Strange, it seemed to me that it would copy over the table I made in Word. I’ll try another way and hope you’ll understand. (Via Convert to text)
It’s simply table 9 on p. 291. It looks roughly like this:
Student/Test|Literature|Composition|Language|English
Yossi|1|1|1|(– )
Reuben|1|?|0|0
Simeon|0|1|(– )|1

And you argued that the only relevant case is when the two empty cells (– ) are 0, because otherwise it would contradict the relations that would result. But how do you know to assume that this was indeed the case? It felt to me like quite a problematic assumption, and I didn’t see that you addressed it.

2) I also had another difficulty understanding your words in the section on induction from a prototype:
Student/Test, Literature, Composition
Yossi, 1, 1
Reuben, 1, ?

You wrote at first that in all the models for filling in (when ?=0/1) they can be represented by a single parameter α or α2, but if so then they have the same level of complexity, so it is hard to see how we can use that measure. Then in the next section you solved this by means of a simplicity criterion taken from graph theory.
I didn’t find that afterward you explained the graph-theoretic criterion again in ordinary verbal terms. So I would be happy to know what the simplicity criterion is in this section as well, and not in graph-topographical form.

3. If I remember correctly, I once read this in By Way of Faith. I don’t understand what the connection is between evolution and this process, but I’ll look again, God willing.

1. That assumption follows from the fact that the Talmud makes an inference of induction from a prototype from two verses. If those cells were 1, nothing could be inferred. So it is clear that we are talking about a case where they are filled with 0. That doesn’t mean 1 couldn’t be there, only that if 1 were there they would not make an induction from a prototype from two verses.
2. I don’t explain the result in words. That’s why I moved to the graphical form. I explained the logic in the graphical criteria. In words this is simply analogy. You can translate the graphical criterion into words fairly easily. The advantage lies in the number of points on the graph. It is obvious that a graph with a larger number of points is less simple.
3. In the article I did not explain the connection to evolution, but showed the fact that Occam’s razor is not a claim about us but about reality. The connection to evolution is obvious. Creatures whose conclusions fit reality in the world less well will survive less.

1. That is clear and understandable, but you argued that this is also how ordinary induction works in the world, and therefore you used not an example from Kiddushin but one from life, like school and students, and there, when you have no information about their grades, you argued that we would fill it in with 0 and not 1. But of course that cannot be claimed.
2. In words you knew how to explain very well the example of an a fortiori inference by saying that behind our understanding there is hidden a quantity of parameters that can explain the model, but in terms of the complexity of analogy you did not manage to explain it in words. I didn’t mean an explanation of the graph I see before my eyes; obviously there it really is easy to see. I meant if not for the graph, but only by looking at the table itself, because there the same parameter of talent exists the whole time.

3. Obviously, if the world is built in a simple way, then this can be explained evolutionarily, in that people whose conclusions do not fit reality in the world will survive less.
So besides the fact that you contradict yourself on two points: a. I thought you accepted Plantinga’s argument against naturalism, and also Taylor’s more complex one… b. You claim that you see that the simple thing is the correct one (intuition), but here you are claiming that you simply evolved to understand it that way.

In any case, what I meant to argue was not why we evolved to understand this, but how it can be that the world is in fact built so that the simple thing is the correct one. That is almost impossible unless there is a higher power and consistent providence. Someone there in the comments gave a very good example of this:
When there is a sick person and there are two diseases that can explain the symptoms (when the probability of each is 1:10) and there is one disease that can explain this (whose probability is 1:100), then according to Occam’s razor there is a clear preference to choose the one disease as causing the symptoms, even though statistically the probability is the same. Someone has to be constantly supervising so that in a case where the probability is equal, the true disease will be the simpler one.

1. Indeed, ordinary induction in the world works that way. And it is correct to do it when those cells are filled with 0. Although even if one does not know what the filling is, it is more reasonable to assume it is 0 because the picture is simpler. Only when one knows the filling is 1 does the inference collapse, and then it really is not correct to make the induction.
2. If it were possible to explain everything in words, mathematics would be unnecessary. I have no interest in unhelpful formalization.
3. In my opinion there is no connection either to Plantinga or to Taylor. The One, blessed be He, is behind evolution, but once we have said that, evolution is a correct theory. Once it is correct, it stands to reason that one who understands reality correctly survives better.
Your final question is a misunderstanding. There is no difference between a question about us and a question about the world. What is perceived by us as simple is the correct structure of the world. It’s not that the world evolved so that it would be structured simply, but that we were built in such a way that what seems simple to us is the structure of the world.

1. Interesting.
2. I didn’t know that; I thought the opposite.
—
3. At least you’ve abandoned your mystical picture of the sixth sense of your mind.
In any case, there still has to be providence that the structure we are fitted to see as simple/correct will also be so in reality.
And you didn’t address the wonderful example I found down there in your comments: that when we have events of equal probability but one is simple and one is complex, then clearly it cannot be claimed statistically that the simple one is the correct one. And if we nevertheless claim that, then there must be some factor ensuring that whatever appears to us as statistically equal is the simple one. And if you are willing to claim that sometimes it is preferable to claim something slightly rarer statistically but simple over something complex and common, then all the more so there must be some factor supervising this.

3. I haven’t abandoned anything, and there is no mysticism here. I am talking about the sixth sense (the eyes of the intellect). As for providence, I discussed that in the third notebook.
I didn’t understand the example. If the two events, one simple and the other complex, then how do you determine that they have equal probability? I claim that the simple one has the higher probability.

What I mean is, suppose it is known from different sources that disease A affects 1:100 and disease B also affects 1:100 people, but disease C affects 1:100000.
Of course we do not know with respect to the case before us. But here we have no a priori reason to prefer disease C over A or B if not for Occam’s razor. Therefore, if you continue to argue that C is the disease that caused the symptoms, that is a sign that there was some coordinating factor behind it.

I think that in this case there is no difference between the possibilities. Occam’s razor is not relevant here when there is a probabilistic calculation. Occam’s razor is applied when there is no possibility of calculating probabilities. For example, a theory with three entities versus a theory with two entities. We have no way to quantify the difference, and nevertheless we will prefer the theory with two entities.

I think that according to your view there still has to be providence behind this. After all, even on your view, when you use the “eyes of the intellect,” you agree that they do not at all reveal the factors themselves in the explanation, but only the connection between the factors. Just as in analogy and a fortiori inference in the examples you wrote in the book Truth and Unstable, we only discover that behind the phenomena there stands a talent of type α, but we do not describe what talent α is.

If so, I think that when we have a phenomenon with three entities α, β, γ, then you would claim that when building a theory one assigns each entity an internal probability P (for otherwise you are begging the question), and therefore when there are three entities the overall probability will be P^3.
But when there are only two entities δ, θ, the probability of the theory is only P^2, and it is significantly preferable to the triple theory.
But surely in truth we have no reason at all to assume this unless there is particular providence that things should be so, because the world is not deterministic from the outset (such that if the initial laws of nature were simple, that would just be a continuation into the current state). In a reality where everyone has free choice, do you understand my argument?

2. I couldn’t understand why, according to your view, one needs to arrive at mystical use of the “eyes of the intellect” regarding Occam’s razor. Why not say that God, through evolution, created us with an alignment for understanding the world such that the simple thing is also the correct one, and otherwise we would have become extinct.
I assume it is somehow connected to what I challenged here, but I don’t see how.

1. The translation into probability is incorrect. There is no way to calculate probability for such a thing (there is no event space), and what you call probability is nothing but another formulation of Occam’s razor.
2. There is no mysticism here. It is possible, and even likely, that evolution created the capacity called intuition. So what?

1. Okay, so if so, my puzzlement only grows. How can it be that the simple is the correct one?

2. Because you argue that intuition is a cognitive sense of Platonic ideas and other claims about spiritual entities in the world. Material evolution has no power to create this, apart from creating the feeling for correct judgment.

Still, the question in (1) remains: how can it be that the simpler option is the correct one, if the world is not deterministic?

Evolution created this capacity, and “I” adopt it and determine that it is reliable. Our brain is a product of evolution, but the soul, of which the intellect is a part, exercises judgment and uses this capacity. The intellect uses the brain in order to think, just as I use my legs in order to walk.

From the beginning, you are claiming that evolution created our potential to analyze what the soul sees in the ideas?? But we see them through the soul, whose place of dwelling is only in the intellect?
Or does the material intellect see the spiritual ideas?

This sounds crazier by the minute.
In any case, you still haven’t explained how the world is built like that.

Me too, from the beginning.
Evolution ensures that the fit survive. How were mutations created (including the fit mutation)? Through the laws of nature, which are themselves a product of providence. There is no need for present-day providence to explain why this works. Beyond that, even if a person makes correct decisions by virtue of spiritual rather than biological senses, his survival is still higher, and that has evolutionary implications.

As for observation of value-ideas (not ideas of material objects, like horseness, which helps us understand the material world around us—which is a derivative of evolution), that may be something a bit different (not necessarily connected to evolution, but perhaps because of the survival value of altruism). But none of this is relevant here. I really can’t understand what problem we are discussing.
The point of present-day providence is unnecessary for any aspect of my words from any angle or any argument. Providence that established the laws of nature that created us—that is something else, of course.

By the way, the intellect is not material. The brain is material. I explained that the intellect is part of the soul, which uses the brain in order to think, just as I use my legs in order to walk.

I think there is a major misunderstanding here.
I have no problem with the ability to understand that Occam’s razor developed evolutionarily / through aliens or hyper-advanced beings / or the eyes of the intellect. (Aside from the fact that you contradicted yourself from the article, where according to your view the distinction is made only because of the eyes of the intellect.)
I also have no problem with the claim that the laws of nature themselves are simple and evolution created in us the feeling to identify them.

My only problem is how reality itself behaves this way at the everyday level.
For example, at a murder scene one can explain the murder by two people or by one person. The reason we choose the possibility of one person is because it is a simpler possibility. But there is no reason to assume that two people would not do it. And here it does not depend on the laws of nature but on their free choice.
And therefore there has to be providence making sure that at murder scenes (where we feel! that there was one murderer) there really was one murderer there.
This example may not be the most precise, but it is enough to illustrate my question.

There is no contradiction at all, and I still can’t understand the problem. There is no law of nature that forever only one person murders and not two. Rather, if you examine all the cases where there are two interpretive possibilities, in most of them it will be one murderer and not two. Here this may even be a matter of probability, and Occam’s razor is not really needed.
As for the behavior of nature, I already explained and will repeat one last time. Nature necessarily does not behave in a simple way; rather, the way nature behaves is what is imprinted in us as simple.
That’s it. My throat is hoarse.

Indeed, I agree that this is mainly a probabilistic calculation, and therefore I prefaced by arguing that this example is not precise as the analogue but only as a parable.
The claim is precisely this: in every place not under the deterministic influence of the laws of nature, there is no reason to assume that what feels simple to us is the correct thing. And there is no way to think so unless there is providence making sure that what feels correct can indeed be realized in reality as correct.
Just as that same idea of interpreting the murder scene can be expressed in another parable about the likelihood of other entities that are not directly affected by our laws of nature. Like a claim about the existence of an “Invisible Spaghetti Monster,” where there is no way for evolution to develop true understanding in such cases. And in order that the understandings we think are correct will actually be correct in reality itself, there needs to be providence behind it causing things that feel complex to us, like these, really not to appear in reality.

I’ll try from a somewhat different direction (which of course has already been said). Choosing between possibilities with different probabilities does not raise the same question for you. Why don’t you ask: what causes the world to behave in the way with the higher probability? Because this is not a claim about the world but about me. The world behaves as it behaves, and probability is my tool for determining what that is. Occam’s razor too is such a tool (which is not equivalent to probability).
Beyond that, I explained that even if there is divine intervention (providence) in the background, it happened when the laws were established sometime in the distant past (at creation), and not at every moment in question.
From me, with all due respect and in tears…

Our master Rabbi Michi, may he live long,
Regarding probability, as you mentioned above, there is a major difference. We derive probabilistic calculation after observing reality, and therefore we also know, roughly speaking, the event space for the calculation. By contrast, Occam’s razor is fundamentally an a priori tool.
But my main claim is that probability generally speaks about things that are fundamentally deterministic and tied to the laws of nature, otherwise it is impossible to calculate their probability. By contrast, Occam’s razor also speaks about situations that are not connected to the laws of nature, and in the most extreme example, explanations by way of the existence of “supernatural” entities (which nature cannot develop understanding against).
In any case, it sounds from your words like you are claiming, as Swinburne says (on the Wikipedia entry):
since our choice of theory cannot be determined by data , we must rely on some criterion to determine which theory to use. Since it is absurd to have no logical method for settling on one hypothesis amongst an infinite number of equally data-compliant hypotheses, we should choose the simplest theory: “Either science is irrational [in the way it judges theories and predictions probable] or the principle of simplicity is a fundamental synthetic a priori truth.”
Only with the distinction in your case that you are uncertain whether the source of the calculation is only internal or also cognitive.

In any case, certainly even if the intervention was at the beginning of creation, it can help only for understandings that follow deterministically from the beginning of creation (like observation of natural laws), but not for anything else. A clear example of this is your dealing with the problem of evil, such as how the Holocaust could have happened, etc.
With all due respect, k the lesser.

Probability is really not derived from reality but from logic. Probability is a field in mathematics, not in the natural sciences. And it has no connection to determinism. There are probabilities even regarding human behaviors that depend on choice, and I discussed this in The Sciences of Freedom.
That’s it, we’ve exhausted the topic.