William M. Briggs

Statistician to the Stars!

Page 4 of 427

Truth, Knowledge, Belief, & Gettier Problems

"Hey, you never know."

“Hey, you never know.”

Proof Isn’t All That

The first section can be skipped for those who know what necessary versus conditional truth is.

I recall an anecdote about John von Neumann which had a fellow asking von Neumann for the proof of some mathematical proposition. Von Neumann asked the fellow which of other several theorems the fellow might already know, and he mentioned two, whereupon von Neumann proved the proposition twice, along the two different paths. Implicit in the story is that he could have proven it upon the other paths as well.

We don’t know what this proposition was, so call it X. Since X is necessarily true, we can have knowledge of it, where knowledge, as some philosopher define it, is “justified true belief.” They’d say the justification comes from the proof and the belief comes from us as an act of our intellect.

But does truth come from the proof? Von Neumann showed there were many different ways of knowing a proposition was true, but the multiplicity did not add to the truth of X. X was just true, and always was, regardless whether anybody knew it or believed it. So there is a difference between the truth of some thing and our knowing it; or rather, there seems to be a difference in the justification of our belief the thing is true and its truth.

Let’s clarify. Take our old standby argument with premises E = “All Martians wear hats and George is a Martian” relative to the proposition Y = “George wears a hat.” Y given E is true; that is to say, we know that Y given E is true, that it follows. We may therefore believe Y given E, as a sort of joint proposition, say, Y-given-E. But Y by itself, sans E, is not a necessary truth. Neither is E by itself a necessary truth. But Y-given-E is. Y therefore is a conditional truth, given or accepting or believing or having faith that E.

A necessary truth is one which is true no matter what. Take non-contradiction. It cannot be true that Z = “X is true and so simultaneously is not-X true”. In other (and confusing words), not-Z is true. There isn’t any way to think that Z (except, as many do, by changing it so that Z is no longer Z, and then forgetting they made changes). Why is Z false? Who knows? God made it that way. Why is it true that W = “For every natural number r, r = r”? I have no idea. God made it that way. What is our justification for believing W? Faith? Or is it that we’re too light in gray matter to discover a proof—or, worse, a counter proof?

Actually, we do have reasons for believing not-Z and W. That we cannot think of how Z is true is a dandy reason for thinking it false, and all experience is that for every natural number r, r does indeed equal r. Induction supplies the rest. From our senses to the truth!

All this is just a sketch, which we needed for the real meat which follows.

Get Gettier

A man hears his wife say she bought him a lottery ticket and he thinks to himself, R = “I now have a chance to win”. Unbeknownst to him, his wife was teasing. We know this, his wife knows this, but the man does not. The man accepts his wife’s word, conditional on which he believes R. R given the premise “Wife bought ticket” is thus a conditional truth. A believable truth, too, given he accepts (unconditionally) his wife’s word. R is not necessarily true, however, as is obvious.

Now Edmund Gettier famously claimed there were situations in which a person has a justified true belief, yet that belief did not meet the test of knowledge. Our lottery situation isn’t quite what he had in mind, because everybody would agree that R is a conditional but not necessary truth. To make this a “Gettier problem”, let’s add the premise “The man’s mother bought him a ticket for the same drawing but told nobody”. It is clear that R is now true, say Gettier followers, and the man is should believe it, but his claim doesn’t rise to the level of knowledge because his accepting R is based on his believing something which is false in fact (his wife’s joke).

But R is still a conditional truth to us and to the mother, who know of her actions. R, being contingent, can never be a necessary truth.

Gettier “problems”, I think, are based on forgetfulness. We forget who knows what and we forget what question is being asked of the evidence. To the man, R is conditionally true based on one set of premises, and to us it is conditionally false based on one set of evidence (just the wife’s statement) true based on another set (adding the mother’s). R is never true is the necessary sense. Plus, there are any number of premises which can exist, and which can be believed, that make it conditionally true. Even conditioned on the premise, D= “I, the man in this example, bought my own ticket” R is still only conditionally and not necessarily true.

In short, Gettier “problems” aren’t. This, incidentally, is one of the few cases where symbolic logic helps; I mean, being able to write the story down in symbols makes it much easier to see what goes where and who knows what, so that it is less easy to slip up.


I’m taking this example from Wikipedia, which (yes) does a good job explaining the set up.

The [justified true belief] account of knowledge is the claim that knowledge can be conceptually analyzed as justified true belief — which is to say that the meaning of sentences such as “Smith knows that it rained today” can be given with the following set of necessary and sufficient conditions:

A subject S knows that a proposition P is true if and only if:

  1. P is true, and
  2. S believes that P is true, and
  3. S is justified in believing that P is true

Recall von Neumann’s example and that X being true and anybody knowing X and the proof or belief of X are not the same thing. And also note that this definition mistakenly forgets to emphasize whether P is a conditional or necessary truth.

Here is a Gettier problem (also Wikipedia):

Smith has applied for a job, but, it is claimed, has a justified belief that “Jones will get the job”. He also has a justified belief that “Jones has 10 coins in his pocket”. Smith therefore (justifiably) concludes (by the rule of the transitivity of identity) that “the man who will get the job has 10 coins in his pocket”.

In fact, Jones does not get the job. Instead, Smith does. However, as it happens, Smith (unknowingly and by sheer chance) also had 10 coins in his pocket. So his belief that “the man who will get the job has 10 coins in his pocket” was justified and true. But it does not appear to be knowledge.

What has gone wrong?

Don’t Say “Natural Variability”


Word that the climate of doom we were promised (repeatedly) has not obtained has begun leaking out. Climatologists have known this for quite some time, but now even environmental activists are beginning to realize the horrible truth that their worst fears have not been realized.

The excuses have thus begun.

We have already learned “Don’t say ‘Hiatus'” because that is to speak nonsensically. Saying there is a “pause” or “hiatus” assumes the models which predicted the doom which did not happen were somehow right after all, and that it is Reality itself that is error.

It cannot be in the models we currently possess, because these models did not foresee what actually happened. The incontrovertible evidence is that these models are wrong. That they should not, in their current state, be trusted. That whatever they say is subject to extreme reasonable rational doubt. That decisions should not be made based upon their predictions (except the decision to produce better models).

To say there is a “pause” is to say that the models were right after all, even though Reality differed from the models. To say there is a “hiatus” is to say Theory is better than Reality. This is to commit the Deadly Sin of Reification.

One of the excuses is that the models were right after all, but the missing high temperature they predicted is actually in hiding. Sort of like in those movies where the Leader sneaks out of his palace or house and mixes with the ordinary people, and thus he learns What’s Really Important. That is, Global Warming has realized that people are important, too, and has given up its nefarious plans. Or something.

Anyway, the “in hiding” excuse can’t be right, not exactly, because the models already swore they took into account all the sources of heat, including the oceans. Obviously the models were wrong and they didn’t take some thing or things into account. What’s wrong, though, is anybody’s guess. Because some thing or things are wrong, however, it does not mean the thing you guess was wrong was the thing that was wrong. To prove it, you’ll have to redo the models and reforecast the future. Then we wait and see. In the meantime, keep quiet.

One thing we know with certainty is that the thing (in error) cannot be natural variability.

Natural variability, sisters and brothers, is what the models said they could predict skillfully. The models did not skillfully predict natural variability. Natural variability just is, in this sense, what the temperature does.

There is another sense of the phrase, though, a kind of enviro-religious sense that people might be using, which is, “What the temperature would do in absence of humans”. Now that is a valid thing to study. Only trouble is, it’s counterfactual. We can produce answers by the grant-load, but we’ll never know, or that is, we can never verify, whether any of them are true.

Because why? Because, of course, we humans are here and have been here. There is no way to remove our influence (or the influence of any species), so there is no way to know with certainty what the climate would be like without us. Of course, we might make reasonable guesses about what a never-were-humans climate would look like. But we would know those guesses are reasonable only after we can create models that can skillfully predict what the climate will look with us. Yet, as said, we’d never be able to verify those guesses because, of course, here we are.

Humans—and ants, aardvarks, and antelopes—are in integral part of the climate. All creatures influence the climate to some degree (get it? get it?). We are thus part of nature, thus part of real natural variability.

It was never a question whether humans influenced climate, for the answer was always yes; instead, the real science lay in understanding how we effect it. And how everything else effects it. And we’ll know we’ve done a good job with those questions—with understanding “natural variability”, that is—one we can produce good forecasts.


The Imposing-Their-Beliefs Fallacy

Perry trying to impose a curious view.

Perry trying to impose a curious view.

Here is an example of the Imposing-Their-Beliefs Fallacy (ITBF), taken from the New Republic article “The Straight, White, Middle-Class Man Needs to Be Dethroned” by Grayson Perry, a self-labeled “artist” (the trick these days is to discover who is not an “artist”):

They dominate the upper echelons of our society, imposing, unconsciously or otherwise, their values and preferences on the rest of the population. With their colourful textile phalluses hanging round their necks, they make up an overwhelming majority in government, in boardrooms, and also in the media.

Incidentally, it was only after reading up on Perry’s background and noting his obsession with the sexual, that I figured out that “colourful textile phalluses” meant ties. Skip it.

The fallacy does not lie in the statement itself, because, of course, it is possible, and even common, to impose one’s beliefs, values, and preferences on another. Indeed, it is even necessary that imposing occur. That fallacy thus lies in stating that it should not or could not.

Since some form of imposition is necessary, the presence of the fallacy, then, is always an attempt to impose beliefs, values, and preferences other than the ones being railed against. First a proof of the necessity, then proof that the fallacy wielder really just wants his own way.

Newborn and infants must have beliefs, values, and preferences imposed upon them, or else they will die. Children, too. The State imposes the belief that killing for fun and profit (of those human beings who managed to escape the womb, at any rate) is wrong, and it further imposes its value that those who kill will be punished; and it expresses preferences for the kinds of punishment. You can dispute that the State should do this, but even insisting on anarchy is to impose beliefs, values, and preferences.

If you say to another man, “Do not steal from me” or “Do not slit my daughter’s throat” you have imposed or are seeking to impose. If you ever say “should” or “ought” you are imposing, and the same is true if you use synonyms of these words like “judgmental” and “hateful” and so on.

It doesn’t even matter if, as Hume insisted, there really is a distinction between “is” and “ought” (and that is disputable), any time you approve or disapprove of another’s actions, you have imposed or are seeking to. The only slim possibility of non-imposition is if you are utterly indifferent to not only your own self, but to all others. That indifference includes the absence of love or hate or any other emotion.

Now evidence that the fallacy is always inverted.

At Truth-Out.org, in the article “Meet the Right-Wing Christian Companies Trying to Impose Their Values on Their Workers“, the author echoes the common complaint that employees not being given (government-mandated) free things because they are employees is an imposition. Which, of course, it is. The employees instead want to impose their belief that they should be given whatever it is they want and to not be required to give anything in return for it. Strangely, and in an indication of how far gone our culture is, the ITBF was convincing to Government.

Think Progress carried the article “Catholic Bishops: ‘Religious Liberty’ Includes Right To Discriminate Against Gay People, Impose Values“, which is seeped in the Imposing-Their-Beliefs Fallacy. Many today have forgotten the (what used to be) obvious fact that imposing beliefs is what religions do, and so to complain about this is a marker of insanity, stupidly, or political Machiavellianism. The Think Progress folks instead want to impose their values in the expected way.

These examples can be multiplied indefinitely, so it is easy to lose the wonder you should feel whenever you encounter the fallacy. But do try to be vigilant.

Now this Grayson Perry who supplied our first example of the fallacy goes on to say that straight, middle-class white men is a “group that punches far, far above its weight.” A curious claim given the list of accomplishments by this “tribe” (to include the computer on which Perry wrote his fallacy and the internet which served it up to his readers).

But it is clear which beliefs, values, and preferences Perry wishes to impose on us (for I can reveal Yours Truly is a member of this suspect group). I wonder if he’ll get away with it.

Summary Against Modern Thought: God Is His Own Essence

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. All posts are under the category SAMT.

Previous post.

The soup thickens. We haven’t learned much about God yet, other than He exists, is not composite, is outside time, is pure actuality and so forth. To go further, we need to expand our vocabulary and introduce the idea of essence. Well summarized by The Catholic Encyclopedia (CE), this is “the radical or ground from which the various properties of a thing emanate and to which they are necessarily referred. Thus the notion of the essence is seen to be the abstract counterpart of the concrete entity; the latter signifying that which is or may be [(in actuality, in potential)], while the former points to the reason or ground why it is precisely what it is.”.

Chapter 21: God Is His Own Essence

1 FROM what has been laid down we are able to conclude that God is His own essence, quiddity or nature.i

2 In everything that is not its own essence or quiddity there must needs be some kind of composition: for since each thing contains its own essence, if a thing contained nothing besides its own essence, all that a thing is would be its essence. Therefore if a thing were not its own essence, there must be something in it besides its essence: and consequently there must be composition therein. For which reason the essence in composite things has the signification of a part, as humanity in a man. Now it has been shown[1] ^1 that in God there is no composition. Therefore God is His own essence.ii

3 Again. Seemingly that alone which does not enter into the definition of a thing is beside the essence of that thing: for a definition signifies what a thing is.[2] Now only the accidents of a thing do not enter into its definition: and consequently only accidents are in a thing besides its essence. But in God there are no accidents, as we shall show further on.[3] Accordingly, there is nothing in Him besides His essence. Therefore He is His own essence.iii

4 Moreover. Forms that are not predicated of subsistent things, whether the latter be taken universally or singly, are not single per se subsistent forms individualized in themselves. For we do not say that Socrates, or man, or an animal is whiteness, because whiteness is not singly per se subsistent, but is individualized by its subsistent subject.iv Likewise natural forms do not per se subsist singly, but are individualized in their respective matters: wherefore we do not say that this individual fire, or that fire in general is its own form. Moreover the essences or quiddities of genera or species are individualized by the signate matter of this or that individual, although indeed the quiddity of a genus or species includes form and matter in general: wherefore we do not say that Socrates, or man, is humanity.v Now the divine essence exists per se singly and is individualized in itself, since it is not in any matter, as shown above.[4] Hence the divine essence is predicated of God, so that we say: God is His own essence.vi

5 Further. The essence of a thing is either the thing itself, or is related to it in some way as cause: since a thing derives its species from its essence. But nothing can in any way be a cause of God: for He is the first being, as shown above.[5]vii Therefore God is His own essence. Again, that which is not its own essence, is related in respect of some part of itself to that essence, as potentiality to act: wherefore the essence is signified by way of form, for instance humanity. But there is no potentiality in God, as shown above,[6] therefore it follows that He is His own essence.viii


iQuiddity: the whatness (as Kreeft says), “the essence that makes something the kind of thing it is and makes it different from any other” (Wesbster, from where we learn the rarely used synonym haecceity).

iiThe essence of you, dear reader, is that you are a man (male or female), which is to say, a rational creature (in Aristotle’s sense). But you are also more than just your essence. Some of you are tall, others are not as blessed. Some have hair on head and some wear hats. That is, as Aquinas says, you are composite, made of more than one thing. But we already know God is not composite, thus He must be His own essence.

iiiHaving or not having hair, or having or not having facial freckles, is an accident. With our without, the essence behind them is still man or woman. The Aristotle reference has him saying (what is obvious) that “we must argue from a definition, viz. by assuming what falsity or truth means.” Else we go nowhere. And that “the essence of a thing is that which is expressed by its definition” (CE, above).

ivWhiteness does not go walking about on its hind legs, i.e. it is not individualized in itself. But my white hat carries on being white because the hat carries on (continues to exist).

vPerhaps it’s obvious, but that humanity exists as an essence, and that Socrates or you is not that essence, but merely examples of it, is one point. How we know it is another, as it always is. Why mention it? Well, Star Trek fans, since our essence is being a rational animal, that essence might come in other accidental packages. See this essay by Fr Schall. Or work by David Oderburg (where’s the link?).

viThis obviously follows from the premises. But on that subject, more next week, when we learn that God’s existence is His essence.

viiBack to the Unmoved Mover, the Unchanging Changer! Chapter 13, that is. See the links from last week’s review.

viiiThis follows simply from above; i.e. don’e forget the second premise “if a thing contained nothing besides its own essence, all that a thing is would be its essence.” And do meditate on the difference between potentiality and actuality. So much flows from this distinction that it isn’t funny (as my old grandma used to say about a related topic).

[1] Ch. xviii.
[2] 4 Metaph. viii. 4.
[3] Ch. xxiii.
[4] Ch. xvii.
[5] Ch. xiii.
[6] Ch. xvi.

A Common, Unfortunate, Avoidable, Devastating Error In Statistics

Smilin' Joe demonstrates our fallacy.

Smilin’ Joe demonstrates our fallacy.

It’s a doozy, this error of ours. So ubiquitous is it that it’s hardly noticeable. Yet it is sinking us into scientism and wild overconfidence.

Every time it appears, both the public and scientists themselves become a tiny bit more over-enamored of science, giving it more honor than it deserves. The effect of any one appearance of the error is small, scarcely noticeable. But when it is repeated ad nauseam the product is deadly to clear thinking.

Of ado, no more. Here’s an example: “conservatives demonstrate stronger attitudinal reactions to situations of threat and conflict. In contrast, liberals tend to be seek out novelty and uncertainty.”

Did you see it? Maybe not. If you thought the corruption lay in the subject matter of the proposition itself, you were understandably wrong. The quote was taken from the peer-reviewed paper “Red Brain, Blue Brain: Evaluative Processes Differ in Democrats and Republicans” by Darren Schreiber and several others in PLOS One1.

What you thought was the main error was instead yet another in a long and growing line of misguided, probably ideologically but unconsciously motivated attempts to demonstrate to the level of satisfaction required by progressive academics that conservatives are biologically different than they are.

Need a hint about the bigger error? Here’s another example, culled from the same paper: “Republicans and Democrats differ in the neural mechanisms activated while performing a risk-taking task.”

Have it yet? Not the content. After an incredible amount of statistical manipulation, such that we can’t really be sure of what we’re seeing, the authors discovered that slightly more registered Democrats had high (statistically derived) activity in their left posterior insula than did registered Republicans (groups which they later re-labeled as liberals and conservatives).

From this, I remind us, they concluded that Republicans and Democrats differed.

This is false. They did not differ; or, at least, not all of them did. Only just enough differed to (after scads of manipulation) provide a wee p-value. But because all of them did not differ, and there is no reason to suppose that in new batches of registered party members, all of them will differ either. The statement is false.

Nor did, as cited above, “conservatives demonstrate stronger attitudinal reactions to situations of threat and conflict” than liberals. Leaving aside the soaring ambiguity in measuring political attitude and the even greater hand waving in defining “situations of threat and conflict”, the statement is still false. It was only found that slightly more “conservatives” than “liberals” answered some questions one way rather than another.

You must have it by now. The error is Irresponsible Exaggeration, which leads inevitably to Gross Over-Certainty. It is a crude mistake, common among the untrained and ill educated (reporters, etc.), and should be rare among scientists, but it increasingly isn’t, as our examples prove (here are many more).

It is now (near?) impossible to read any public report of research without this error—let us call it the Statistical Exaggeration Fallacy. Reports are lazy, harried, or not intelligent enough to realize they are making the mistake. But it’s surprising that it is never corrected by scientists.

Now as proved here, the purpose of statistics is not to say anything about what happened in a particular experiment, but what that experiment might mean in the future. The future must necessarily be less certain than the past, where the experiment lives (proved here). And not only that, it is a consequence of the crude statistical methods used by researchers, but their results are even less certain than implied even without the Statistical Exaggeration Fallacy (are all Republicans “conservatives”?).

I mean, relying on p-values already guarantees over-certainty, which is multiplied in the presence of the SEF. And by the presence of over-extended definitions, like calling Republicans “conservatives”, and conflating the answers on some questionnaire with some deep-seated and real psychological tendency.

Your help needed

What I’d like you to do, sisters and brothers, when you have the time, is to note in the comments whenever you see an instance of the Statistical Exaggeration Fallacy. It is well to have a large, contemporaneous collection of these to prove my claim of its non-rarity.

Of its harmful effect, well, if it is not obvious to you, it will be after you read the examples.


1A curiosity of this journal. They put the Results and Discussion before the boring, who-really-needs-to-read-it Methods section, which appears at the bottom.

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