Monthly Archives: June 2015

Study: Even Atheists Distrust Atheists

From Epiphenom:

Leah Giddings and Thomas Dunn, of Nottingham Trent University in the UK, set out to replicate some of the earlier work on atheism and trust, but with a twist.

They gave a group of 100 people a short story to read about Richard. It’s the same story that’s been used in previous research, and it goes as follows:

Richard is 31 years old. On his way to work one day, he accidentally backed his car into a parked van. Because pedestrians were watching, he got out of his car. He pretended to write down his insurance information. He then tucked the blank note into the van’s window before getting back into his car and driving away.

Later the same day, Richard found a wallet on the sidewalk. Nobody was looking, so he took all of the money out of the wallet. He then threw the wallet in a trash can.

Half the participants were asked whether they thought Richard was a teacher, or a teacher and a Christian. The other half were asked whether he was a teacher, or a teacher and an atheist.

Now of course there’s nothing in the story to indicate Richard’s spiritual beliefs, so if they claim it does that’s evidence of prejudice.

As expected, Christians were likely to be prejudiced against atheists. But once again, so were the atheists (albeit to a lesser degree – nearly 50% of atheists and over 75% of Christians associated atheism with untrustworthy behaviour).

So here we are in one of the most secular countries on earth, and even atheists think that other atheists aren’t to be trusted.
To follow on from this, the researchers gave the participants some statistics on the number of atheists in the country. Some of them got accurate statistics, and some got statistics that inflated the number of atheists.

It didn’t make much difference. Pro-christian prejudice went down, but anti-atheist prejudice did not.

As usual, this is only one study, so don’t take it as the definitive say on the matter. It’s more likely that this study is descriptive for the environment it was created in instead of it describing some fundamental human nature.

I only put that caveat there because I kinda get tired of people pointing to one study and claiming it is the be all end all of all argument. But this study is still interesting nonetheless!

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Posted by on June 26, 2015 in cognitive science


Διαριθμέω / Diarithmeo

to list

A roundup of some stuff I found interesting pertaining to religious belief!

Meditation has slightly different effects on the brains of men and women:

To our knowledge, this is the first study examining potential modulating effects of biological sex on hippocampal anatomy in the framework of meditation. Our analyses were applied in a well-matched sample of 30 meditators (15 men/15 women) and 30 controls (15 men/15 women), where meditators had, on average, more than 20 years of experience (with a minimum of 5 years), thus constituting true long-term practitioners. In accordance with the outcomes of our previous study of meditation effects on hippocampal anatomy by pooling male and female brains together (Luders et al., 2013b), we observed that hippocampal dimensions were enlarged both in male and in female meditators when compared to sex- and age-matched controls. In addition, our current analyses revealed that meditation effects, albeit present in both sexes, differ between men and women in terms of the magnitude of the effects, the laterality of the effects, and the exact location of the effects detectable on the hippocampal surface.


Although existing mindfulness research seems to lack sex-specific analyses—at least with respect to addressing brain anatomy—the observed group-by-sex interactions seem to be in accordance with a recent study reporting sex-divergent outcomes when assessing the impact of a mindfulness intervention on behavioral measures/psychological constructs (de Vibe et al., 2013). More specifically, administering a 7-week mindfulness-based stress reduction (MBSR) program, that study detected significant changes in mental distress, study stress and well-being in female students but not in male students.

The hippocampus is a small brain structure integral to the limbic (emotion-motivation) system. It plays important roles in learning, mood, and the formation of memories.

Meditation and prayer have some of the same effects on the brain, so we might see the same results with people who pray regularly. This might be another reason why men are less religious than women: Do women benefit more from religious practices?

Next, type of belief in free will linked to performance in self-control based tasks:

The first task used to measure self-control is known as the “Stroop task,” which requires participants to resist the urge to name a word on a colored background rather than simply saying the name of the color, which requires a degree of self-regulation to stifle the incorrect response. The second, an anagram test, gave participants seven letters and unlimited time to make as many English words as they could with the letters, which measures persistence despite boredom or fatigue.

Both tests are considered “seminal indices of self-control,” according to Clarkson, although the skills required to perform each are different.

“So it is not simply a matter of conservatives being more efficient or liberals being overly analytical,” he said.

In their performance on both tasks, however, conservatives outpaced their liberal counterparts. At the same time, both groups were shown to have similar levels of motivation and effort.


[Next], a group of study participants was told that the belief in free will has been shown to be detrimental to self-control by causing feelings of frustration, anger or anxiety that inhibit concentration. Under these circumstances, the effects were reversed. Liberals outperformed conservatives, suggesting that a belief in free will can undermine self-control under certain conditions.

“If you can get people to believe that free will is bad for self-control, conservatives no longer show an advantage in self-control performance,” Clarkson said.

So, if one believes in free will then one will perform better on tasks that test free will. But if you poison the concept of free will, and you believe you have this poisoned trait, then you’ll do worse on tests of free will! Pretty wild stuff. Reminds me of stereotype threat and growth mindset.

Next on the rationality front, expert philosophers are just as irrational as the rest of us [pdf]:


We examined the effects of order of presentation on the moral judgments of professional philosophers and two comparison groups. All groups showed similarsized order effects on their judgments about hypothetical moral scenarios targeting the doctrine of the double effect, the action-omission distinction, and the principle of moral luck. Philosophers’ endorsements of related general moral principles were also substantially influenced by the order in which the hypothetical scenarios had previously been presented. Thus, philosophical expertise does not appear to enhance the stability of moral judgments against this presumably unwanted source of bias, even given familiar types of cases and principles.

Seems as though expert philosophers are subject to framing effects just like some other experts in their field. This is why one needs to learn to just shut up and multiply. But not too much.

And then, being told about naive realism and experiencing an optical illusion makes people doubt their certainty:

Nearly 200 students took part and were split into four groups. One group read about naive realism (e.g. “visual illusions provide a glimpse of how our brain twists reality without our intent or awareness”) and then they experienced several well-known, powerful visual illusions (e.g. the Spinning Wheels, shown above, the Checker Shadow, and the Spinning Dancer), with the effects explained to them. The other groups either: just had the explanation but no experience of the illusions; or completed a difficult verbal intelligence test; or read about chimpanzees.

Afterwards, whatever their group, all the participants read four vignettes about four different people. These were written to be deliberately ambiguous about the protagonist’s personality, which could be interpreted, depending on the vignette, as either assertive or hostile; risky or adventurous; agreeable or a push over; introverted or snobbish. There was also a quiz on the concept of naive realism.

The key finding is that after reading about naive realism and experiencing visual illusions, the participants were less certain of their personality judgments and more open to the alternative interpretation, as compared with the participants in the other groups. The participants who only read about naive realism, but didn’t experience the illusions, showed just as much knowledge about naive realism, but their certainty in their understanding of the vignettes wasn’t dented, and they remained as closed to alternative interpretations as the participants in the other comparison conditions.

“In sum,” the researchers said, “exposing naive realism in an experiential way seems necessary to fuel greater doubt and openness.”

I imagine doing something like this, and then teaching some other rationality concepts (like my feeling of certainty) might be a good overall teaching tool. Might.


Posted by on June 25, 2015 in cognitive science, rationality


Probability: The Logic of the Law

bayes theorem

While poking around on JSTOR (thanks, grad school!) I found an interesting article in the Oxford Journal of Legal Studies called “Probability – The Logic of the Law“. In it, Bernard Robertson and G. A. Vignaux argue that probability is, you guessed it, the logic behind legal analysis and arbitration.

So not only do we have arguments in favor of probability being the logic of science (Jaynes), probability being the logic of historical analysis (Tucker, Carrier), but we now have an argument that probability is the logic of the legal world, too.

Here’s how Robertson and Vignaux derive Bayes Theorem in the article:

It has been argued that the axioms of probability do not apply in court cases, or that court cases out not to be thought about in this way even if they do apply. Alternatively, it is argued that some special kind of probability applies in legal cases, with its own axioms and rules… with the result that conventional probability has become known in the jurisprudential word Pascalian… In practice one commonly finds statements such as:

The concept of ‘probability’ in the legal sense is certainly different from the mathematical concept; indeed, it is rare to find a situation in which these two usages co-exist, although when they do, the mathematical probability has to be taken into assessment of probability in the legal sense and given its appropriate weight

This paper aims to show that this view is based upon a series of false assumptions.

The authors then go into some detail about common objections to the “mathematical” view of probability and why people think it doesn’t apply to the law:

1. Things either Happen or They Don’t; They Don’t Probably Happen

An example of this argument is provided by Jaffee: ‘Propositions are true or false; they are not “probable”

2. A Court is Concerned not with Long Runs but with Single Instances

Thus, descriptively:

Trials do not typically involve matters analogous to flipping coins. They involve unique events, and thus there is no relative frequency [my emphasis] to measure

And normatively:

Application of substantive legal principles relies on, and due process considerations require, that triers must make individualistic judgements about how they think a particular event (or series of events) occurred

3. Frequency Approaches Hide Causes and Other Relevant Information which Should Be Investigated

For an extended example of this argument see Ligertwood Australian Evidence (p14)

4. Evidence Must Be Interpreted

The implicit conception [in the probability debate] of ‘evidence’ is that which is plopped down on the factfinder at trial… the evidence must bear its own inferences… each bit of evidence manifests explicitly its characteristics. This assumption is false. Evidence takes on meaning for trials only through the process of being considered by a human being… the underlying experiences of each deliberator become part of the process, yet the probability debates proceed as though this were not so

5. People Actually Compare Hypotheses

Meaning is assigned to trial evidence through the incorporation of that evidence into one or more plausible stories which describe ‘what happened’ during events testified to at trial …The level of acceptance will be determined by the coverage, coherence and uniqueness of the ‘best’ story.

6. Assessment of Prior Odds ‘Appears to Fly in the Face of the Presumption of Innocence’

7. The Legal System is Not Supposed to be Subjective

Allen refers to

the desire to have disputes settled by reference to reality rather than the subjective state of mind of the decision maker

As you can see, a lot of the objections to probability here are continually raised in the frequentist vs. Bayesian interpretation of probability. But following in the steps of E. T. Jaynes, Robertson and Vignaux demonstrate that probability can be derived from some basic assumptions about propositional logic.

The authors then go on to explain the different “types” of probability, which is probably (heh) sowing confusion:

A priori probability refers to cases where there are a finite number of possible outcomes each of which is assumed to be equally probable. Probability refers to the chance of a particular outcome occurring under these conditions. Thus there is a 1 in 52 chance of drawing the King of Hearts from a pack of cards under these conditions and the axioms of probability can be used to answer questions like: ‘what is the probability of drawing a red court card?’ or ‘what is the probability of drawing a card which is (n)either red (n)or a court card?’

Empirical probability refers to some observation that has been carried out that in a series Y event X occurs in a certain proportion of cases. Thus surveys of weather, life expectancy, reliability of machinery, blood groups, will all produce figures which may then be referred to as the probability that X will occur under conditions Y.

Subjective probability refers to a judgement as to the chances of some event occurring based upon evidence. Unfortunately, Twining treats any judgement a person might make and might choose to express in terms of ‘probability’ as a ‘subjective probability’. This leads him to say that subjective probabilities ‘may or may not be Pascalian’.


This analysis of probability into different types invites the conclusion that ‘mathematical probability’ is just one type of probability, perhaps not appropriate to all circumstances… The adoption of any of the definitions of probability other than as a measure of strength of belief can lead to an unfortunate effect known as the Mind Projection Fallacy. This is the fallacy of regarding probability as a property of objects and processes in the real world rather than a measure of our own uncertainty. [my emphasis]

An instance of this fallacy is something called the Gambler’s fallacy. Indeed, in that post of mine I pretty much wrote what I emphasized in the quote above.

The authors then point out something pretty obvious: That flipping a coin is subject to the laws of physics. If we knew every single factor that went into each coin toss (e.g., strength of the flip, density of the air, the angle in which it was flipped, how long it spins in the air, the firmness of the surface it is landing on, etc.) we would know which side of the coin would be facing up without any uncertainty.

However, we don’t know every factor that goes into a coin toss, or drawing cards from a deck, or marbles from a jar (including the social influences of the marble picker). So there is a practical wall of separation between epistemology and ontology; a wall between how we know what we know and the actual nature of what we’re observing.

The authors continue with three minimal requirements for rational analysis of competing explanations:


1. If a conclusion can be reasoned out in more than one way, then every possible way should lead to the same results.

2. Equivalent states of knowledge and belief should be represented by equivalent plausibility statements. Closely approximate states should have closely approximate expressions; divergent states should have divergent expressions.

The only way consistently to achieve requirement 2 is by the use of real numbers to represent states of belief. It is an obvious requirement of rationality that if A is greater than B and B is greater than C then A must be greater than C. It will be found that any system which obeys this requirement will reduce to real numbers. Only real numbers can ensure some uniformity of meaning and some method of comparison.

3. All relevant information should be considered. None should be excluded for ideological reasons. If this requirement is not fulfilled then obviously different people could come to different conclusions if they exclude different facts from consideration.

Clearly the legal system does exclude evidence for ideological reasons. Rules about illegally obtained evidence and the various privileges constitute obvious examples. It is important therefore, that there should be some degree of consensus as to what information is to be excluded in order to prevent inconsistent results. It is also important that we are explicit about exclusions for ideological reasons and do not pretend to argue that better decisions will be made by excluding certain evidence. This pretence is one of the justifications for the hearsay rule, for example, and it is clear from these cases from a variety of jurisdictions that judges are increasingly impatient with this claim.

The next section I will try to sum up where possible:

Rules to Satisfy the Desiderata:

1. The statement ‘A and B are both true’ is equivalent to the statement ‘B and A are both true’.

2. It is certainly true that A is either true or false.

The statment ‘A and B are both true’ can be represented by the symbol ‘AB’. So proposition 1 becomes ‘AB = BA’.

This is the basic rule for conjunction in propositional logic. P ^ Q is equivalent to Q ^ P.

How do we assess the plausibility of the statement AB given certain information I, symbolically P(AB | I)?

First consider the plausibility of A given I, P(A | I), then the plausibility of B given I and that A is true, P(B | A, I)… Thus in order to determine P(AB | I) the only plausibilities that need to be considered are P(A | I) and P(B | A, I). Since P(BA | I) = P(AB | I) (above)… [c]learly, P(AB | I) is a function of P(A | I) and P(B | A, I) and it can be show that the two terms are simply to be multiplied. This is called the ‘product rule’.

And because of the product rule, and because of requirement 2 above, the numbers we should assign to our certainties of “absolutely true” and “absolutely false” are 1 and zero, respectively.

Next, since we know that absolute certainty is 1, then the statement P(A, ~A) — that is, the probability that A is true or false — should be 1. And from that it follows that if P(A) + P(~A) = 1, then however much P(A) increases, P(~A) is equal to 1 minus P(A). This, the authors call the addition rule.

We may wish to assess how plausible it is that at least one of A or B is true…

P(A or B) = P(A)P(B | A) + P(A)P(~B | A) + P(~A)P(B | ~A)

Now, the first two terms on the right hand side can be expressed as:

P(A)P(B | A) + P(A)P(~B | A) = P(A)P(B or ~B | A) = P(A, B or ~B) = P(A)

And the third term, P(~A)P(B | ~A) as P(B, ~A) by the product rule.

Hence P(A or B) = P(A) + P(B, ~A).

This means that if we are interested in a proposition, C, which will be true if either (or both) A or B is true we can assess the probability of C from those of A and B. Thus, if the defendant is liable if either (or both) of two propositions were true then the probability that the defendant is liable is equal to the union of the probabilities of the two propositions. Courts appear to find this rule troublesome. The Supreme Court of Canada applied it correctly in Thatcher v The Queen but in New Zealand the Court of Appeal failed to apply it in R v Chingell and the High Court failed to apply it in Stratford v MOT.

3. If P(A | I)P(B | A, I) = P(B | I)P(A | B, I) (the product rule) then if we divide both sides of the equation by P(B | I) we get

P(B | I)P(A | B, I) / P(B | I) = P(A | I)P(B | A, I) / P(B | I)

The two P(B | I)’s on the left hand side cancel out and we have

P(A | B, I) = P(A | I)P(B | A, I) / P(B | I)

This is Bayes’ Theorem.

Cue Final Fantasy fanfare!

From here, the authors begin going over objections to probability and its utility in the law; objections that are borne of the misconceptions about probability and its utility outside the law. Most of these objectsions, in fact, are due to a frequentist view of probability; thinking of probability as a fundamental aspect of the object or event we’re looking at instead of a description of our uncertainty. As a matter of fact, that view should be put to rest by the authors’ demonstration of only using logic to derive Bayes Theorem. At no point did they use frequencies or any appeal to the nature of an object.

I did read one response to this article in the same publication in JSTOR, but it amounted to basically “This would be really hard to do” and not “this is invalid and/or it doesn’t follow from the rules of logic”.

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Posted by on June 16, 2015 in Bayes


The Rules Of Logic Are Just Probability Theory Without Uncertainty

bayes theorem

So it looks like for the summer I won’t be having any grad courses. Which means I can go back to blogging a bit and commenting on the multitude of things I find dealing with religion and/or rationality that I come across on the web. Maybe even finish reading some books I’ve bought and blogging about them too!

One thing I read on Quora is an intersection of religion and rationality: Using Bayes Theorem in history. Unfortunately this won’t be a post praising the argument; rather, it’ll be one explaining the author’s fail at rationality:

To begin with, it’s illustrative to note who uses Bayes Theorem to analyse history and who does not. In the first category we have William Lane Craig, the conservative Christian apologist, who uses Bayes Theorem to “prove” that Jesus actually did rise from the dead. And we also have Richard Carrier, the anti-Christian activist, who uses Bayes Theorem to “prove” that Jesus didn’t exist at all. Right away, a curious observer would find themselves wondering how, if this Theorem is the wonderful instrument of historical objectivity both Craig and Carrier claim it to be, two people can apply it and come to two completely contradictory historical conclusions. After all, if Jesus didn’t exist, he didn’t do anything at all, let alone something as remarkable as rise from the dead. So both Carrier and Craig can’t both be right. Yet they both use Bayes Theorem to “prove” historical things. Something does not make sense here.

Yes something doesn’t make sense here, and one can tell what that is by inference from the title of this current blog post.

As I wrote above, logic is just probability without the attendant uncertainty. Which should sorta be uncontroversial since logic and math are highly interconnected, just like math and probability are interconnected. I’m also not the first to point this out; I first read this connection in Jaynes.

But let me offer a couple of demonstrations. How about the basic syllogism with a conjunction as the major premise:

1. P ^ Q (true)
2. P (true)
Therefore Q

If I give a probability value to the major and minor premise, we can find out what conclusion follows:

1. P ^ Q (100%)
2. P (100%)
Therefore Q (100%)

This follows both logically and mathematically / probabilistically. If P * Q is 1, and P is 1, then Q must also be 1. So the answer is the same for both the formal logic formulation and the probabilistic formulation. Another example, using the same format:

1. ~(P ^ Q)
2. P (true)
Therefore ~Q

So if you can’t understand the fancy symbols, this reads that if you have a conjunction P and Q that is false, and you also know that P is true, then it follows necessarily that Q is false. The same conclusion will follow if we substitute probabilities:

1. P ^ Q (0%)
2. P (100%)
Therefore Q (0%)

This reads if the probability of P and Q is 0%, and we know that P is 100% then it must mean that Q is 0%. It’s a straightforward algebraic solve-for-x deal. The conjunction of the major premise of this case can be converted into a disjunction using DeMorgan’s law:

1. ~P v ~Q (true)
2. P (true)
3. Therefore ~Q

Does using probability yield the same conclusion?

1. ~P v ~Q (100%)
2. P (100%)
3. Therefore ~Q (100%)

Since this is a disjunction, we are no longer using multiplication to find the answer.

The point with this is that the underlying mechanisms are the same: conjunctions in propositional logic have the same “mechanism” for finding conclusions that math/probability do. The main difference between logic and probability is that logic is binary (yes/no) whereas probability is comparative. If we know that A is greater than B, and B is greater than C, then A must be greater than C. The shortcut for those sorts of comparisons is using numbers. And more relevantly, if history is about comparing explanations — which is a measure of uncertainty — the only clear way to do so is by using numbers: Probability.

So let’s substitute “Bayes theorem” with “propositional logic” in the original quote and see if this still makes sense:

To begin with, it’s illustrative to note who uses [modus tollens] to analyse history and who does not. In the first category we have William Lane Craig, the conservative Christian apologist, who uses Bayes Theorem to “prove” that Jesus actually did rise from the dead. And we also have Richard Carrier, the anti-Christian activist, who uses [modus tollens] to “prove” that Jesus didn’t exist at all. Right away, a curious observer would find themselves wondering how, if this [propositional logic] is the wonderful instrument of historical objectivity both Craig and Carrier claim it to be, two people can apply it and come to two completely contradictory historical conclusions. After all, if Jesus didn’t exist, he didn’t do anything at all, let alone something as remarkable as rise from the dead. So both Carrier and Craig can’t both be right. Yet they both use [modus tollens] to “prove” historical things. Something does not make sense here

And there we have it. It is indeed true that both Carrier and Craig have attempted to use propositional logic to defend their cases. This must mean that historians need to do away with using formal rules of logical inference because they can lead to different, contradictory conclusions. Clearly, this now means that the whole gamut of logical fallacies is now in play to argue anything one wants in historical analysis!

This reminds me of how Creationists and other anti-science types think that the scientific enterprise is wholly corrupt because sometimes the scientific method produces two contradictory studies.

But yes. Both probability and logic (and science) follow the GIGO rule: Garbage in, garbage out. We can’t argue against a tool just because it follows GIGO.

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Posted by on June 5, 2015 in Bayes

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