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June 21, 2018 | 1 Comment

Chapter 1 Excerpt from Uncertainty: The Soul of Probability, Modeling & Statistics

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Necessary & Conditional Truth

Given “x,y,z are natural numbers and x>y and y>z” the proposition “x>z” is true (I am assuming logical knowledge here, which I don’t discuss until Chapter 2). But it would be false in general to claim, “It is true that ‘x>z‘.” After all, it might be that “x = 17 and z = 32“; if so, “x>z” is false. Or it might be that “x = 17 and z = 17“, then again “x>z” is false. Or maybe “x = a boatload and z = a humongous amount”, then “x>z” is undefined or unknown unless there is tacit and complete knowledge of precisely how much is a boatload and how much is a humongous amount (which is doubtful). We cannot dismiss this last example, because a great portion of human discussions of uncertainty are pitched in this way.

Included in the premise “x,y,z are natural numbers and x>y and y>z” are not just the raw information of the proposition about numbers, but the tacit knowledge we have of the symbol >, of what “natural numbers” are, and even what “and” and “are” mean. This is so for any argument which we wish to make. Language, in whatever form, must be used. There must therefore be an understanding of and about definitions, language and grammar, in any argument if any progress is to be made. These understandings may be more or less obvious depending on the argument. It is well to point out that many fallacies (and the best jokes) are founded on equivocation, which is the intentional or not misunderstanding double- or multiple-meanings of words or phrases. This must be kept in mind because we often talk about how the mathematical symbols of our formulae translate to real objects, how they matter to real-life decisions. A caution not heard frequently enough: just because a statement is mathematically true does not mean that the statement has any bearing on reality. Later we talk about how the deadly sin of reification occurs when this warning is ignored.

We have an idea what it means to say of a proposition that it is true or false. This needs to be firmed up considerably. Take the proposition “a proposition cannot be both true and false simultaneously”. This proposition, as I said above, is true. That means, to our state of mind, there exists evidence which allows us to conclude this proposition is true. This evidence is in the form of thought, which is to say, other propositions, all of which include our understanding of the words and English grammar, and of phrases like “we cannot believe its contrary.” There are also present tacit (not formal) rules of logic about how we must treat and manipulate propositions. Each of these conditioning propositions or premises can in turn be true or false (i.e. known to be true or false) conditional on still other propositions, or on inductions drawn upon sense impressions and intellections. That is, we eventually must reach a point at which a proposition in front of us just is true. There is no other evidence for this kind of truth other than intellection. Observations and sense impressions will give partial support to most propositions, but they are never enough by themselves except for the direct impressions. I explore this later in the Chapter on Induction.

In mathematics, logic, and philosophy popular kinds of propositions which are known to be true because induction tells us so are called axioms. Axioms are indubitable—when considered. Arguments for an axiom’s truth are made like this: given these specific instances, thus this general principle or axiom. I do not claim, and it is not true, that everybody knows every axiom. The arguments for axioms must first be considered before they are believed. A good example is the principal of non-contradiction, a proposition which we cannot know is false (though, given we are human, we can always claim it is false). As said, for every argument we need an understanding of its words and grammar, and, for non-contradiction specifically, maybe the plain observation of a necessarily finite number of instance of propositions that are only true or only false, observations which are consonant with the axiom, but which are none of them the full proof of the proposition: there comes a point at which we just believe and, indeed, cannot do other than know the truth. Another example is one of Peano’s axioms. For every natural number, if x = y then y = x. We check this through specific examples, and then move via induction to the knowledge that it is true for every number, even those we have not and, given our finiteness, cannot consider. Axioms are known to be true based on the evidence and faith that our intellects are correctly guiding us.

This leads to the concept of the truly true, really true, just-plain true, universally, absolutely, or the necessarily true. These are propositions, like those in mathematics, that are known to be true given a valid and sound chain of argument which leads back to indubitable axioms. It is not possible to doubt axioms or necessary truths, unless there be a misunderstanding of the words or terms or chain of proof or argument involved (and this is, of course, possible, as any teacher will affirm). Necessary truths are true even if you don’t want them to be, even if they provoke discomfort, which (again of course) they sometimes do. Peter Kreeft said: “As Aristotle showed, [all] ‘backward doubt’ terminates in two places: psychologically indubitable immediate sense experience and logically indubitable first principles such as ‘X is not non-X’ in theoretical thinking and ‘Good is to be done and evil to be avoided’ in practical thinking”.

A man in the street might look at the scratchings of a mathematical truth and doubt the theorem, but this is only because he doesn’t comprehend what all those strange symbols mean. He may even say that he “knows” the theorem is false—think of the brave soul who claims to have squared the circle. It must be stressed that this man’s error arises from his not comprehending the whole of the argument. Which of the premises of the theorem he is rejecting, and this includes tacit premises of logic and other mathematical results, is not known to us (unless the man makes this clear). The point is that if it were made plain to him what every step in the argument was, he must consent. If he does not, he has not comprehended at least one thing or he has rejected at least one premise, or perhaps substituted his own unaware. This is no small point, and the failure to appreciate it has given rise to the mistaken subjective theory of probability. Understanding the whole of an argument is a requirement to our admitting a necessary truth (our understanding is obviously not required of the necessary truth itself!).

From this it follows that when a mathematician or physicist says something akin to, “We now know Flippenberger’s theorem is true”, his “we” does not, it most certainly does not, encompass all of humanity; it applies only to those who can and have followed the line of reason which appears in the proof. That another mathematician or physicist (or man in the street) who hears this statement, but whose specialty is not Flippenbergerology, conditional on trusting the first mathematician’s word, also believes Flippenberger’s theorem is true, is not making (to himself) the same argument as the theory’s proponent. He instead makes a conditional truth statement: to him, Flippenberger’s theorem is conditionally true, given the premise of accepting the word of the first mathematician or physicist. Of course, necessary truths are also conditional as I have just described, so the phrase “conditional truth” is imperfect, but I have not been able to discover one better to my satisfaction. Local or relative truth have their merits, but their use could encourage relativists to believe they have a point, which they do not.

Besides mathematical propositions, there are plenty other of necessary truths that we know. “I exist” is popular, and only claimed to be doubted by the insane or (paradoxically) by attention seekers. “God exists” is another: those who doubt it are like circle-squarers who have misunderstood or have not (yet) comprehended the arguments which lead to this proposition. “There are true propositions” always delights and which also has its doubters who claim it is true that it is false. In Chapter 2 we meet more.

There are an infinite number and an enormous variety of conditional truths that we do and can know. I don’t mean to say that there are not an infinite number of necessary truths, because I have no idea, though I believe it; I mean only that conditional truths form a vaster class of truths in everyday and scientific discourse. We met one conditional truth above in “x>z“. Another is, given “All Martians wear hats and George is a Martian” then it is conditionally true that “George wears a hat.” The difference in how we express this “truth is conditional” is plain enough in cases like hat-wearing Martians. Nobody would say, in a general setting, “It’s true that Martians wear hats.” Or if he did, nobody would believe him. This disbelief would be deduced conditional on the observationally true proposition, “There are no Martians”.

We sometimes hear people claim conditional truths are necessary truths, especially in moral or political contexts. A man might say, “College professors are intolerant of dissent” and believe he is stating a necessary truth. Yet this cannot be a necessary truth, because no sound valid chain of argument anchored to axioms can support it. But it may be an extrapolation from “All the many college professors I have observed have been intolerant of dissent”, in which case the proposition is still not a necessary truth, because (as we’ll see) observational statements like this are fallible. Hint: The man’s audience, if it be typical, might not believe the “All” in the argument means all, but only “many”. But that substitution does not make the proposition “Many college professors are intolerant of dissent” necessarily true, either.

Another interesting possibility is in the proposition “Some college professors are intolerant of dissent,” where some is defined as at least one and potentially all. Now if a man hears that and recalls, “I have met X, who is a college professor, and she was intolerant of dissent”, then conditional on that evidence the proposition of interest is conditionally true. Why isn’t it necessarily true? Understand first that the proposition is true for you, too, dear reader, if we take as evidence “I have met X, etc.” Just as “George wears a hat” was conditionally true on the other explicit evidence. It may be that you yourself have not met X, nor any other intolerant-of-dissent professor, but that means nothing for the epistemological status of these two propositions. But it now becomes obvious why the proposition of interest is not necessarily true: because the supporting evidence “I have met X, etc.” cannot be held up as necessarily true itself: there is no chain of sound argument leading to indubitable axioms which guarantees it is a logically necessity that college professors must be intolerant of dissent. (Even if it sometimes seems that way.)

We only have to be careful because when people speak or write of truths they are usually not careful to tell us whether they have in mind a necessary or only a conditional truth. Much grief is caused because of this.

One point which may not be obvious. A necessary truth is just true. It is not true because we have a proof of it’s truth. Any necessary truth is true because of something, but it makes no sense to ask why this is so for any necessary truth. Why is the principle of non-contradiction true? What is it that makes it true? Answer: we do not know. It is just is true. How do we know it is true? Via a proof, by strings of deductions from accepted premises and using induction, the same way we know if any proposition is true. We must ever keep separate the epistemological from the ontological. There is a constant danger of mistaking the two. Logic and probability are epistemological, and only sometimes speak or aim at the ontological. Probability is always a state of the mind and not a state of the universe.

May 23, 2018 | 11 Comments

Atheists Will Have No Excuse

There is a class of the argument of God’s existence that depend entirely on your thoughts. The one that interests us is Joseph de Maistre’s “no excuse” argument. It is less well known than Saint Anselm’s “ontological argument”, which is worth a few moments puzzling over.

Anselm’s argument runs like this:

  1. You have some idea, even as an atheist, about who God is;
  2. God is “a being than which none greater can be imagined”; that is, it is impossible to think of a being greater than God;
  3. Beings that exist in reality are greater than those that exist only in the mind, because existing itself is a good;
  4. But we cannot think of a greater being than God;
  5. Therefore, God exist.

Everybody agrees with the first step in the argument. And there seems to be no real controversy in the second and third steps. It’s step four that makes us think some sleight-of-hand has been pulled.

It’s true existence is a good, and it seems to be true we can’t think of any being greater than God. So we can’t think of a being who actually exists greater than God. So it must be this being about whom none greater can exist must himself exist. Right?

Many think not. The argument seems to conjure God’s existence out of our thoughts, or even hopes. It’s hard to escape the notion that a circularity or flaw is buried somewhere, but it’s tough to finger.

But not impossible. Saint Thomas Aquinas said that since we cannot know the full nature of God, it is thus not possible for us to absolutely think of “a being than which none greater can be imagined”. This necessarily limited understanding of God is the reason Anselm’s argument fails to be completely convincing.

We can’t know all about the nature of God—we are, after all, limited creatures. But we can know some of God’s nature. What does that imply? De Maistre said it led to another argument of God’s existence.

Joseph de Maistre was a Catholic reactionary chased from France after Napoleon came to power. And not because he was a friend to the French Revolution. Nor was he keen on the then-fresh scientific materialism of Francis Bacon, which de Maistre perceived would lead to rampant atheism.

Bacon didn’t think much of Anslem. Bacon thought it “absurd” the claim that “men have found by reason the existence of a being of which they cannot form any idea.” (All quotes are from de Maistre’s An Examination of the Philosophy of Bacon in the chapter “Of God and Intelligence”.)

De Maistre countered “To maintain that we have no idea of God because we cannot have a perfect idea, and that it is absolutely the same thing not to know what he is, or if he is, is not only blasphemy against God himself, it is also a blasphemy against good sense.”

That’s step one of de Maistre’s proof. Step two is the truism “we can affirm nothing of what does not exist.” To affirm is greater than to state. Thus we can affirm facts about horses, but we can only make statements about unicorns. There is no way to affirm anything about unicorns, because they do not exist.

Now the man who says “I have no idea of God, contradicts himself without knowing it; for it is precisely as if he said that he has an idea of which he has no idea.” And “The very fool who says God is not affirms that he has an idea of him, for no mind can deny an unknown existence.” To deny God is to deny something.

De Maistre needs only one more step. “How could man receive a new truth if did not carry within him an interior truth, an innate rule by which he judges the other?” Any teaching, human or divine, is a revelation—a revealing. We must have inbuilt a (even if flawed) sense of which arguments work and which not. To deny that is to affirm it. This sense must be of divine origin. De Maistre of course does not say, but this sense could not be biological in origin and simultaneously trustworthy: you could never know if your genes were lying to you.

In a word, the goal of revelation is only to lead the human mind to read in itself what the divine hand has traced there; and revelation would be worthless if reason, after the divine teaching, was not rendered capable of demonstrating to itself the revealed truths, just as mathematical teaching, or any other human teaching, is only recognized as true and legitimate when reason, examining the theorems on the eternal rule hidden in the depths of its essence, says to the human revelation, YOU ARE RIGHT, that is to say, you are reason.


God speaks to all men by the idea of himself that he has placed in us by this idea that would be impossible if it did not come from him, he says to us: IT IS I! Those who are called atheists reply: How could this be you, since you do not exist?

De Maistre concludes: This is why they will be inexcusable. This of course echoes St Paul: For the invisible things of him, from the creation of the world, are clearly seen, being understood by the things that are made; his eternal power also, and divinity: so that they are inexcusable.

This bumps against St Thomas’s other rebuttal of Anselm: “No one can mentally admit the opposite of what is self-evident; as the Philosopher (Metaph. iv, lect. vi) states concerning the first principles of demonstration. But the opposite of the proposition ‘God is’ can be mentally admitted: ‘The fool said in his heart, There is no God’ (Psalm 53:2). Therefore, that God exists is not self-evident.”

That we cannot know God does not exist, given we can have some idea of God, is self-evident, says de Maistre. A fool can always reject a truth, out of mere stubbornness or petulence if nothing else.

Even so, it’s hard to escape the notion that de Maistre—besides his excellent point about partial knowledge—assumed what he sought to prove, here in the step where he asserted our reason must be God-given. This is surely true, but there might be a way to bring in CS Lewis’s famous argument against biological confirmations of reason to support de Maistre at his weak point.

May 17, 2018 | 5 Comments

Book Sneak Peek: Imposing Your Beliefs Fallacy

All, here is a segment, a tease, a mere fragment, of an upcoming book of popular fallacies from the chapter Imposing Your Beliefs Fallacy. The Imposing Your Belief Fallacy occurs when somebody says, “You should not impose your beliefs”, which, of course, is an attempt to impose the speaker’s beliefs, and so the command is self-contradictory.

A pertinent example. In the early fall of 2017, Senator Diane Feinstein, a secular Jew (which needs mentioning because of the role religion and group identity plays), in her official role of senatorial inquisitor was questioning Amy Coney Barrett, a law professor from Notre Dame, who was nominated for a federal appellate court judgeship. Feinstein is in favor of would-be mothers having the “right” to kill the lives that live inside them, ensconced (in the United States anyway) in the law decided in Roe v. Wade. Barrett is a Catholic who professes belief in that religion’s stance on abortion, which is that such killing is always immoral, akin to murder, and thus the height of selfishness.

Feinstein was concerned Barrett would draw upon her religious beliefs in making future rulings about abortion. Feinstein said,

You are controversial. Let’s start with that. You’re controversial because many of us who have lived our lives as women really recognize the value of finally being able to control our reproductive systems, and {\it Roe} entered into that, obviously. You have a long history of believing that your religious beliefs should prevail.”

She also said, “The dogma lives loudly within you, and that’s of concern.” We grant that Feinstein had no choice but to live her life as a woman. Incidentally, to prove Americans still have a sense of humor, Feinstein’s “dogma lives loudly” quip instantly became of catchphrase among traditionalist Catholics. Almost before she was done speaking, t-shirts with the slogan were available.

Now there are all sorts of legal quibbles about Feinstein’s method of questioning that are of no concern to us; for instance, whether it is legal or “Constitutional” to ask the question Feinstein did. The point here is that Feinstein implicitly invoked the Imposing Your Beliefs Fallacy. She did not want Barrett’s values to be used in deciding legal questions pertinent to abortion, but she did want her (Feinstein’s) values to be used. Feinstein was anxious to continue to impose her beliefs on the nation.

Feinstein’s tactic of highlighting Barrett’s religious beliefs fails because there is no point in which a religious person’s life is not touched by her religious beliefs, no matter how weak that touch. Of course Barrett’s views on abortion, informed as they are by Catholic dogma, will be used by her in deciding abortion litigation and on matters regarding human life. One can imagine purely bureaucratic or technical rulings associated with abortion which are, at best, faintly religious; for instance, deciding what date hearings will be scheduled on an abortion matter and so forth. But it is just as obvious Feinstein’s religious views arising from her secular background inform her own votes on these matters. Somebody has to win these debates and decide the law of the land; therefore, somebody’s views will be imposed.

It does not matter that Barrett ran from Feinstein’s accusatory fallacy and hid behind the law saying she would “follow unflinchingly all Supreme Court precedent.” That is, it would not matter except if she meant by that that she would readily abandon her Catholic beliefs in the face of precedent, or that she meant she didn’t really hold her religious views strongly. All that matters is Feinstein believed Barrett’s faith was genuine, and thus Feinstein’s line of argument was fallacious.

Because Feinstein’s argument was fallacious, Feinstein bullied Barrett, or tried to. A practice which, I hasten to add, is well accepted in politics. If Barrett estimates she will once in office be unable to blunt these barbs, she ought to consider remaining in the Ivory Tower. Feinstein’s bullying revealed the majority position of the ruling elite (a view which may not be the majority belief of the entire populace). Feinstein argued from a position of strength, taking her own views on abortion as granted and accepted by right-thinking people. So natural are these beliefs to her that she did not see that she imposes her own beliefs. Her failure to recognize minority viewpoints is why the Imposing Your Beliefs Fallacy exists.

May 15, 2018 | 43 Comments

The Why & Frequency Of Miracles: Shapiro’s The Miracle Myth Reviewed — Part IV

Read Part I, II, III.


That miracles have supernatural origins is true by definition (see below, in PROBABILITY). Miracles, like all events, must have causes, and cause has motive as part of its aspect, as we have already seen. God must have wanted to perform the miracle, else, of course, He would not have done it.

Shapiro, however, says “God’s intentions, desires, habits, and so on are simply not available to us. Whatever we assume about God’s nature is purely speculative—guesses, really.” Both statements are (at least sometimes) false. If Jesus were God, as he claimed, then his intentions, desires, and habits were known and available. Deciding whether the miracle that a man can be God is a separate question. God’s, or Jesus’s, or even the Holy Spirit’s motives are not always plain, of course, but when a man begs Jesus to be made well, and Jesus heals the man miraculously, the motivation is clear.

If Shapiro is not an empiricist, he plays one in the book. Divining God’s attributes and nature, in our limited and fractured way, is the topic of theology and metaphysics (did you see the bad joke?). It can be done, and has. (See this series.) So Shapiro is wrong because he insists on measurement even when it cannot be had. He asks, “But how do we verify assumptions about God’s characteristics and ‘personality’?” How do we verify there are an infinity of numbers? Answer: we do not and cannot verify it. But we all believe it. An empiricist cannot believe it, however, because he cannot verify it. No math for Shapiro, then, nor logic. Also, that we cannot measure God’s attributes, though we can deduce some, is not a refutation that all our deductions are wrong. Instead, empiricism is a fallacy.

Shapiro’s main conclusion is that since, he says, we cannot discern God’s motives, then we “have no justification for believing” in miracles. “[B]ecause,” he says about one instance, “verification of either assumption [about motives] is impossible—we can’t simply ask Oprah to sit down with a divine entity for an interview about it goals and methods—we’re not justified in believing either of them.” And from this supposed lack of knowledge of motive he concludes “inference to supernatural causes is never justified.” That is the gist of the entire book.

Shapiro fails to see that motive can be guessed about some miraculous claims: that motivation is part of the inference to an explanation, when it accounts for the assumed metaphysics and theology. Shapiro assumes his own, but fails to see they are his own. He also did not acknowledge that we cannot always know motives even in mundane events, such as the example which opened this review (ball on table placed by Alice, Bob, or Charlie). Alice could walk in the room and say “I did it”, but Shapiro would have to reject her claim because he did not learn why she did it. With God, we sometimes do know the motive: He tells us. And sometimes we do not. In any case, that we do not does not mean what happened did not happen (if something indeed happened).

That Shapiro has fallen into these errors is because he attempts to divorce his metaphysics, about which he is mainly wrong, from his epistemology, which he misuses.


Shapiro thinks an event’s improbability is what, in part, makes it a miracle: “the more unlikely the occurrence, the more reason to believe that something supernatural is taking place,” and “miracles should be extremely improbable.” His two criteria for miraculous are:

1. Extremely improbable: a miracle should be unlike anything we have seen before. It should be contrary to everything we know about how the world works.

2. Supernatural: a miracle can’t have a natural explanation. It must be the product of supernatural and typically divine agency.

The second criterion is not controversial. The first is incorrect. The problem is, no miracle has a probability: no event does either. Nothing has a probability, not even the roll of a die. That means improbability cannot be used to judge the veracity of a miracle, or of anything.

Probability is only defined with respect to assumed evidence. Miracles, then, are more or less probable depending on the evidence for or against them that it accepted or assumed. It is the same for any event.

It is however easy to see why Shapiro (or anybody) would say why walking on water is improbable. It is because he gathers evidence of his experience and discovers, in relation to that, such events do not often happen, or have never happened (to his knowledge), and that he cannot think of a cause. Gathered, that evidence makes the event improbable. But, at least to to the one doing the act, who knows the cause (in all its aspects), the probability is certain.

Think about a non miracle, like being struck by lightning. Nobody “has” a probability of being struck by lightning. A golfer standing on hillock in a thunderstorm is under different circumstances than an office drone seated as his desk in a skyscraper. Their circumstances differ in ways we know to be related to the causes of lightning, hence their probabilities differ. Indeed, the golfer and drone may be the same man at different times. Probability solely depends on the evidence believed or assumed. If the evidence changes, the probability changes.

Shapiro finally attempts to turn improbability into a reason not to believe miracles, by referencing the base rate fallacy. This is a real fallacy and old saw (regular readers will well recognize it), introduced in every elementary probability book when Bayes’s theorem arrives. How worried should an asymptomatic women, aged 40-60, with no family history of breast cancer, be when the mammogram comes back positive, considering the mammogram is right (say) 99.9% of the time? Not that worried, as it turns out, because conditional on the information assumed the base rate of cancer is small. A positive mammogram adds to the evidence and increases the probability of cancer only a little. The fallacy comes in supposing the probability of cancer is close to the accuracy of the test.

This is applied to claims of miracles by first assuming miracles are rare, and then assuming claims of miracles are imperfect to some degree in the same way medical tests are. Going only by the “base rate” of miracles, the probability of a miracle is small. Add to the evidence a good but possibly imperfect report, and the probability of the miracle does rise, but it still remains small overall. “The absolutely crucial point is that when we are faced with testimony about something very improbable, such as an alien abduction, we have to ask ourselves one question: What is more likely—that the event really happened, as the witness reports, or that some other explanation for the testimony is true?”

This is a fine question which, as Shapiro says, should and must be asked. Notice it relates to cause, both of the purported event but also of the motivation of the reporter. Not everybody who relates a miracle properly interpreted what they saw, and not everybody tells the full truth. It therefore makes sense to examine every claim critically. And it even makes sense, as in the case of alien abductions, to ignore the claim, given the base information that heretofore all such events critically examined have proved false, or were very likely false (given the evidence accumulated in the investigation). This is because, given all past events like this were false or probably false, we judge the probability high that the newest claim will also be false or likely false.

But this does not work for miracles, for three reasons. One, not all claims of miracles have been proven false or likely false; some have been proved true or likely true. Thus, it is worth investigating substantial new claims, and worth ignoring insubstantial ones (like faces in burnt toast; notice how decision is wrapped up in this). Two, since miracles do not have probabilities, they do not have base rates. Their probability only makes sense with respect to assumed evidence. We can assume their near impossibility, making them immune to any report à la Hume, but this becomes a circular argument (a well known criticism). Three, if we were to rule out any report of unlikely events, nobody who (say) claimed (say) to win the lottery could ever be believed (lottery probabilities specify the precise evidence with which to calculate their probabilities).

Shaprio is finally wrong again, because rarity does not define miracles. Every day in tens of thousands of churches, the miracle of transforming nature of bread in the Body of Christ occurs. Another miracle is the universe being held in existence from moment to moment. Rarity doesn’t enter into it.