Why Fallacies are False -- 05, Conjunctions and Documentary Hypothesis

Here’s why I love Linda. A quick review.

The Linda problem starts with a dataset describing Linda. You can say anything you want about her. It doesn’t even have to be true.

Then you try to decide which of two statements is more likely to be true:

1.     Linda is X.

2.     Linda is X and Y.

There is no relationship between X and your dataset about Linda. There is also no relationship between Y and your dataset, or between Y and X. So you can’t prove that either X or Y is true about Linda based on the dataset.

To turn this into math, assign a probability of truth to each of X and Y, and the number is between zero and one because you can’t prove anything about either one. Because statement #2 is a conjunction, the math is multiplication. When you multiply two fractions smaller than 1, the product is always smaller than both of them. That means it will be smaller than X. So the product of #2 is smaller than #1 and 1 is more likely to be true. But most people get that wrong the first time they meet a Linda problem, and even after you explain it, some people still get it wrong the next time they meet one.

Linda’s formal name is the Conjunction Fallacy.

Here’s why I love Linda. The Documentary Hypothesis has a dataset describing at least four putative documents. DH claims that my Torah is made up of these four documents. You may have heard of JEDP; that’s them.

DH denies that “Linda is X” is possible. They reject any description of “Linda” that allows the simple statement.

DH insists that “Linda is Y and Z and…”, that is, Torah is made up of four sources. The probability of that is the product of the probability that each assignment to one of the four putative documents is correct. There is no hard evidence that any of them existed, so the probability of every assignment is between zero and one.

To get the answer, you must multiply because DH says you can’t assign the whole thing to just one document, and you must have enough assignments to achieve the whole Torah. So the answer is some fractions between zero and one multiplied together, with at least four terms contributing to the product.

Now, it would be one thing if DH took each book in Torah and assigned the whole book to one of the four sources. So your terms might be J for Genesis, E for Exodus, P for Leviticus, D for Deuteronomy, and then one of them again for Numbers. 

But the dataset doesn’t support that because the descriptions of the documents don’t match any one book. P gets the closest to Leviticus and assignments to P would have the highest value – if they were all restricted to Leviticus.

But they aren’t. DH splits all five books. Numbers is split up the most, with parts from each of the four documents. Leviticus is split up the least, but some of its putative sources are not even JEDP. But at any rate, the number of terms is larger than four and you might as well make it ten for starters. That would be the outcome if Leviticus was all P and Deuteronomy was all D, and each of the other three books was split between two or three of the documents.

But they’re not. All right, then, it would be nice if DH took each narrative in Torah and said it came from one of the four sources. That means you’re looking at each narrative as one term and assigning it a probability of coming from one of those four documents. I’ve counted some 80 narratives in Torah, stories with plots and characters and action (and this feeds into something I will talk about later). So there are at least 80 terms in the calculation, all of them fractions between zero and one. And then you have to consider the non-narrative portions, which fall between the narratives. So the number of terms is larger than 80; we could set it at 100 for starters. A fraction between zero and one to the 100th power is infinitesimal, down around 10 to the minus 61. (For comparison, the Planck length is 10 to the minus 35 meters.)

But that’s not what DH does. DH splits some narratives up and assigns part to one document and part to another. So you can’t count on two verses that are sequential, being assigned to the same document. Actually, it’s worse than that, because DH splits some verses up, assigning part of the words to one document and part to another. But let’s ignore that last bit, because what I’m saying is that DH has a probability that is the product of some fractions between zero and one. You have the same number of terms as the number of pieces in the DH assignment, something more than 100 terms.

Since you can’t count on a chunk of verses to all be assigned to the same document, you have to consider every verse a separate term. 

Torah has 5845 verses.

DH’s probability calculation has at least 5845 terms. Each term has a value between zero and one. If every term had the same value, the answer would be that value raised to the 5845th power. The answer is infinitesimal. The probability of DH being true is vanishingly small.

Now, DH will say that it is not a Linda problem, there IS a connection between the dataset and the assignments, the dataset describes the four documents to which they are making the assignments. But as I said, we don’t have hard evidence (yet) that they ever existed. What’s worse, I show on my blog that the descriptions themselves have a basis in fallacies, two of which I will get to later. Worst of all, I show that, from the start, DH relied on false data. This is what I wrote about last time: even if you don’t have a real conjunction fallacy, but your dataset contains falsehoods, you’re wrong. The whole concept has a zero probability of being true.

My science author was a trained biochemist but, like I said last week, that doesn’t mean he had training in logic. And even if he did, it's obvious that he didn't subject DH to a probability calculation. He had no training in the Bible, he admits that. It was one of a number of instances where a scientist writes about something they’ve never researched, and the work gets attention because of who they are, not because they know what they’re talking about. I won’t go into that rant here.

Knitting -- ripple throw with piped edging from leftovers

I haven't posted on this page in a long time because I was working on a project which I just finished.

A long time ago, when I was in high school, there was this craft thing where you had a wooden spool with a large hole down through it. At one end there were some thin nails, and you used a crochet hook to loop yarn around them and then sort of knit a long snake. 

Technically this is called an I-cord. You can use it for the edge of a coat; I used to have a coat pattern with this specific design.

Otherwise the only other thing you might do with an I-cord is make a piped edging for housewares, like a cushion cover. So let's do that.

I worked this project on straight needles because my circular needles already had projects on them. The leftovers were Comfy fingering and I used size 3 needles.

This being a throw, it was rectangular. I used Arne and Carlos' video to make the I-cord for the bottom edge, but I used 3 stitches not 4.

https://www.youtube.com/watch?v=T_lQU5QsdNs

I made an I-cord that was 312 stitches long. This gave me 306 stitches for the pattern and 3 stitches on each end for the side I-cords.

I used a crochet hook to pick up the Vs on the top of the I-cord to work the yarn into them, then I knitted the next row with I-cord stitches at each end.

ON RIGHTSIDE ROWS:  K1, yarn forward, slip purlwise, yarn back, K1. Work across in pattern, then K1, yarn forward, slip purlwise, yarn back, K1.

ON WRONGSIDE ROWS: slip 1 purlwise, yarn back, K1, yarn forward, slip purlwise. work across, and then yarn forward, slip purlwise, yarn back, knit, yarn forward and slip.

The pattern is an 18-stitch repeat. Work this between the 3-stitch edges. If you want to work it stand-alone, K1 at the edges on R3 and R4.

R1 & 2: Knit  

R3: *[(Knit 2 sts together) x 3, Knit in front and back of next stitch, (Yarn Over, K1) x 4, K in front and back of next stitch, (Knit 2 sts together) x 3], repeat * till last stitch.

R4: Purl to last stitch.

Work this until all your leftovers are gone -- or at least until there's not enough of the leftovers to do a full 4 row pattern. I used 4 sets of the pattern for white and 5 sets for all the other stripes. One skein of yarn made three sets of 5-pattern runs. I had one full skein of that dark pink and it made all three of those stripes.

When your leftovers are used up, add in the color for the top piping. I recommend working the same  number of repeats at the top as you did after starting the bottom I-cord.

For the top bind-off work another series of piping.

Put a DP needle into the three stitches of the edge piping and pick up the first stitch of the top row. Using another DP, K1, yarn forward, slip purlwise, yarn back, K2TOGTBL leaving the last stitch on the needle. Now pick up the loop in front of the next stitch and do this again. Pick up that same loop and K2 K2TOGTBL. This turns your corner.

Use one of your DPs to * pick up the next stitch of the top row, K2 K2TOGTBL leaving the last stitch on the needle * until all the top row stitches have been worked. Slip two stitches over to make an edge for the piping and work the end of the yarn into the wrong side. Notice that this will roll the I-cord just as on the bottom since you never turn the work over and purl.

If you want to work this in bulky, make a bottom I-cord 132 stitches long.

This I-cord piping makes another nice non-curling edge and, as you see, it follows the ripple in the pattern. 

Why Fallacies are False -- 004, Conjunctions and Conspiracy Theories

You’ve had a week now to think about that last statement. If you teach somebody about the Conjunction fallacy and why it’s wrong, at a later time they will STILL PICK THE WRONG STATEMENT AS MORE LIKELY.

If they’re looking at the exact same problem, that’s nuts. But it could be a matter of not realizing they’ve seen it before. We all forget things.

To show you how conspiracy theories are Linda problems, I have to emphasize some things.

First, it is irrelevant whether the “description of Linda” dataset is true or false. It’s just a bunch of information labeled Linda for the purposes of the explanation. You could collect sports statistics and that would still be your dataset.

The important thing is that both your X and your Y have nothing in common with that dataset. In the classic problem, the first statement is that Linda is a bank teller. Nothing in the description says she went to business school or college or any training sessions that qualify you to be a bank teller. If your conspiracy theory is a real conjunction fallacy, the simple statement will have nothing in common with your dataset. If it does, you’re not looking at a conjunction fallacy.

The same is true about your Y, which in the classic problem says Linda is a feminist. Again, there’s nothing in the dataset that says she belongs to NOW or any other woman-oriented organization, or that she participated in feminist protests. Again, if the “Y” in your conspiracy theory actually has a relationship to your dataset, it’s not a real conjunction fallacy.

At that point, it becomes important if your dataset has falsehoods in it. If you can prove that, you prove the whole thing is probably not true. Same if the connection between the dataset and the other statements have fallacies in them.

Once you pick your dataset, be careful with the claims you make about it. Prove the X claim. Then you can go on and prove the Y claim. And THEN you can make a probably true conjunctive claim involving X and Y. And this is exactly what science does, under Cartesian method: break a problem up into pieces; prove one piece true at a time; and then combine them.

Highly educated people face as much risk of creating and believing conspiracy theories as people who left school in their teens. Higher education does not force people to learn about fallacies. Even if STEM training teaches Cartesian method, they don’t necessarily lecture on the connections to logic or why the Method helps prevent creation of fallacies.

Next time I’ll give an example of a conjunction that never heard of Cartesian method.