[Thursday night] I don’t generally stay up all night for tikkun leil hoshanah rabba. So I feel awfully guilty about staying up tonight for a certain non-learning event that begins at 2:19 AM Israel Standard Time. I offer the following divrei torah by way of penance.

[Next morning] My intention was to type up the torah while I watched the game but I couldn’t tear myself away from the game screen (I don’t have cable so I watch on mlb.tv). So I ended up scribbling notes on paper and am typing it up now. (I should mention that vasikin was scheduled for 5:00 and at 4:54, Guillermo Mota went to a 3-0 count to the potential tying run with Albert Pujols on deck. The tragic unfolding of events, including my screwing up my last chance at redemption by blowing vasikin, seemed utterly inevitable. But, B”H, Mota found the plate, got out of the inning and I headed off to shul. Hamtakas hadin mamesh.)

Anyhow, the torah I want to discuss involves a practical question that an old friend tells me has been troubling a prominent talmid chakham in the U.S. It seems that there is a well-established statistical rule that some non-negligible fixed percentage (let’s say 10%) of milking cows turn out to be treifos, though at the milking stage we have no way of knowing whether a given cow is one of the treifos or not. In any given randomly selected large herd of cows it is very likely that the percentage of treifos is close to 10%. The combined milk of the herd therefore includes a high proportion of non-kosher milk, certainly well in excess of 1/60. Should industrial milk therefore be ruled non-kosher?

The question is interesting not only because of its practical consequences but also because it forces us to think about some very fundamental questions involving the use of uncertain reasoning in halacha. The first thing that should be pointed out is that no legal system actually employs probabilistic reasoning in the ideal sense that a math geek might think appropriate. A math geek might reason as follows: conclusion C holds if and only some specified combination of premises hold. So let’s estimate the probabilities of each of the premises and glom them together to compute some estimated probability for C. In real life, this never works. First of all, except for certain very artificial cases, we don’t have plausible numerical estimates of probabilities for the premises. Second, we can only easily glom the probabilities together by assuming that the various premises are independent of each other, which they almost never are. These kinds of calculations are invariably messy, if not downright intractable, and the results (“there’s a 78% chance that he is the killer, is of sound mind and did not act in self-defense”) don’t often sound especially convincing.

What we actually do in real life is to look at each premise in isolation and decide if the probability that it is true is reasonably close to 1 or not. Once we do that the rest is simple. This is pretty much how it works with halacha as well. Let’s consider some increasingly difficult examples of how halacha deals with uncertainty and mixtures, until we can reach some conclusion about the milk. (I admit that I could just jump right to the end but I want to make some important intermediate points anyway.)

Let’s start with a herd of cows about which we know for certain that exactly 10% of the cows are treifos, though we don’t know which. We milk a single cow. Is this milk kosher? In any case of a mixed (e.g., kosher and non-kosher) set of objects, in which we can’t distinguish the kosher objects from the non-kosher ones, the first question is whether we can simply determine that the minority is batel. If so, our problem is solved. All the objects have the majority status. Let’s start numbering our conclusions:

1. When bitul applies, each object has the status of the majority.

Let’s think about how seriously, we are willing to take this conclusion. Surely, if we choose a single object from a set consisting of mostly kosher objects, there is a chance greater than half that it is kosher. But what if we start eating our way through the set? The odds that at least one of the ones we eat is non-kosher are very high and getting higher with each bite. Nevertheless, there is no dispute that as long as we are not necessarily eating a non-kosher object, we can keep eating. More precisely:

2. If bitul applies to a set with k non-kosher objects (and more than k kosher objects), we can certainly eat all but k objects from the set.

Can we eat more than that? To take it to an extreme, can we eat all the objects in the set? The argument against is obvious: even if we don’t know which of the objects is non-kosher, it is impossible that we would not be eating some non-kosher object. (Let’s call this argument the “impossibility argument”.) The argument in favor is that once bitul is invoked the entire set is regarded as a kosher set. This point is disputed by the rishonim with the Tosfos Rid taking the stringent view and the Rosh taking the lenient view.

3. If bitul applies to a set with k non-kosher objects constituting the minority, there are varying opinions regarding whether we can eat more than all but k objects from the set.

In any event, in the particular case of the cows that we consider here, bitul does not apply because bitul does not apply to sets of animals.

What do we do when bitul does not apply? Here we need to distinguish between two cases. If we select a cow from an “integral herd”, then the principle of kavua is invoked. Selecting a cow from an integral herd means that the herd has not broken up and the cow selected had not broken off from the herd prior to its selection. According to this principle, the set itself is irreducibly mixed (it contains both kosher and non-kosher objects) and the object selected inherits that mixed (mechtza al mechtza) status. In most cases, including our cow scenario, this would mean that we rule stringently.

4. If we select a cow from an integral mixed herd, the cow inherits the “mixed” status of the herd and, in practice, its milk would be forbidden.

Suppose now that three quarters of the herd has remained whole and we select one cow from this depleted herd. Do we still invoke kavua? If we are willing to ignore certain complicating factors, we can answer unequivocally that kavua does not apply here because the depleted herd lacks the crucial property of ischazek issura. That is, we can’t say that the herd is a mixed herd since it may be that all the cows in the depleted herd are kosher. (Recall that we know that exactly 10% of the cows in the original herd are non-kosher and that 25% of the cows in the original herd have left the herd.)

5. If we select a cow from a herd that does not necessarily include any non-kosher cows, the principle of kavua is not invoked.

So let’s consider now what happens in those cases where kavua is not invoked. If we do not select a particular cow from an integral set, that is, if the herd has broken up prior to our selecting a cow or if the cow in question had wandered off prior to our selecting it (parish), then the selected cow is simply deemed to be from the majority (mi-ruba parish).

6. If we select a cow that originated in the herd but is not, at the time of selection, part of the mixed integral herd, then the cow is deemed to be from the majority.

Now let’s consider the analogs of cases 3 and 4 above. Can parish be applied multiple times to cows from the same set? It is clear (see, for example, Zevachim 74b) that in fact it can. I don’t know of any source, however, that deals with the application of parish to eating the entire set, but it is reasonable to assume that, at least in this regard, the application of parish would be analogous to the application of bitul.

7. Principle 2 and 3 above apply to parish as well.

Until now we have discussed the cows as discrete entities. Let’s now complicate matters yet further by considering what happens when the milk from the cows is mixed. Imagine that we have milked all the cows in the herd and mixed the milk together. First, suppose that the herd remained an integral herd, so that the principle of kavua applies to the herd. The milk would certainly be forbidden.

8. If the principle of kavua applies to the herd and the combined milk includes more than 1/60 non-kosher milk, the milk is not kosher.

Suppose, though, that less than 1/60 of the milk is from non-kosher cows. If we simply apply the rules of bitul relevant to homogeneous (e.g., liquid) mixtures, the milk would be permitted. However, the milk was obtained from a set to which kavua applies so that each animal in the set actually has the status mechtza al mechtza. That is, we don’t actually have more than 59/60 of the herd that are kosher but rather an entire herd of cows with a status of “mixed”. In such a case, I conjecture (somebody please prove either that I’m right or that I’m wrong) that the rule of bitul in homogeneous mixtures does not apply.

9. If the principle of kavua applies to the herd, the combined milk is not kosher even if less than 1/60 is from non-kosher cows.

Now suppose that we have mixed the milk of the entire herd but that kavua does not apply, but rather either bitul or parish applies. Does the dispute mention in principle 3 (and implicitly in principle 7) above apply here as well?

Let’s begin with the easy case in which it is known that less than 1/60 of the milk is not kosher. The Rosh, who would permit eating the whole herd, would certainly permit the milk. And, in fact, even Tosfos Rid, who ruled that one couldn’t eat the whole herd, would simply invoke the rule of bitul in homogeneous mixtures to permit the milk.

10. If kavua does not apply to the herd and less than 1/60 of the combined milk comes from non-kosher animals, the milk is kosher.

But now let’s consider the harder case where more than 1/60 of the mixture is from non-kosher animals. Tosfos Rid, who ruled that one couldn’t eat the entire herd, would certainly have no grounds to permit drinking the milk. But the Rosh, who permits eating the whole herd on the grounds that every cow in the herd is treated as kosher once bitul/parish has been invoked, could argue that the milk of the whole set is kosher by extension. This would be analogous with the conjectured principle 9 in that once the status of the set has been established, the rule of bitul in homogeneous mixtures does not apply.

11. If kavua does not apply to the herd and more than 1/60 of the combined milk comes from non-kosher animals, the dispute of principle 3 applies.

Let’s complicate matters a bit. Suppose now that we have not taken milk from the entire herd but rather from some fraction of the herd. Assume that the fraction of milked cows is small enough that there are not necessarily any non-kosher cows among those that are milked. Then two conclusions follow. First, kavua cannot be invoked with regard to the milked set since we do not have ischazek issura (see principle 5). Second, Tosfos Rid could not invoke the impossibility argument. Thus, according to both Tosfos Rid and the Rosh, once the cows in the milked set are deemed kosher, the milk would be kosher by extension and the rule of bitul (or non-bitul) in homogeneous mixtures would not be applied here.

12. If a majority of the herd is kosher and a given sub-herd does not necessarily include any non-kosher cows, then the combined milk of that sub-herd is kosher.

Until now, we have considered cases where we know the proportion of non-kosher cows in the herd. Suppose now that we only know that generally about 10% of cows are non-kosher but we know nothing about this particular herd. Which of the above principles carry over?

Let’s start with a simple question. Can we drink milk from a random cow given that we know nothing about the kashrus of this cow other than the general statistical rule that 90% of cows are kosher? Indeed we can since this statistical rule is a recognized ruba de-leisa kaman [=RDLK]. (In those cases where there is a presumption that contravenes the RDLK and the RDLK is not an overwhelming one, R. Meir would not invoke the RDLK (chaishinan le-miuta). But there is no contravening presumption in our case and, in any event, we don’t hold like R. Meir.)

13. If most cows are kosher and we know nothing special about a given cow (or about any closed set of which it is a member), we can presume that the cow is kosher.

Suppose that we have some very large herd of cows about which we know nothing special beyond the general statistical rule that 90% of cows are kosher. For convenience, let’s call this herd H. Is the mixture of milk obtained from H permitted? This is the very practical question from which we began. It is true that it is highly likely that the mixture contains about 10% non-kosher milk. But does this matter?

In fact, this case is almost identical to the case of principle 12. Think of the herd in principle 12 about which we know the proportions of kosher and non-kosher cows as being the set of all (real or hypothetical) cows. Think of H as being the sub-herd in principle 12 from which we have taken our milk. Regardless of how large H is, it is theoretically possible that H includes no non-kosher cows. Thus, there is no ischazek issura, no kavua, and no impossibility argument. Consequently, just as in principle 12, the entire set H is regarded as a kosher set. We simply apply the RDLK to each cow in H. As in principle 2 (for bitul) and it’s analog in principle 7 (for parish), there is no limit to the number of times that we can invoke RDLK so long as it does not run up against an impossibility argument.

14. If most cows are kosher, then the combined milk of a herd of cows is kosher.

This resolves the practical question that motivated this discussion. I should add that, since I am utterly unqualified to pasken anything, I would never be so audacious if not for the fact that Rav Usher Weiss has paskened leniently on this exact matter. See Minchas Usher on Sefer Shemot, Siman 43.

Also, I have no negiah here. I am lactose intolerant and have not eaten dairy products in 18 years.