored. Keller: “Chile took their foot off the gas.”

If you’re going to flame out, best do it with a cliché. People might not remember it was you.

Too many American announcers sound like self-loving too-serious IRS accountants loudly explaining minute changes to the tax code while strictly limiting themselves to a vocabulary of 100 words.

I watched what games I could on Univision.

Speaking Of Foreknowledge

And also speaking of Chesterton…I recall an article by him, “The Perfect Game”, from his Tremendous Trifles, in which Chesterton engaged a fellow in a game of croquet. After “patting [his opponent] affectionately on the head with a mallet,” he explains to him why a lack of skill enlivens the game.

“[I]t is logically possible, to play croquet too well to enjoy it at all. If you could put this blue ball through that distant hoop as easily as you could pick it up with your hand, then you would not put it through that hoop any more than you pick it up with your hand; it would not be worth doing. If you could play unerringly you would not play at all. The moment the game is perfect the game disappears.”

What is true of the playing, is true of the watching. Once we know the score and highlights of the game—perhaps somebody tells us the result before we get home to watch the recording—there is far less joy in watching it. There is still the artistry to admire, true, but the thrill of anticipation is removed. The emotional landscape is flattened.

The same thing happens when zealots try to remove all ambiguity, such as when they insist on instant-replay judgments, justifying it with words like, “This game is important” and adding, circularly, “Why not get it right?” It is circular because the “why not” is what is in question.

Anyway, the answer is easy: removing the ambiguity in judgment alters the nature of the game. It is against tradition. It makes it less interesting, and surely less memorable. Nobody but a dullard would say, “I remember that instant replay back in ’14. Took over four minutes watching from all angles to get the call right. I was able to take Sparky outside for a bathroom break.”

This is not to say we should blindfold the referees. There is a balance. Yet I shiver to think that soccer will descend the same road taken by other major sports. Where would the game be without its manos de Dios?

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