I have started working my way through the Shaarie Yosher, specifically the sections that deal with Majority.

He actually starts in the beginning with a discussion about Uncertainty in general, and Double Uncertainties in particular.

He makes the point (in Shaar 1, Perek 19) that there are 2 reasons we can use to explain the principle that by Double Uncertainties we are lenient: 1) by Double Uncertainties, between the each Uncertainty we wind up with a Majority of reasons to be lenient (i.e., the Odds of one Uncertainty are 50/50, and on the side that the thing is forbidden we have another 50/50 Uncertainty, that leaves us with a 75/25 towards leniency), and 2) that we only forbid things when we are Uncertain about whether something is definitely forbidden, but if we're uncertain about something that in and of itself we are uncertain about, we would not forbid it.

The difference between these two explanations would be in the following case: let's say that the secondary uncertainty has a Majority to forbid it and only a minority to permit it. According to this first explanation, we would still be able to say that there is a Majority to permit: 50/50 on the initial uncertainty, and on the side that it may be forbidden we have another 90/10 Uncertainty (90% forbidden, 10% permitted), but still the 10% chance that it is permitted should join to the 50% chance of the first Uncertainty and thus leave us with a total Majority to permit!

But according to the second reason, since by the second Uncertainty there is a Majority to forbid it, and since we treat Majority like certainty, it means that the second Uncertainty is not actually an uncertainty at all, but rather a certainty that the thing is forbidden, and if so then we are left with a single uncertainty about something which if the side to forbid it is correct then it will be 100% certainly forbidden! Therefore, the principle of Double Uncertainty would not apply!

The Shaarie Yosher maintains that the second explanation is the correct one. And then he ingeniously applies it to what would appear to be a completely unrelated matter: namely, the issue of why everyone agrees that by Rabbinic Uncertainties we are lenient - even though by Biblical Uncertainties we are strict - in spite of the fact that we are Biblically commanded to follow the rulings of the Sages from the verse of "Lo Sassur"! So since the commandment to follow the Rabbis is itself Biblical, then every time we have a Rabbinic Uncertainty we should really treat it like a Biblical Uncertainty and be strict!!

The Shaarie Yosher says that based on our explanation of Double Uncertainties we can answer this quite well. Because we have established a principal that when we are directly uncertain about whether something is definitively forbidden, then and only then do we say that we are strict. But here, our uncertainty is not regarding the commandment of "Lo Sassur", rather it's about something else - it's about whatever the specific ruling that the Rabbis commanded us on. Therefore, since the Uncertainty is not directly on something which is definitively forbidden, we do not apply the principle of defaulting to being strict.

This distinction the Shaarie Yosher makes about how "Lo Sassur" is a separate thing, actually fits in quite nicely with how I generally explained the topic of Majority, where I maintained that by the courts our uncertainty is not about the judges themselves, but rather about the case that that has come before them; and this is true even when they are ruling about something which did not come before them, for example when they are debating whether to apply a penalty of whipping, where there is no external uncertainty, I argued that we would still say that the uncertainty is not directly about the judges, rather the uncertainty is the external fact that the Torah commands us "Lo Sassur" and thus these commandments compete and we do not know which is the correct ruling, and the Voting of the Sages acts as an external Majority (similar to the meat stores where the meat was found on the street).

I finished Shmaatza 4 of the Shev Shmaatza today!

It's the biggest of all the Shmaatza's and it deals primarily with Majority.

There were a couple of chapters that were not directly relevant to Majority, and these sections I only skimmed. But in general this has given me a really solid foundation on the mechanics of majority and the context around it.

I intend to review it soon, but I think before that I want to dive into the Shaarie Yosher and get a feel for how he addresses Majority. I don't know if he will have a self-contained section dealing exclusively with it the way the Shev Shmaatza did, but I intend to find out.

I want to take the opportunity to summarize the major thrusts of my piece of Majority.

I'm thinking about even incorporating it into the piece itself as an official summary of sorts. Regardless, it will be very useful to review what I have learned so far and what my general takes on the topic of Majority are.

I'm not sure if I should place the summary before or after the piece. I'm leaning towards including at the end of the piece because I want the reader to first work their way through the piece before they reference a summary.

Majority is applied in the following scenarios:

1) Uncertainty

a) Voting

b) Odds

2) State

Uncertainty has a principle of Majority Negates Minority.

The source for Uncertainty is from a verse that deals with a court Voting (this is also the source for Odds).

The source for State is from a verse that deals with Nazir.

We can derive Odds from the verse of judges Voting because the verse deliberately uses ambiguous language that could be read as referring to either the court or the general public; thus, this deliberate ambiguity may be designed to teach us that there is a general principle of Odds that the public must follow.

We can derive Majority Negates Minority from the verse of judges Voting because the verse clearly maintains that by capital cases we err on the side of caution even in the face of a majority of one, and yet by a majority of two we do not err on the side of caution; the only way to explain why suddenly by a majority of two we do not err on the side of caution is if the mechanics are via Majority Negates Minority and thus the minority is completely negated and there is nothing for us to err on the side of.

There are two kinds of Uncertainty:

1) Experimental Probability

2) Theoretical Probability

The source for Experimental Probability is from judges Voting.

The source for Theoretical Probability is from several different verses.

I saw something incredible in the Shev Shmaatza today.

In Shmaatza 4, Perek 24, he quotes the opinion of Tosafos who explains why it is that by monetary matters we say that we do not follow the Majority; Tosafos explains the reasoning, that by monetary matters, the minority joins up together with the Status Quo and together they beat the Majority.

Now this is clearly difficult to understand, because we know that there is a general principle that whenever a Majority conflicts with a Status Quo, we always follow the Majority. If so, then why would monetary matters be any different? According to Tosafos that the whole reason we do not follow Majority by monetary matters is because the Status Quo beats it, then how do we reconcile that with the principle that universally a Majority always beats a Status Quo?!

The Shev Shmaatza says a tremendously novel answer. He argues that there are different classes of Status Quo. The classic type of Status Quo is what we can refer to as a Historical Status Quo: it saying that since something was a certain way in the past, we can assume it is still that same way now. For example, if a husband goes missing and we are unsure if he is alive or dead, we say his wife is not allowed to get married because the Status Quo was that last we knew he was alive, and if so we continue to assume that he's alive now.

However, there is another class of Status Quo. We can call this a Current Status Quo; meaning, that the Status Quo is not something historical, rather something which is current right now. For example, by monetary matters, where Person A says that Person B owes them money, currently it is Person B who is holding onto the cash, and thus they have a Current Status Quo. And it goes without saying that a Current Status Quo is far stronger than a Historical Status Quo; so much so that it will even beat a Majority!

Now, the reason I find this take so striking, is that according to the Shev Shmaatza the universal principal that Majority beats Status Quo is actually only true with regards to Historical Status Quos. But anytime we come across a Current Status Quo - even if it is not by monetary matters - the Status Quo will beat the majority.

Just a random, nice insight I had.

We know the principle that by Biblical uncertainties we are always strict and err on the side of caution. There is a debate whether the requirement to do so is itself Biblical or just Rabbinic. Those who say it is Biblical deduce it from a verse.

But my question is, that even according to those who maintain that the requirement to be strict is itself Biblical, why do we need a verse to teach us to be strict? Wouldn't that be simple common sense? After all, as I write in my piece on Majority, if God says not to do something and there's even the faintest possibility of my violating His commandment, isn't it basic common sense that I should have to refrain from any chance of violating God's command?

It must be, therefore, that there is something really profound here. The Torah and the Rabbis clearly do not view erroring on the side of caution as being common sense! Meaning, that they must maintain that it isn't "fair" to expect somebody to refrain from doing something on the chance that the may be doing something wrong according to the Torah. The Torah view is that by common sense we would assert that unless the Torah is definitively forbidding something, then there's no reason for me to refrain from doing it!

I would suggest that this is an application of the famous principle: "It's ways are pleasant". Namely, that a fundamental principle of the Torah is that we can expect its laws to be "fair" and "pleasant"; and being forbidden from doing something on the off-chance that it may be wrong wouldn't be fair and pleasant!

Yesterday I asked how we can understand the principle that Established Things Have 50/50 Odds.

I saw today in the Shev Shmaatza, Shmaatza 4, Perek 18, that he gives an outline of a way to understand it. By the case where I bought meat from the store and I'm just not sure which one, the uncertainty is directly connected to the store, since I am the one who bought it and I am unsure which store it was bought from. Whereas by the case where I found meat on the street, the person who dropped the meat may very well know exactly which store the meat came from, therefore the uncertainty does not stretch back all the way to the store itself, but rather to when it was dropped by the person who bought it.

So when the uncertainty extends all the way back to the store, that is where we say that since the minority non-kosher store is an Established Thing, the majority of odds cannot negate it. Whereas when the uncertainty only starts from long after the meat has left the store, the fact that the minority is an Established Thing is not directly relevant to the uncertainty at hand.

While I do think this is a clever distinction, I still don't understand WHY this distinction makes a difference! Who cares when the uncertainty was birthed? Why does that make a difference? The fact is that by the case where I found the meat on the street, I am uncertain about which store it came from, and the stores are Established Things, so why can the majority negate the odds?

And even more puzzling, is that once again this appears to be a purely logical construct with no Biblical source. The Rabbis must have seen this as an obvious logical statement. So what am I missing??

I FOUND THE SOURCE!

I saw that Rashi in Yoma 84b points to the Talmud in Sanhedrin 79a, where the Talmud quotes the verse in Devarim 19:11: "And when there will be a man who hates his fellow and lies in wait for him and rises against him and strikes a fatal blow and kills him, and then flees to one of these [sanctuary] cities. The elders of his city shall send and retrieve him from there, and put him in the hands of the blood-avenger and he shall die." The Talmud deduces from the emphasis on "him" in the words "wait for him and rises against him", that in order to be held liable for murder, the murderer must specifically have intentions to kill "him", i.e. a specific individual. But let's say that someone randomly threw a deadly rock into a group of people with the general intention to kill, but not intending to kill any specific person, we cannot hold them liable for murder. This is the opinion of Rabbi Shimon. But the Sages maintain that as long as the group of people were composed of people for whom I would be liable to the death penalty, then I would be held liable for murder.

The Talmud asks, that according to the Rabbis why is the verse emphasizing "him"?

The Talmud answers, that we know that capital punishment is reserved for the murder of Jews (since in Jewish law we are extremely reluctant to apply the death penalty and only do so extremely rarely, and in general we find that the Torah limits the application of the death penalty to as small an area as possible), so if the majority of the group that one threw the deadly rock into was gentile, there is no need for this verse since I would say to follow the Majority and thus we would consider the entire group gentile and the thrower could say that as far as they were concerned they were following the majority and thus should not be held liable for murder. And even if the group was half Jewish and half Gentile, since we say that be capital cases we always rule leniently, we would also say that the thrower here could not be held liable for murder, and once again there is no need for a verse. Says the Talmud, it must be that the scenario where we need the verse is where there is a majority of Jews and a minority of gentiles. Without the verse I would say that we should follow the majority and consider the group as if it were all Jewish and thus as far as the thrower is concerned he should be held liable for murder. Therefore, according to the Sages, the Torah wrote this verse to teach me that Established Things Have 50/50 Odds!

Now of course we have to figure out how the Talmud deduces this principle from the verse. At first glance I would say that this principle must just be an Oral Tradition, and that the Rabbis were just finding a place in the Torah where we can attach an inference (this concept is explain in the Mevo HaTalmud). But I still want to chew on this and see if we can determine some method that the Talmud had in why it felt that the mechanism of this verse was via Established Things Have 50/50 Odds.

A couple interesting questions occurred to me today:

We know that there is a principle that Established Things Have 50/50 Odds. But if this is the case, then why by Scach do we say that Majority Negates Minority?! After all, the non-kosher Scach is as established as it's possible to be! It's literally present right now physically inside our uncertainty!!

This is a HUGE question. I really cannot overemphasize how big of a deal it is. There must be something really big I'm missing in my understanding of Majority.

I think my misunderstanding has to do with my lack of clarity on how the rule of Established Things works. I'm not even clear what it means to be an Established Thing. Does it mean merely that it is a physically present object? Does it mean that it's the kind of physically present object that does not move from it's place? I actually lean toward the second understanding even though it makes a lot less sense logically than the first (not that the first makes a great deal of sense either); the reason is first of all that the language of "Established Things" is more commonly used in the Talmud in this sense of not moving from it's place, and also that the way the Talmud talks about it, like for example by the case of the 10 meat stores, we are discussing things which don't move from their place. Although actually, from the fact that we also apply it by the case of a woman who may have visited a man and had relations with him, where we consider the uncertainty regarding the man as him being an Established Thing, that the concept of Established Thing is not literally that it doesn't move, but rather that as regards our uncertainty it was fixed in place and our uncertainty "came to it" in a manner of speaking.

I just saw a fantastic question from the Mordechai in Chullin (quoted by the Machatzis Hashekel, Orach Chaim, Raish Tzadi Ches, Yud Gimmel, Gimmel), that how is is that by the courts we follow the Majority, aren't the judges considered Established Things and thus should force 50/50 odds?! The Mordechai gives an answer that seems incredibly difficult to understand: he argues that the words that leave the sages mouth are what we are uncertain about, and those words are considered like an item that has separated from a Majority and thus is not considered as an Established Thing. The Machatzis Hashekel does try to give the answer some more context, by explaining that the Mordechai must mean that the speech of the court is considered its own distinct entity, separate from the physical judge that the speech came from. But this is still very difficult to understand.

(Let me note, that per my understanding of the topic, that the uncertainty by the court is not just which group of judges we should follow, but really the uncertainty is regarding the actual uncertainty of the case that was brought before the court, then we could answer the Mordechai's question by saying that similar to the principle of Separated Things Come From The Majority, this is also "Separated" in the sense that the uncertainty here is not really about the court itself rather about the case that came before the court, and thus it is not the kind of case where we can apply the principle of Established Things.)

But all this aside, I simply cannot for the life of me make sense of the underlying logic here. Why is it that Established Things Have 50/50 Odds? I don't believe (or I haven't seen anywhere) a verse mentioned as the source. Thus, it appears to be a purely logical construct. But if so, WHAT EXACTLY IS THE LOGIC??!!

I saw one interesting avenue to explain why Established Things Have 50/50 Odds: the Shailos Veteshuvos Bais Ephraim on Even HaEzer Siman 5 gives an interesting take (to my understanding): he says that lets say by the case of the 10 meat stores where we find the meat on the street, the uncertainty is specifically on this individual piece of meat: did it come from the majority of kosher stores or the minority of non-kosher ones? In this kind of case the majority and minority are both directly interacting and positing overlapping claims on the origination of this meat; therefore, since the majority and minority are both directly fighting each other we follow the majority. Whereas by a case of 10 meat stores where the uncertainty is whether I entered a kosher or non-kosher store to purchase my meat, the uncertainty is not on any specific object and therefore the majority and minority are not directly competing over the status of a singular thing, and thus the majority cannot win and we are therefore left with 50/50 odds.

However, according to this explanation, it's difficult to explain why by the courts we don't say that they are Established Things, since by the courts we also are not in a state of uncertainty over a singular object - ie the "court" itself - but rather we are faced by two distinct commandments to follow the sages, and we don't know which set of sages to follow. Perhaps the Bais Ephraim understands the court as being a "singular" thing, and we are unsure which single ruling the court is issuing and thus the Majority and minority are directly competing.

This is also problematic re our question by Scach: by the Scach there isn't a "singular" object upon which I have an uncertainty whether it originated from the majority or minority; rather, here by Scach we are unsure if right now the area I'm sitting under is either in this moment kosher or non-kosher Scach, but the two options are not directly interacting over a singular, separate object, rather we are trying to determine for every inch of Scach whether it is the kosher or the non-kosher Scach. So since they are not directly competing over a singular, separate item, we cannot say that the Majority overpowers the minority, and thus we should be left with 50/50 odds. And you can't say that it is a singular thing, since the uncertainty is regarding the Succah as a whole, since the Kehillos Yaakov clearly demonstrates that by the case of Scach we do not treat it as a singular uncertainty regarding the Succah as whole!

Another question that occurred to me, is that if Majority Negates Minority - meaning that we consider the Odds as being 100% certain - then how does the principle of Double Uncertainty work?? The first uncertainty is resolved 100% by the Majority of Odds, and the same for the second one. Each one on its own should be considered as 100% certainty and therefore there should be no way for us to apply the principle of Double Uncertainty!

One potential answer, is that first of all Double Uncertainty does not mean that one uncertainty occurred after the other; rather, it means that we are simultaneously faced with one uncertainty, and on the majority side of that uncertainty we are faced with another uncertainty. Thus, we consider both uncertainties simultaneously as part of the same general uncertainty. Given this, perhaps the logic here is that there is a flaw in the initial Majority: meaning, that the initial Majority has within it yet another uncertainty, and thus it is only by pure Majorities that do not contain any uncertainties that we can say that the Majority applies.

Another potential answer, is that we have established that according to the Rogachover (as well as many, many proofs) the entire concept of following the Odds is a Legal principle and not a Physical principle: meaning, that the Torah is not telling us that Odds are telling us what actually happened and as a result we can follow that conclusion, rather that the Torah is simply giving us a guideline of what to do when we are uncertain, which is to follow the Majority. Given this, then it means that the rule of Majority Negates Minority is also purely a legal construct; which means that the actual minority of Odds never disappears, rather it is just that we consider it like it is not present at all. Given this, then it makes sense to say that in the case of Double Odds we apply a different ruling, since from a physical, real-world viewpoint the minority was always present, and now we have a double minority, and thus our legal rule can take that into affect and change.

I saw a great sheet on Sefaria that brings a lot of great sources as regards the topic of Majority: https://www.sefaria.org/sheets/532755

As per the Shev Shmaatza, the reason that we do not apply Majority to financial matters, is because the minority joins up together with the Status Quo (of whoever currently has possession of the money), and together they beat the Majority, thus preventing us from following the Majority and taking the money away from the person who currently has it.

The question on this, of course, is that elsewhere the Shev Shmaatza asserts that Majority is more powerful than either the principle of Status Quo or Proximity. In a head to head of Majority vs Status Quo or Majority vs Proximity, Majority wins every time! Given this, then why are financial matters any different? Why specifically and only by financial matters do we say that the Status Quo beats the Majority.

By capital cases I can understand why we don't follow Majority: it's because we have a singular stringency when it comes to taking human life, that we require an extremely high level of proof and thus Majority alone does not suffice. But by financial matters, the reasoning that the Shev Shmaatza gives is not that financial matters require a higher level of proof, but rather that the Status Quo joins up with the minority to beat the Majority.

The Shev Shmaatza in Shmaatza 4, Perek 16, asserts that Majority actually does work even by financial matters, and even where the Majority is not simply establishing an earlier set of facts but rather is directly influencing our uncertainty. The case would be where the Status Quo (which we join up to the minority) is a weak Status Quo, for example where we have an uncertainty who an item belongs to, and one of the plaintiffs grabs it away from the other in front of the court. True, he is currently in possession of the item, and thus he has a Status Quo, but since the way he got the Status Quo was in a disputed manner, we say it is a weak Status Quo and we will not rely on it against a Majority. Contrast that to a case where everyone agrees that a legal transaction occurred and that an item was then possesses by Person B, if we now have an uncertainty regarding the ownership going forward, we do make use of the legally obtained Status Quo to join it together with the minority to beat the Majority.

Per this understanding, the Shev Shmaatza has now established a much more finely drawn set of lines regarding when we do and do not follow Majority. In order of strongest Majority to weakest we have:

1) By a case of forbidden and permitted things:

a) If it is an Experimental Probability:

aa) We do follow the Majority even if there is a Status Quo or Proximity opposing it

bb) We do follow the Majority even if there is a 100% certainty that the forbidden thing will be violated (as long as the forbidden and permitted things are fully mixed together until they are indistinguishable)

b) If it is a Theoretical Probability:

aa) We do follow the Majority even if there is a Status Quo or Proximity opposing it

bb) We do not follow the Majority if there's a 100% certainty that the forbidden thing will be violated (this is my understanding; I have not seen an explicit discussion of this, and it is possible that such a case by Theoretical Probability is impossible)

2) By a case of heritage (meaning establishing who can marry whom), everything is the same as by forbidden and permitted things, except that instead of one Majority being sufficient, we require two Majorities (because we are extra careful)

3) By a case of financial matters:

a) If the Majority is establishing facts from before the uncertainty was created, then we do follow the Majority

aa) Both by Experimental and Theoretical probabilities (as we find by capital cases, thus certainly here as well)

bb) I believe even against Status Quo or Proximity (since the whole reasoning why we do follow Majority before the uncertainty was created is because it is no longer a financial matter, then we should fall back on the standard rule that Majority is superior to both Status Quo and Proximity)

b) If the Majority is establishing facts from after (or simultaneous with when) the uncertainty was created

aa) If the financial Status Quo is not in and of itself 100% undisputed (for example the plaintiff grabbed the item in front of the court), then we do apply Majority

bb) If the financial Status Quo is in and of itself 100% undisputed, then we do not apply Majority

4) By capital cases, everything is the same as by financial matters

Back to the Shev Shmaatza!

Nice and warm learning, perfect for a bitter cold day.

The Shev Shmaatza in Shmaatza Daled, Perek Vav, begins an extraordinary piece that extends across multiple perakim.

He starts asserting the well known principle that we do not follow Majority when it comes to money.

He then quotes the Tosafos in Sanhedrin 3b who asks, that it is well known that we do follow Majority when it comes to capital cases. If so, then certainly we should follow Majority when it comes to monetary matters! After all, if the principle of Majority is good enough to rely on when taking away a person's life, then it certainly should be good enough to take away their money!

And he points out that not only do we apply Majority to capital cases when we are dealing with Experimental Probability, but even when we are dealing with Theoretical Probability!

The Shev Shmaatza gives a brilliant answer, that lays down for us some key foundational points in the topic of Majority.

He starts by discussing the well known fact that to convict someone in a capital case you require the testimony of two witnesses. However, there is another requirement to convict: in addition to there being witnesses that the murder was committed, the court must also be sure that the defendant was warned that murder was wrong! In Jewish law we require absolute certainty that the person knew that what they were about to do was something that was forbidden. Given this, I would assume that we would also need 2 witnesses to say that the defendant was warned that it was forbidden to kill. However, the fact is that even if a single witness testified that the defendant was warned that murder is forbidden, the testimony is considered sufficient and we will execute the defendant based on that testimony!

So it must be, that while it's true that when it comes to directly influencing a capital case - like testifying that the action of murder was committed by the defendant - we do require 2 witnesses, when it comes to indirectly influencing the case - like testifying that before the action of murder even occurred the defendant was aware that murder is forbidden - then even a single witness suffices.

Given this, the Shev Shmaatza establishes a broad principle: the principle of Majority can only directly resolve an uncertainty when we are dealing with matters of permitted and forbidden things; but when we are dealing with capital cases or even with financial matters, Majority can only indirectly influence the matter at hand when it comes to establishing the facts before the uncertainty ever occurred!

I haven't been writing as frequently here because I've been heads down in the Shev Shmaatza.

I started at Shmaatza Aleph and made my way through a nice chunk until I realized that the Shmaatzos are labelled as to what they are discussing, which led me to the realization that Shmaatza Daled is really what I'm interested in right now since it discusses: "the laws of Majority and Proximity, and Majority by money, and the laws of Status Quo".

My plan is to build a critical mass of knowledge around the Shev Shmaatza's opinions re Majority, and then circle back to plug it back into my piece of Majority.

One concept I have been struggling with in the topic of Majority, is that the Sages differentiate between the following cases:

1) There are 9 kosher meat stores and one non-kosher, and we find meat on the street, we say that we can rely on the majority to say that the meat is kosher

2) There are 9 kosher meat stores and one non-kosher, and I am unsure which store I bought this piece of meat from, we do not rely on the majority and instead we say that the meat is forbidden!

The reasoning why in case #2 it's forbidden, is described as "Something which is Established is treated like 50/50 odds".

Meaning, that the meat stores - including the non-kosher one - are considered Established; and my "uncertainty" was birthed inside the store, since my uncertainty is regarding which one I purchased it from; and something which is Established can not be negated by a Majority, but rather forces the uncertainty to be considered as if the odds were 50/50.

Whereas in case #1 where I find the meat on the street, the uncertainty was birthed right now as the meat was discovered on the ground, and thus no part of the uncertainty extends to the actual store that the meat originated from.

Now to be honest, I am trying as best I can to phrase the distinction to the best of my understanding, but really this is a tough distinction for me to grasp.

I mean, by the case where I find the meat on the street, what does it mean that the uncertainty does not extend to the store itself. If you think about it, the entire uncertainty is regarding which store it came from! So how is it different from the case where I'm uncertain which store I purchased the meat in: there as well, the uncertainty is about which store the meat came from!

Clearly, the underlying thrust of the argument here is to say that an Established minority cannot be so easily negated.

I'm thinking out loud, but maybe the idea is similar to a point I made in the piece on Majority, that we can distinguish between an Intrinsic Majority and an External Majority: that an Intrinsic Majority is like by the Scach, where the Uncertainty and the Majority are both from the self-same object, whereas an External Majority is like by the case where meat is found on the street where the Uncertainty is regarding the meat but the Majority is tied to the stores.

Maybe a similar distinction is present by the cases we've described above: by the meat on the street the Uncertainty is External and thus we can ignore the fact that the minority is Established; but maybe by the case where we are unsure which store we purchased the meat from, the Uncertainty is much more directly about the store itself and thus is an Intrinsic Minority and therefore something that we cannot simply brush aside and therefore it forces a 50/50 odds.

(I don't much like this approach though; it feels like a stretch.)

There is an interesting comment made in the Talmud in Baba Basra 24a. The Talmud asserts that when it comes to the menstrual process there are three body parts involved: 1) the Source, where the blood originates from, 2) the Corridor, which is the passage the blood takes from the Source to go outside the body, and 3) the Attic, which is an area above the Corridor which is not a part of the menstrual process but which sometimes bleeds as well.

Abbaya points out that if blood is found in the Corridor, we assume that it is menstrual blood from the Source, even though the Attic is physically closer to where the blood was found in the Corridor. This, says the Talmud, is evidence that Majority is more powerful than Proximity, since even though the blood found in the Corridor is in a closer Proximity to the Attic, still, since the Majority of blood is found in the Source we say that Majority is more indicative than Proximity and thus we follow the Majority and say that the blood likely came from the Source.

Rava responds to Abbaya and says that you cannot bring a proof from this case that Majority is stronger than Proximity, because in this case of menstrual blood there are actually two principals that are informing us that the blood came from the Source: 1) the fact that the Majority of blood is found in the Source, and 2) the fact that there is a greater Frequency of blood passing through the Corridor that originates from the Source. Meaning to say, that we are not only relying on Majority here, but also on the fact that the blood that does run through the Corridor comes mostly from the Source!

Now the language that Rava is using here is quite interesting. He refers to this new principal he's introducing as "Frequency", but refers to the other established principal as the classic one of "Majority". My question is, that the Frequency under discussion here is also a type of Majority! What I mean by that, is that we have established that there are two types of Majority: 1) an Experimental Probability, and 2) a Theoretical Probability (or in the terms of the Talmud, a Tangible Majority and an Theoretical Majority). A Theoretical Probability is where the Majority is based purely on logic and conjecture, with no hard, certain physical facts present before us. Given that, then both the fact that there is a Majority of blood found in the Source, and the fact that the greater Frequency of blood that runs through the Corridor is from the Source, both of these are just types of Theoretical Probability! So why is Rava calling one of them "Frequency" as if it was something completely different from "Majority", when in fact it just a regular kind of Theoretical Probability?!

The simplest understanding is that Rava is simply saying that Abbaya has no proof from here that a single Majority is superior to a single Proximity, since here that are actually two Majorities!

And in fact, see Tosafos there (U'Shma Minah) at the end where he makes the point that when the Talmud here says that we learn from Rabbi Chiya that Majority is Biblical, it can't be referring to an Experimental Probability since we know that already from the verse of "After the majority to incline", rather it must be referring to a Theoretical Probability. Meaning to say, that the type of Majority we are discussing here is a Theoretical Probability which is exactly as I answered above.

I have begun to work my way through the Shev Shmaatza.

Here are some general principles that he has established thus far (I am up to Shmaatza Aleph, Perek Hey):

He starts by establishing that there is a general disagreement between the Rambam on one side, and the Ran and Rashba on the other side, as to whether by uncertainties that involve Biblical matters are we required to be strict because of Biblical law or because of Rabbinic law. Meaning to say, that according to the Rambam, Biblically we can be lenient whenever we are unsure about a Biblical matter (and it is only the Rabbis who instituted that we should be strict by Biblical uncertainties). However, according to the Ran and the Rashba, the Torah itself requires us to be stringent by Biblical uncertainties.

The Shev Shmaatza proceeds to make an interesting distinction between two types of uncertainty: 1) Original Uncertainty, and 2) Novel Uncertainty. Original Uncertainty is where the uncertainty we have is not something newly arisen, but rather is something that has been present from the very beginning. For example, if we are uncertain if an animal was born with a terminal flaw, that is an uncertainty which dates back to the very origin of the object under discussion. However, lets say we are dealing with an uncertainty over whether divorce papers were served, in that case the uncertainty involves a new change - not something that has been present since the very beginning.

And what difference does this distinction make?

Well, says the Shev Shmaatza, it makes a huge difference according to the Ran and Rashba who maintain that the Torah itself requires us to be stringent by Biblical uncertainties. Because even the Ran and Rashab only maintain their position when we are discussing a case of Original Uncertainty; but even they would agree that in a case of Novel Uncertainty the Torah does allow us to be lenient!

The Shev Shmaatza also points out that this disagreement between the Rambam and the Ran/Rashba is actually a Tannaic dispute between Abbaya and Rava. Rava maintains that the Torah requires us to be strict by Biblical uncertainty, and Abbaya says that the Torah allows us to be lenient.

He brings a brilliant question that the Ramban asks. The Ramban begins by asking: how do we know that we have to obey what the Sages tell us to do? The Ramban points out that we know this from the verse in Devarim 17:11 that "You shall not deviate from the matter that shall be instructed to you [by the Sages] either to the left or right". This is a commandment to obey the Sages. Given that Biblically we are commanded to follow the ruling of the Sages, then how can we possibly say that by Rabbinic uncertainties we are lenient: every Rabbinic uncertainty is actually a Biblical uncertainty of "You shall not deviate"?!

The Shev Shmaatza brings the opinion of the Zohar Rekiah who answers that when the Sages instituted their rulings, it must be that they did so while explicitly excluding their ruling from applying whenever there is uncertainty! Thus, when we are faced with a Rabbinic uncertainty, we must consider it like there is no Rabbinic ruling at all, and therefore no verse of "You shall not deviate" can be applied.

The Shev Shmaatza asks on this a wonderful question. How can we apply this logic to a case where the Sages are disagreeing over a matter of Rabbinic law itself?! Where one side is maintaining that you must do A, and the other side maintains you must do B: why is it that we say that we follow the lenient opinion, over here it's impossible to say our logic that the Sages explicitly did not apply their rulings to uncertainty; over here there is no Rabbinic ruling at all, rather a disagreement over whether there should even be a ruling in the first place!

The Shev Shmaatza also maintains as a general principle, that when a Chazaka or a majority or a "nearness" are applied to an uncertainty, we consider it like there is no uncertainty at all, and that we have a certain resolution.

I want to clean up the big section where we deal with the Maharitz Chiyus.

At this point, I don't even understand how he could ask that we should apply Majority Negates Minority to Nazir, since by Nazir there is no uncertainty at all. I want to rewrite the section - and even rewrite the opening of the whole piece on Majority - to make it clear that by a court we are uncertain about the matter at hand, not about which judges to follow, and that the Torah is just saying that when there is a pre-existing uncertainty we can follow the laws of probability and rely on the majority of judges or the majority of meat stores.

However, I just came across an enormous can of worms that I must dig deeply into.

The Rambam in Mishne Torah (Maachalos Asuros 8:11) says that if there were 10 meat stores and 9 sold kosher and one sold non-kosher, and I purchased meat from one of the stores and I just don't remember which one, then the meat is forbidden to be eaten! The Rambam gives as the reasoning that whenever the minority is firmly established, we consider the uncertainty to be 50/50 and therefore forbidden. However, says the Rambam, if the meat was found on the street then we would follow the majority of kosher stores to say that it is permitted.

The Talmud as well it Kesubos 15a says that there is a principle that "anything that is seperated, we can assumed that it was seperated from the majority". And the Talmud there also references the same ruling that the Rambam mentions, that any time the minority is firmly established then we consider the odds to be 50/50.

Another fascinating topic is that the Talmud there discusses that sometimes for very sensitive issues like determining parentage, the Rabbis required that there be 2 majorities present!

The Talmud there also attempts to bring a Biblical source for those who maintain that established minorities are considered like 50/50 odds both to indict and to acquit: they quote the verse in Devarim 19:11 that says regarding a murderer "And lie in wait for him, and rise upon him", which indicates that the murderer must have been deliberately waiting for the victim. This is all the Talmud says. The problem of course is how in the world to understand this!! What in the world does this verse have to do with an established minority??

Another juicy topic is the Talmud in Baba Kama 27b that says that by monetary matters we do not follow the majority. (Rashi does explain there that what it means is that if the defendant is giving us a reasonable explanation for how they currently posses the money, we cannot take it away from them simply based on a majority, since it is the claimant's responsibility to provide solid proof.)

Another interesting vein of discussion is the Talmud in Chullin 28b that discusses what happened if exactly half the trachea or esophagus was cut during ritual slaughter: the Talmud says it is an Amoraic dispute whether it is kosher or not, since one side holds that the Torah requirement is that the majority cannot be in a state of "uncut", and the other side maintains that the requirement is that the majority be in a state of "cut".

The Shulchan Aruch in Yoreh De'ah 98:2 says that if a minority of drink was mixed together with a majority, then only if the 2 drinks were of the same type do we apply the law of Majority Negates Minority, but if they were of two different types of things then we do not apply this principle.

For the life of me I cannot rediscover the source I originally saw that said that if there are 3 pieces of meat and 2 are kosher and 1 is not and you don't know which is which you can rely on Majority Negates Minority and it is permissible to eat all 3. Obviously it is critical I find this source again since it is a key part of the piece of Majority. Maybe I saw it brought from a source in the Artscroll Talmud?

Continued from yesterday...

Let me first write the actual language of the Maharitz Chiyus' answer and then I will explain how I understand it.

The literal text of his answer is as follows: "But the truth is, that by this where we follow the majority, nobody argues since the verse says 'After the majority to incline'. But from the verse it is not apparent what would happen if the minority is apparent to counter the majority, and the minority is like it is non-existent, and if an action is done with the majority like here where he shaved and two hairs remained, we say Majority Equals Entirety and it is as if the action were performed on the minority as well."

I believe that the Maharitz Chiyus is making a profound distinction here. By standard cases of Odds, all we need is for the minority to disappear and be non-existent, leaving us the Majority as the sole remainder and thus we follow it. But here by the Nazir, the law is that every single hair on the head must be shaved! We cannot just say "well, we will consider the minority as not existing", since the law is that every hair must undergo shaving, and these minority hairs have not been shaved! Rather, it must be that the Torah has to introduce a new principle, that by the Nazir we do recognize the very real existence of the remaining hairs, but we say that since the majority had an action of shaving performed on them we will consider the minority as having had an act of shaving performed on them as well! And that is why the Torah had to specify that by Nazir every hair must literally be shaved because otherwise I could rely on this new principle of Minority Acts Like Majority!

Over Shabbos I came across an amazing Maharitz Chiyos which threatens to blow my entire understanding of Majority out of the water.

The Talmud in Nazir 42a quotes the verse in Bamidbar 6:9 which discusses a Nazir and says regarding the end of the Nazirite period that "He should shave his head on the day he becomes pure, on the seventh day he shall shave it". The Talmud infers from the repitition of mentioning shaving that the Torah is teaching us that the Nazir must shave his entire head. The question of course is why would the Torah need to teach me this: if it just said "He should shave his head", wouldn't we assume he has to shave the whole thing? From this the Talmud deduces that it must be that there is a general principle that Majority Equals Entirety, and that therefore by a Nazir if he merely shaved the majority of his head it would suffice, so therefore the Torah comes to tell me that he must shave the entire thing.

What emerges from this is that per the Talmud the verse in Bamidbar is teaching us 2 things: 1) Majority Equals Entirety, and 2) a Nazir does not suffice with shaving the majority of his head but must shave the whole thing.

This is where the Maharitz Chiyus asks a question: we know that everyone agrees that by an Experimental Probability we apply the verse of "After the majority to incline" to say that we follow the majority; if so, what is the difference here by a Nazir? Why is this not the same thing as an Experimental Probability since here, too, all the information is right in front of us: we see right here that most of the hair on his head is shaved and we see that some remain. So why is there any need for a dedicated verse here at all?

My understanding of the Maharitz Chiuyus's answer is as follows:

To interrupt from the Steipler for a minute...

As I'm working my way through his piece, I'm realizing that I need to rewrite the opening to the piece on Majority.

Instead of starting with the distinction between Voting and Odds, there is actually a more fundamental distinction between Uncertainty and State. Meaning, that Uncertainty is what we've been discussing until now: what to do when we are unsure of something: and the answer is we follow the majority. But State is something different: when the requirement is that something must be in a certain State, does being mostly in that State count? And the answer is yes, we follow the majority and consider the entire object to be in the majority state.

An example of State is the requirement to cut the trachea and esophagus when slaughtering an animal. They have to be in a State of "cut". So what is their state if only the majority of each one is cut? Do we say that a majority of "cut" is enough to consider it "entirely" cut or not? This is where we do apply majoroty and say that we consider the entire thing is considered as "cut".

The question with State is: what is the mechanism for considering that Majority Equals Entirety? Is it via Majority Negates Minority where we are saying that we consider it like there is no minority here at all, or is it via fiat, that we just say "even though we still recognize a minority here, still it has to be in one of two states, so lets just pick the majority state".

I would say that logically it does not work via Majority Negates Minority for two reasons: 1) since the Minority is visibly apparent and distinct it is difficult to say logically that we can consider it like its not there at all, and 2) we saw by Voting that we had to deduce the principle of Majority Negates Minority from the verse which implies it is something that requires a verse and not something that we can just casually apply.

Continued from yesterday...

The Steipler brings Rashi from the Talmud in Succah 9b which discusses the case of where a person bent down the branches of a tree until they were mixed in with the kosher Scach, where Rashi explains that the branches from the tree must be thoroughly mixed in until they are indistinguishable from the kosher Scach and thus the majority of kosher Scach negates the minority of invalid Scach.

So clearly we see that Rashi's own opinion is that we can only apply Majority Negates Minority when the elements are thoroughly mixed together. Given this, how can Rashi in Succah 2a maintain that the Scach and the gaps between the Scach are subject to Majority Negates Minority: by definition they are not mixed together and are distinguishable!

The Steipler says that there are two different type of ways in which a majority can negate a minority: 1) basic Majority Negates Minority by uncertainty, 2) Majority Equals Entirety which applies even when there is no uncertainty.

#1 is what we have already discussed at length.

#2 is quite an interesting principle. The idea is that when we are trying to define the status of something, if the majority of that thing is one status and the minority another - and there can only be a single status for the whole thing - we assign the status of the majority. For example, when slaughtering an animal, the trachea and esophagus must be cut: what happens if only the majority of each one is cut? Meaning, the majority of the trachea has a "cut" status, but the minority does not (and similarly for the esophagus). Based on the principle of Majority Equals Entirety, we would say that the entire status of the trachea is a "cut" status and therefore the animal is considered properly slaughtered.

These two concept are NOT RELATED AT ALL.

They are derived from completely different verses.

Concept #1 is derived from "After the majority to incline", and concept #2 is derived by the Talmud in Nazir 42a that quotes the verse that discusses a Nazir and says that on the seventh day of their completion ceremony they should completely shave their head; we derive from the Torah's added emphasis of "completely shave their head" that specifically by Nazir must the entire head be shaven, but everywhere else a simple majority would be considered Entirety.

And as the Steipler points out, a key difference with #2 is that the minority is not all mixed in with the majority. By the cut trachea and esophagus we can clearly distinguish between the segment that was cut and the segment that was not. The same goes for a Nazir, we can clearly distinguish between the cut and uncut hair. And yet we still say that we follow the majority!

I've finished the section on Experimental and Theoretical Probability.

The question is what to tackle next.

I still think that I'd like to address the big underlying principles before diving into more minutia in depth.

So far we've covered: the source of Majority by Voting, the source of Majority by Odds, the source of Majority Negates Minority, Experimental and Theoretical Probability and their sources, Artificial Uncertainty, and Resolvable Uncertainty.

I think what I need to do for the next couple days is to just work my way through some of individual sugyas and record my notes in these daily logs. Through the process of doing this I will be able to extract fundamental principles and work them into the primary Majority piece.

The Steipler in the Kehillos Yaakov brings the Talmud in Succah 2a where the Mishna says that if the sunlight in a Succah is greater than the shade (meaning that there are more gaps in the Scach than actual Scach), then the Succah is invalid. Rashi on that Mishna elaborates that the reason the Succah is invalid is because the minority of Scach is negated by the majority of gaps, and that therefore it is as if there is no Scach here at all.

The Steipler asks, how does Rashi make any sense? We know that the rules of Majority Negates Minority only applies when two things are completely mixed together and indistinguishable - thus creating uncertainty; but here, the Scach and the gaps between the Scach are clearly distinguishable and separate from each other!

Well, Trump won it.

Anyway...

As I dig into the topic of Tangible and Theoretical Majorities, it seems to be that we can define 5 levels of Majority in order of how impactful the "Majority" aspect is:

  1) A Blended Majority: this is where the Majority is literally blended together with the minority (for example kosher and not kosher wine that's mixed together and we literally cannot tell what part is what)

  2) An Embedded Majority: this is where we can distinguish between the majority and minority elements but they are still mixed together (I would say that a court is like this since we can tell which justices maintain which opinions)

  3) A Separated Majority: this is where the item under discussion is physically separate from the majority (like the 10 meat stores example where the meat is currently on the street)

  4) A Current Statistical Majority: this is where the item under discussion is NOT connected at all with some physical majority like in the examples above, rather the majority is a statistic (for example, statistically most people do not possess a Mortal Defect and thus if they are murdered the perpetuator is liable)

  5) A Future Statistical Majority: this is where the item under discussion has no majority associated with it at present, but statistically we know a majority will apply at some point in the future (for example, we allow two children who perform a Levirate marriage to stay together even though one of them may end up not hitting puberty since statistically the majority of children do attain puberty)

This is my own list of distinctions, but some of them do not seem to be generally recognized. For example, I consider the court to be an Embedded Majority, but the commentaries seem to treat it as a Blended Majority (I do not even think they recognize the idea of an Embedded Majority unless we say that that is what Majority Equals Entirety is). The Talmud also does not seem to distinguish between a Current Statistical Majority and a Future Statistical Majority, since in Chullin 11a/b examples of both are given for the singular concept an Theoretical Majority (even though to me a Future Statistical Majority is CLEARLY different since there is no physical majority at present whatsoever).

I had a huge breakthrough today!

I decided to try and research the general mathematical principles of probability. Now, as I'm sure that anyone with a decent mathematical education is going to already know, it turns out that the mathematical field of probabilities recognizes 2 types of probability: 1) Experimental Probability, and 2) Theoretical Probability.

Experimental Probability deals with tangible odds. For example, you spin a real coin 10 times and it comes up heads three times, you would conclude that the odds are 30% that a coin flip will result in heads.

Theoretical Probability is where you calculate the odds abstractly using math and logic. You look at the coin and determine that since it has two sides, a random flip would result in a 50% chance that it will result in heads.

It seems clear to me that THIS is the distinction that the Talmud is making!

A Tangible Majority is an Experimental Probability: we know for sure that in this specific case there are 10 stores, one of which is not kosher and nine that are kosher.

However an Theoretical Majority is a Theoretical Probability: there is not an exact, physical set of numbers we can examine, rather we calculate a logical and mathematical odds, for example that most children reach puberty.

So to summarize, the key difference is that by a Tangible Majority we know EXACTLY what the odds are, whereas by an Theoretical Majority we're just guessing.

And now it makes sense that an Experimental Probability is considered stronger than a Theoretical one: real life is complex and with a Theoretical probability there's always the chance that we're wrong somewhere in our reasoning, but with an Experimental Probability we are 100% certain as to what the odds are right here right now!

Election day!

Not to be discussed here since politics have seeped into everything else and I'd like to keep at least this separate.

I have finished the key pieces where I work out the sources for Voting and Odds, and additionally the sources for the general application of Odds and Majority Negates Minority.

The question is: what comes next?

I'd like to continue working methodically. Now we've established the fundamental basis of Majority, I'm not sure where to go next.

Should I tackle the Talmudic distinction between Tangible and Theoretical Majority? I would consider doing this if only I had a clear understanding of why this is such a fundamental distinction to the Talmud. Maybe if I understood the principles of the distinction better than I could start by first discussing what common sense would dictate should lead to the distinction, and then work my way step by step through the Talmud.

But first I'm going to have to tackle the Talmudic discussion there.

Alternatively I could jump straight to the Steipler's question from Succah, but I feel like that is diving in too quickly before establishing core principles.

Maybe I should discuss the concept of Majority Equals Entirety, but honestly I see that so far as really a completely different concept which based on the fact that it has a completely difference Torah source implies it is actually distinct from the principles of Majority discussed thus far.

So what I'm going to do is deep dive into Tangible and Theoretical Majorities and make sure I first gain a really solid understanding of the fundamentals there.

It's going to require a deep dive into the Talmud in Chullin 11a/b and working my way through the commentaries (I know that the Shev Shmatsa famously has some really solid pieces on this, I just have to work up the nerve to drop down into it).

Chilly today!

For the first time since the summer. I guess winter is real after all.

I had an interesting thought - pretty exciting approach actually - to how the Sages deduced Majority Negates Minority from the verse of "Incline after the majority".

What would we have done if the verse never existed? What decision would we make when there's a disagreement between Sages in court?

There is a concept in the Talmud that some things do not require a verse: rather, they are common sense logic. Without an explicit verse, what should I - an individual - do when a court minority is ruling one way and a majority is ruling another way?

Now, there are some cases maybe where you could say "Just be careful and follow the stricter opinion", but that would only work when the court is disagreeing over a matter of religious practice for example, but not if the disagreement is regarding ruling who some money belongs to let's say.

So for the cases where I can't just "be strict", what would I have done?

I would suggest that the Sages would have said that common sense would compel me to follow the odds!. Since I have to pick one of the two sides, the most logical thing to do would be to pick the side that the majority of Sages stand behind.

If so, then it can't be that the verse of "Incline after the majority" is just teaching me to simply follow the majority, since I would have known that already from common sense!

It must be that the verse is covering the scenario that common sense would not cover: meaning the cases where we could just "be strict" and follow the minority opinion where the minority is the stricter ruling, just to be safe.

So it must be that the verse is teaching me that legally when we are presented with a majority, we consider it like there is no minority opinion at all!.

I think this is a beautiful proof.

I've started trying to write up the concept of Tangible and Theoretical Majorities, but honestly it's hard simply because the distinction even in its simplest form in the Talmud seems so arbitrary.

The Talmud in Chullin 11a defines the case of a Tangible Majority. It's where someone finds meat on a street where there are 9 stores selling kosher meat and 1 selling non-kosher meat.

An example of Theoretical Majority is given as the assumption that minors attain puberty. Presumably the term "Theoretical" is because there are not physically present before us a majority of children who attain puberty; instead, the majority is just an abstract fact.

My problem with this is that while it's true that in the case of 10 meat stores they are physically near us on the street, but when it comes to children attaining puberty there are physically on the planet a majority of children who do attain puberty! Why does it make a difference whether the odds are constrained to a street or the planet? And if for some reason the geographical size does make a difference, then what's the cutoff point? What if there are 10 meat stores in the borough? The city? The country?

Maybe the terms are referencing the fact that by the 10 meat stores, the meat used to be physically inside one of the stores and now is moved out, which is a physical trigger that raises the question about odds. Whereas by a child attaining puberty, there was no "trigger" of "leaving" something which would cause us to have doubts and resort to determining odds. But why would this distinction be meaningful?

I know there are a lot of heavy hitters who discuss this (especially I believe the Shev Shmattsa), so I just need to go through the topic.

I just need the simplest possible way of explaining the distinction.

I ran out today to buy a set of Birchas Shmuel.

I only really need the first volume where he has a piece at the end of Yevamos on Majority.

Alas, I am unprepared to spend $69.99 on the 4 volume set. Maybe I'll pick it up later, but for now I'll have to go through a copy at BMG.

I had nearly finished the piece, which I saw in a sefer that was lying around on Simchas Torah, but essentially he deals with a Majority as regards a mikva that has invalid water mixed in.

Apropos of that topic he also delves into Majority as a more general topic, and I belive it was here that I got turned on to the source in Chullin 11a/b that deals with a "Tangible" Majority and an "Theoretical" Majority.

On a side note, it's interesting that Rashi and the Rosh are so quick to ascribe the verse of "After the majority to sway" to be the source of the concept of "A majority negates". After all, we see in Chullin 11 that we are only able to deduce from this verse as regards a Tangible Majority, but we have to search for a different source for an Theoretical Majority, presumably because of the concept that we try to minimize novelty when interpreting a verse. But if so, then why not say the same for the novel concept that a majority does not merely provide a ruling but even negates the minority. Why do we not need a separate dedicated verse for this novel concept?

It was also in the course of re-reviewing the sources I've been compiling over Simchas Torah, that I realized that the fundamental categorization I've made with Majority between Voting and Odds is incorrect. There is another element which needs to be included, and that is State: for example, when the majority of the neck of an animal is cut we consider the state as if the whole thing was cut and thus it is a valid slaughtering. This obviously does not slot into either Voting or Odds. And in fact, the Kehillos Yaakov says the source is different as well (not from the verse of "After the majority to sway", rather from the verses regarding a Nazirite who must shave all their hair after their Naziritic period is complete, from which we deduce that here is where all the hair must be cut, but that elsewhere a majority would suffice since a majority is like the entirety).

Although now that I re-read the preceding paragraph, I suppose that State could be slotted under Voting. They can both be considered Ruling I suppose, which is maybe the better term for it. I'll need to think about it. (And if this is the case, then it's interesting to think how it may relate to the Rogatchover's opinion that Majority is a "legal" device for "Rulings" and not merely a scientific "odds").

I just thought of a proof to the Rogatchover.

The Mishna in Sanhedrin 1:6 says that from the verse in Shemos 23:3 that "You shall not follow a majority to do ill" we deduce that you should follow a majority to do good. Meaning, that already from the first part of the verse we can lay down a rule that so long as the majority is not convicting (which is how the Mishna is interpeting "ill"), we do have to follow the majority rule. The Mishna says that if so, however, then why does the verse end with "Incline after the majority" (meaning, why repeat the rule to follow the majority)? Answers the Mishna, it must be that the second statement is teaching me that we always follow the majority, even when it comes to convicting; and the first half of the verse is just telling me that I cannot follow a simple majority of one to convict, rather I must follow a majority of at least two.

The thing is, that if Odds are just straightforward logic, then how does having a majority of 2 make any difference?!

If it was a supermajority like 90% or something then I could understand, but a majority of 2 (especially in the context of the court size that the Mishna is discussing which is 23) is surely not adding any real improvement on the odds!

This is much better though according to the Rogachover's understanding, that Odds are purely a "legal" matter, a construct commanded by the Torah and not based on the physical logic of actual odds