Paul --- A Rock and a Hard Place

[Peter Kirby]

A simple example of a binomial distribution that you see everywhere is the number of boys and girls that are the children in a family. That follows a binomial distribution, where you could consider "p" the probability of having a girl as a child, "n" the number of children, and "k" the number of girls in that family.

We don't exactly think of these conception events as "repetitions of an experiment" or as "Bernoulli trials" or as being "repeated using identical conditions," to be sure, but if we chose to use a binomial distribution to model this, we'd be spot on.

The idea of doing the same with "arguments" is certainly more exotic... and, I think, not really that helpful... but I don't think we should stumble on all the terminology being used. Again, what appears to be happening is that you've gotten the impression (after a superficial skimming) that a "Bernoulli trial" is some very particular thing. It isn't. If you can speak of an event happening (or not) with probability "p," you've satisfied all requirements of the definition.

(If you have more than one of these and if their probabilities are independent, with identical probability "p," you've satisfied the conditions of a binomial distribution.)

[Peter Kirby]

PS-- You're plagiarizing.

Gee, that's stiff. He's just not attributing. It's hard to get people to cite. Besides, plagiarizing only really applies to more formal circumstances. Have a chocolate.

I'm not convicting his ethics. I don't care about that. But I don't know if he knows what he's talking about. Partly, because he's plagiarizing...

[iskander]

The letter n denotes the number of trials.. and the six arguments of Bernard cannot be accepted as an n.

Great. Maybe I agree -- surely more than willing to agree here (having come to this conclusion, apparently, by a different route). But you haven't explained why you say that. Equivalently, what do you believe about "arguments" that means that they cannot be considered "Bernoulli trials"? Is it the idea that they have a "probability" at all? Is it the idea that we have this probability figure? Is it independence?

PS-- You're plagiarizing. Makes me wonder how steady you are on all of this.

https://cnx.org/contents/vGiLOvP5@6/Bin ... stribution

There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1.
The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial.

The n trials are repeated using identical conditions in order to replicate the experiment n times. The six arguments do not replicate anything and hence the binomial distribution is the wrong tool to apply to the problem.

PS What I posted is what can be found in any textbook on the subject.

PS2 sexual intercourse can produce a boy or a girl with a constant probability of 0.5 each one. Every sexual intercourse replicates the previous one as to the outcome
PS3 You are substandard. Bye

[Bernard Muller]

Arguments that "prove a point" with a probability of X% (where X is low) is a slippery concept. I'm guessing that this means (using one interpretation of probability), that whenever arguments like this are used, they are found to prove the point 20% of the time. But that seems extremely odd.

I agree assigning probability (regarding proving a point) is suggestive, even if in some case, the use of algorithm can reduce the suggestive part.

It comes back to what it means to "prove a point," which is to make it certain or virtually certain that it is true. I could understand saying that I have an argument that has a 99% probability of proving a point. Maybe, in the past, arguments like this one (which have been checked by two people) have stood the test of time as arguments that prove the point.

In the domain of research on earliest Christianity, it is quasi impossible to have arguments with high probability, because for each one which has been made, there has been a lot of debate and opposition. So trying to make several independent arguments to "prove" the same point is not only commendable but also necessary.
Too many times, scholars (and others) are "proving" their point on only one argument (from one piece of evidence).
And also, many would reject a point because each one of several arguments (used to "prove" the point) is not accepted as "proving" beyond doubts, on its own, the point. However, I think it is wrong reasoning, according to the aforementioned equation.

But 20% chance to prove a point?

Yes, but if one has a large number of independent arguments with similar probabilities in order to make the same point, then those several 20%, put together, are not so ridiculous.

It's a startling claim. It's like saying that there's a 2% chance that a baby can guard Fort Knox with absolute security, or that there's a 4% chance that a crook can be trusted with your money with absolute confidence. It just sounds silly. Only under contrived circumstances could you make statements like that (e.g., maybe a computer program generating arguments could spit out proofs with probability 20% or something).

I agree trying to make a point with a 2% or 4% probability is ridiculous, even with a large numbers of arguments "proving" (with the same low probabilities) that same point.
First, if the probability is so low and cannot be accurately assessed (like most of assigned probabilities not resulting from strictly factual quantified data), it can as well be a 0%.
Second, you would need a lot of arguments in order to arrive to about an overall 50% total probability "proving" the same point; I calculated around 32 of these arguments with, for each, probability of 2%.
But for arguments with 20% probability, we need only three of them to arrive around this 50% overall probability.

Maybe there's a way to make use of these arguments, maybe a way to do so in a probabilistic setting, but they are not "proving" any points.

I would say, I have 6 independent arguments proving the same point. If you accept these arguments as 1) being independent, b) at least having a probability for each of 20% to prove the point, then these arguments put together will show my point is more likely to be valid than not valid.

We too easily allow ourselves to make contradictory assumptions when we let in slippery concepts like this. Imagine two people arguing a topic. Each has six arguments. Each of them insists that they must be allowed at least a 20% chance for their respective arguments to succeed (and that these probabilities are independent). Well, that's a contradiction. And our concept of "proving a point" or not, will not allow us to have an easy time of weighing both sides of an argument where there is uncertainty (not proof, only indications of a probabilistic nature).

I think the critique of the arguments of the first person made by the second person should lower the probabilities of the arguments from the first person. The second person should try to make his own arguments "showing" the opposite of the point made by the first person, and taking in account the critique of these arguments by the first person, which would lower the overall probability of the opposite point. That would be the best methodological way to discuss a point.

Math doesn't say anything about how we have to model things like this. (Creating a model requires domain knowledge. It's not just math.) So I'm not going to say that there is just one solution. But I think we need to try harder than this.

I think that equation makes a lot of sense, and is a part of a solution about proving points, but because it requires input data which are largely suggestive and debatable (cannot be scientifically quantified), it is far from being foolproof. But it allows to link mathematically different arguments about the same point.

Cordially, Bernard

[Bernard Muller]

About my arguments on 1 Thessalonians 1:10 (which as "Son" (of God) in it) which I highly suspect to be an interpolation:
For the later Paul's epistles:
In 2 Corinthians, there are about 4477 words, 108 occurrences of Christ or Lord, and 3 occurrences of "Son" or Jesus having God as his Father.
In Philippians, there are about 1629 words, 50 occurrences of Christ or Lord, and 0 occurrences of "Son" or Jesus having God as his Father.
In Philemon, there are about 335 words, 13 occurrences of Christ or Lord, and 0 occurrences of "Son" or Jesus having God as his Father.
In Galatians, there are about 2230 words, 36 occurrences of Christ or Lord, and 4 occurrences of "Son" or Jesus having God as his Father.
In Romans, there are about 7111 words, 105 occurrences of Christ or Lord, and 8 occurrences of "Son" or Jesus having God as his Father.

For the earlier Paul's epistles:
In 1 Thessalonians, there are about 1481 words, 108 occurrences of Christ or Lord, and 1 occurrence of "Son" or Jesus having God as his Father.
In 1 Corinthians, there are about 6830 words, 69 occurrences of Christ or Lord, and 2 occurrences of "Son" or Jesus having God as his Father.

In the latter epistles, there are about 15777 words, 273 occurrences of Christ or Lord, and 15 occurrences of "Son" or Jesus having God as his Father.
That's about 1 occurrence of "Son" or Jesus having God as his Father for 18 occurrences of Christ or Lord.
That's about 1 occurrence of "Son" or Jesus having God as his Father for 1052 words.

In the earlier epistles, there are about 8311 words, 140 occurrences of Christ or Lord, and 3 occurrences of "Son" or Jesus having God as his Father.
That's about 1 occurrence of "Son" or Jesus having God as his Father for 47 occurrences of Christ or Lord.
That's about 1 occurrence of "Son" or Jesus having God as his Father for 2770 words.

Ratio of about 2.6 for both cases between earlier and latter epistles.

That's one reason why I suspect the three "Son" in the early letters as part of interpolations.
One main objection for all that is the dating of Galatians, which would kill my arguments if that epistle had been written early on. But I did my homework on that in http://historical-jesus.info/hjes3xx.html, complemented by some of the content of http://historical-jesus.info/appp.html.

Cordially, Bernard

[Peter Kirby]

The n trials are repeated using identical conditions in order to replicate the experiment n times. The six arguments do not replicate anything and hence the binomial distribution is the wrong tool to apply to the problem.

PS What I posted is what can be found in any textbook on the subject.

PS2 sexual intercourse can produce a boy or a girl with a constant probability of 0.5 each one. Every sexual intercourse replicates the previous one as to the outcome
PS3 You are substandard. Bye

You seem upset with me, but the root of the problem was that you weren't being clear about what you were saying (among other things). I was pushing you to be clear about what you were saying. I'm not going to defend Bernard, but... A brittle understanding of what's going on with the binomial distribution has been demonstrated.

If you have events X1, X2, X3, X4, X5, X6, and if each event X1 .. X6 has the same individual probability to occur ("p"), and if these probabilities are independent... that that would be sufficient to use the distribution. Hypothetically these events could be anything and the "conditions" of these events could be anything or nothing.

Now, nobody can say that Bernard's "arguments" meet those assumptions. Identical probabilities? Why? Also... independence? Why? But if the assumptions were met, it wouldn't matter whether the additional verbiage regarding "trials" "repeated" "using identical conditions" "to replicate the experiment" was also true. That kind of wording is not inherent to the math here.

PS - I'm amazing.

[Ben C. Smith]

One main objection for all that is the dating of Galatians, which would kill my arguments is that epistle had been written early on. But I did my homework on that in http://historical-jesus.info/hjes3xx.html, complemented by some of the content of http://historical-jesus.info/hjes3xx.html.

For whatever it may be worth, here is Goodacre's impression of the order of Paul's letters:

1. 1 Thessalonians
2. 1 Corinthians
3. Galatians
4. 2 Corinthians
5. ?Philippians
6. ?Philemon
7. Romans

[Peter Kirby]

Here's a simple example.

Bob and Alice play a game.

Bob tosses a normal six-sided die. Each face has probability 1/6 to appear. These events are independent (of each other and with respect to Alice's, below).

Alice tosses a coin and then spins a wheel. The coin has 1/2 chance of heads and 1/2 chance of tails. The wheel has 1/3 chance of red, green, or blue. These events are independent. She codes these results as:
Heads Red - 1, Heads Green - 2, Heads Blue - 3, Tails Red - 4, Tails Green - 5, Tails Blue - 6.

Each of these is a "turn" of the game.

If there are 13 turns taken in the game (by Bob or Alice - in total), what are the odds that "1" never appears? (For example.) That can determined from a binomial distribution where success (appearance of "1") has a chance of 1/6 and failure has a chance of 5/6.

The events, however, can be different. They just need to have the same probability of success and to be independent of each other, in order to apply a binomial distribution to find the answer.

This simple example could be expanded further to describe six entirely different events, each with the same probability for success... hopefully the reader has the ability to see that.

[Peter Kirby]

Maybe there's a way to make use of these arguments, maybe a way to do so in a probabilistic setting, but they are not "proving" any points.

I would say, I have 6 independent arguments proving the same point. If you accept these arguments as 1) being independent, b) at least having a probability for each of 20% to prove the point, then these arguments put together will show my point is more likely to be valid than not valid.

At the moment, I am questioning (b) for one thing. I don't think it makes sense to talk about an argument "proving" a point with a probability of 20%. Also, yes, I don't think the concept of "proof" has much use for us here. And since these are not "proofs" with any known probability... I don't think we can describe them in the way that is hoped. (This was the point of my previous post. I don't think it was well-understood; the examples in it were intended to show the incongruency of numbers like 20% with the concepts of guarantee or proof, but you saw the examples as a point of contrast instead, because of the particular number made up for the example.)

Note that even if we don't like other ideas (such as those involving the "B"-word), that doesn't mean we have to use this one... this one is problematic at least.

Possibly you could rephrase these arguments as logically deductive arguments and talk about the probability of the premises being true. I don't know; I haven't tried this.

[Ben C. Smith]

Maybe the numbers themselves were not the point of Bernard's post. Maybe the whole thing was just a heuristic device designed to show how, though each individual datum might not mean much on its own, they combine to create an overall probability that the event in question happened. Historians use a sort of procedure like this sometimes, which Gilbert J. Garraghan calls "cumulative evidence" on page 305 of A Guide to Historical Method:

Cumulative or converging evidence is virtually circumstantial. It is "a heaping up" (L. cumulus) of bits of evidence, individually never more than probable, and often only slightly so, until they form a mass of evidence, the net result of which is certainty. But, as already noted, the resulting certainty does not issue directly from the mass or cumulus of probabilities, since no number of mere probabilities added together can logically produce certainty. To produce such effect, one must invoke the "principle of sufficient reason," by arguing that the only possible explanation why so many bits of evidence point to the same alleged fact, is that the fact is objectively true.

I am not sure how valid any given numbers would be for such an exercise.

To me it seems that the use of numbers would best involve a heavy use of statistics. For example, one might accumulate data until it becomes apparent that 85% of the time, when an ancient author uses term X, s/he fails to also use term Y, a synonym to X. But this or that Pauline letter uses both term X and term Y, so perhaps one of those terms belongs to an interpolation. Even here, though, I am not sure what that number, 85%, would mean. Would it mean that there is an 85% chance that Paul penned only one of those terms, and thus only a 15% chance that the other term is not part of an interpolation? Would it be that direct? It does not seem so to me, but my experience with statistics is pretty limited.