Volume 11 Number 92
Produced: Mon Feb 21 17:55:05 1994
Subjects Discussed In This Issue:
Safek and Rov
[Mitch Berger]
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From: <mitch@...> (Mitch Berger)
Date: Mon, 31 Jan 94 15:57:23 -0500
Subject: Safek and Rov
Moshe Goldberg (in v11n39) writes that "Only recently was the meaning of
"rov" changed to mean 'most'." Harry Weiss (in V11n46) continues on a different
direction about "bitul b'rov". I wanted to contribute my 2cents (okay, it's
more like $1.50) about the mechanics of rov. It's hard to know how much
knowledge of boolean algebra the typical reader of this forum knows. While
boolean algebra is certainly not a prerequisite for following mail-jewish,
it usually is for getting a computer account. Well, if I lose somebody,
please drop me a line (<mitch@...>).
There are several indications that the logic used by the sages of
the Gemara was not Aristotelian. In particular they allowed both a
proposition and its negation to be (at least partially) true at the same
time.
For example, T. Succah discusses using the esrog used for the
mitzvah for something secular. The conclusion the Gemara reaches is
that you can use your esrog starting the evening after the day you used
it for a mitzvah. The reason given is that once the esrog is prohibited
for personal use part of the day, it is prohibited the entire day. So,
the esrog is prohibited the entire day you used it. This would include,
to be stringent on a biblical prohibition (safek d'Oraisa lichumra) bein
hashmashos (twilight - a period of time which is "safek yom safek
laylah", perhaps day, perhaps night, i.e. part of the next day) of that
day. Bein hashmashos might be part of the next day also. Once it is
prohibited bein hashmashos, it is prohibited the entire next day.
How can you be strict and say simultaneously that it is part of the
first day AND part of the second? It is tarta disasrei (a paradox)? It
seemed to me that the phrase "safek yom, safek layla" meant that it is
simultaneously both day and night. This to sounded to me like a
superposition of states in quantum mechanics. Further, if this is what
safek means in this context, can safek mean that in general? Usually
safek is taken to mean "I know it is either A or B, but I don't know
which." Perhaps it means "A and B in superposition" in other cases also!
When R. JB Soleveitchik taught this piece of Gemara in Boston during
the 1982 Yarchei Kallah Shiur, he commented that it appears that
Talmudic logic is not boolean. Similarly, in Ish HaHalachah he comments
that it is not a simple two-valued logic.
Another case, given in Chullin, is that of three pieces of fat, one
of them is non-kosher, although we don't know which (but not kavu'ah,
established, see below). Each of the three pieces can be eaten.
(According to one opinion in the Gemara, they can all be eaten together
in one dish!) Had safek meant that we treat each as an uncertainty, this
result would be illogical - since we know for sure that he ate
non-kosher meat at some time. Instead, we view each piece as
kosher+treif in superposition. Each piece can be eaten since each has
the kosher state in greater intensity than the non-kosher state.
This idea is seen in philosophy as well as halachah. We have a
principle in the Gemara that "eilu va'eilu divrei Elokim Chaim hein".
There was a protracted debate between Beis Shammai and Beis Hillel. A
voice came from the heavens saying "both these and those are the words
of the Living G-d (or perhaps: the G-d of life)" (Eiruvin 13b).
Similarly, the Gemara (Hagigah 3b) is concerned about the person who
will note when "those [Rabbis] prohibit, yet those [authorities] permit
[the very same thing]... how can I possibly learn Torah today?" The
answer is found in the words of Koheles 12, "Nasnu meiRo'eh echad - both
views were given by the same shepherd." Whenever there is a disagreement
between two opinions reached by valid halachic methods, both are valid
(even though only one becomes the accepted ruling).
It is hard to understand how two people can disagree, one says A,
the other says not-A yet both are correct. Somehow the Will of the
Creator is a superposition of the two conflicting opinions, and each
Rabbi stresses a different piece of a more complex whole.
There are two rules for resolving doubt (in the absence of evidence)
in the Gemara. The first says: Kol diparish meirubah parish - whatever
separates itself [from the group], [can be assumed to be] separated from
the majority. When in doubt, follow the majority. The other is: Kol
kavu'ah kimechtza al mechtza dami - All [doubts related to] things that
are established are as though they were 1/2 and 1/2. A doubt is an
unknown, and we live it unresolved - with no consideration of majority.
To give the examples used in Chullin 95a: In a town where 9 butchers
sell kosher meat and one sells non-kosher meat, the Kashrus of some meat
in in doubt.
1- If the meat was purchased, but the person forgot from whom, it is
forbidden;
2- If it was just found on the street, it is kosher.
This is because in the first case, the meat is kavu'a. Therefor, it is
treated as a 50:50 doubt, and since the issue is of Divine origin
(Mid'Oraisa), we play it safe. In the second case, we follow the
majority, and say it is kosher, just like in the case of three pieces of
fat we quoted earlier.
Similar to to the halachah of kavu'ah is the way we treat a case in
which eidim, witnesses are presented. The Shev Shma'atsa (Shma'atsa 6,
Ch. 22) says about cases where each side presents eidim in its support:
Since we have two [eidim] and two [eidim] in all cases our
safek is an equal safek, even where we have a majority.
Perhaps based in this is another odd feature of the laws of eidim.
Trei kimei'ah - two witnesses have the same credibility as 100. A case
can not be decided by who has the plurality of witnesses. Again, we
don't follow majority.
There is a principle in the laws of Beis Din call migo. If a person
has a choice of two claims to win his case, and he makes the weaker of
the two, he is believed. We say that had he wanted to lie, he would have
chosen the other alternative. For example, one litigant claims that a
person borrows money without a contract. The other says he borrowed it
and returned it. We believe the second person, since had we wanted to
lie, he would have said the loan never occurred.
The exception to this is where there are eidim. Migo bimakom eidim
lo amrinan - we don't say the law of migo in opposition to eidim.
Witnesses have greater credibility than that gained be migo.
Tosafos, on Baba Kama 72b, rule in the case where one side has eidim
in its support, and the other has both eidim and migo. Had the second
side come with two sets of eidim, he would have no more credibility than
the first (trei kimei'ah). Migo is weaker than eidim, so they conclude
that adding migo to his case would not help him.
The last notion we will look at is that of Chazakah. There are two
types: Chazakah Dimei'ikarah, in which we assume that something did not
leave its original state; and Chazakah Disvara, where something is
assumed based on nature or human nature.
For example, if a mikvah is found to be deficient of the required 40
se'ah of water, we do not say that everyone who used the mikvah since it
was last checked must return to the mikvah. Rather, by chazakah
dimei'ikarah, we assume that the mikvah remained in its kosher state as
long as possible.
Examples of Chazakah Disvara are such truisms as: A person would not
since, for gain that is not his; People don't pay debts before they are
due; etc...
According to the Shev Shma'atsa (ibid), only chazakah dimei'ikarah
has authority in the face of two conflicting pairs of eidim. Chazakah
disvarah, like rov, adds no credibility.
Kiddushin 64a rules if one side has a migo supporting it, and the
other one a chazakah dimei'ikara, we follow the migo. So, Chazakah
dimei'ikara is weaker than migo. This leads to a very strange result.
Chazakah dimei'ikara, the weaker, has authority in the case of
contradicting pairs of eidim, and migo, the stronger, doesn't!
To explain this, R. Akiva Eiger (Sh'eilos Utshuvos Ch. 136)
delineates between two types of deciding principles. (As a warning, this
is the only idea taken from R. Akiva Eiger. Where ever the S"S
disagrees, I had chosen the S"S's ruling. I didn't work over the details
of how my model could accommodate other opinions.)
He distinguishes between rules for determining what actually
happened from rules that determine how to act when we can't resolve what
happened. Migo and eidim determine reality, therefor we can say trei
kimei'ah for either. Chazakah dimei'ikarah works on a different level,
and therefor can not be compared.
R. Tzadok HaKohein, in Resisei Leylah (sec. 17), applies similar
reasoning to the mechanics of halachic debate. On the subject of debates
he notes, "Whenever a new thing found about the Torah by any wise
person, simultaneously arises its opposite.... When it comes to the
realm of action (po'al) it can not be that two things true
simultaneously. In the realm of the mind (machshavah), on the other
hand, it is impossible for a man to think about one thing without
considering the opposite."
From this we can classify two logics: Safek logic (SL) to be used to
represent reality, and Kavu'ah logic (KL) that is used in analyzing
halachah. SL is superpositional, and takes into account rov - which
state is more intense. KL is boolean, and applies after the object has
been observed - either by eidim, or because it is kavu'ah. The observed
reality, to my mind, belongs in R. Tzadok's realm of po'al. A person can
not witness two opposite things occurring. Yet, the unobserved is pure
speculation. It lies within the world of machshavah.
Once eidim testify on a given issue, it operates within KL. Therefor
the issue of majority is irrelevant. We don't listen to one side because
it has more eidim - t'rei kimei'ah.
Migo works within SL. It tells you which side is more likely to be
telling the truth. Chazakah dimei'ikarah, in KL. It is only a rule to
tell you how to act. Therefor, if we have one side supported by a migo,
we resolved the safek before we need to rely on KL. The chazakah
presented by the other side is irrelevant, so the migo is believed.
However, once eidim come, the doubt is forced to KL. We don't care
about which side is more likely. Once we are within KL, the migo is the
one that is irrelevant, and the chazakah dimei'ikarah is primary.
Once we say that safek is a valid answer, and not just a way of
saying that the answer is unknown, we have to understand what is meant by
a sfek sfeka. In a sfek sfeka, the status of a case is subject to two
doubts. If the resolution of either doubt were "mutar" the ruling as a
whole is mutar. Sfek sfeka is much like the Boolean logic notion of OR.
Boolean logic takes the approach that logic could be understood as a
type of algebra. The complex statement "A OR B" is true if either "A" or
"B" (or both) is found to be true. This is usually shown as a table,
much like the addition or multiplication tables:
OR || false | true
=====++=======+=======
false|| false |
-----++-------+
true | true
Aside from "OR", it defines other operators, like "AND" (true only
when both clauses are true), "NOT", "NOR", etc... Like algebra, it
defines distributive rules, associative rules, and so on - way of
simplifying our "expression". One pair which we will look at is de
Morgan's rules.
de Morgan showed that (NOT A) AND (NOT B) = NOT (A OR B). It helps
to give an example. Saying "I am not going to the store, and I am not
going to the school" is equivalent to saying "I am not going to the
store or to school." Similarly, (NOT A) OR (NOT B) = NOT (A AND B).
In much the same way, we can make a more complicated table for our
5-state SL. To make this table, I used the rules that "mi'ut bimakom
safek - a minority in a situation where there is already a doubt, k'man
dileisi dami - is as though it does not exist", and sfek sfeka.
OR || asur | mi'ut | safek | rov | mutar
========++=======+=======+=======+=======+========
asur || asur | | | |
--------++-------+ + safek + rov +
mi'ut || mi'ut | * | * | * - mi'ut k'man
--------++-------+-------+-------+-------+ dileisi dami
safek || safek * | rov |
--------++-------+-------+-------+
rov || rov * | mutar
--------++-------+-------+
mutar ||
Negation (NOT) is defined intuitively, the gemara assumes a majority
indicating A is equivalent to a minority indicating not-A.
| not
========+=======
asur | mutar
--------+-------
mi'ut | rov
--------+-------
safek | safek
--------+-------
rov | mi'ut
--------+-------
mutar | asur
In parallel to sfek sfeka toward leniency is a sfek sfeka as grounds
for stringency. In this case, something is mutar only if both A AND B
were resolved mutar. It seems to be the direct reflection of the sfek
sfeka we outlined above. The notion in boolean logic:
NOT (A AND B) = (NOT A) OR (NOT B)
de Morgan's law holds for SL as well.
The distributive law, however, doesn't. In boolean algebra,
(A and B) or (A and C) = A and (B or C)
Let A = B = C = SAFEK.
The left describes two sfek sfekos lichumrah. The right is A and a sfek
sfeka likulah.
ASUR or ASUR =/= SAFEK and MUTAR
ASUR =/= SAFEK
This is reassuring, since the quantum mechanics defies also defies the
distributive law.
If you're interested, I'd check out (aside from the things I had
full references for):
Shev Shma'atsa - too many places to list, check the index
Higayon, Edited by Moshe Koppel, Ely Merzbach; Aluma, Jerusalem
1989 (others if there are other editions)
Safek and Sfek Sfeka, R. Dr. Leon Ehrenpreis, Gesher Vol 8,
Yeshiva University, New York
Mathematical Logic, Martin Gardner (Discusses multi-valued logic.)
Micha Berger
<mitch@...>
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End of Volume 11 Issue 92
