This will only be of interest to people who care about Symbolic Logic and about the rules of birur, of resolution of doubt in halachic questions. But I found something I wrote back in 1994, and didn’t want to lose it, so I’m blogging it here. Hopefully I will have time to put up something of more general interest in the near future.

Earlier in this series we distinguished between cases that are qavu’ah and those where we say kol deparish, that is, those doubts that are between established items, and those where we have an amorphous set of items. The basis of my theory is a statement by Rabbi Aqiva Eiger (shu”t #136) that there are two types of doubt, and each has its own mechanism for birur, for clarification.

The case of qavu’ah is one where the reality was once established. So in principle, there is a specific halakhah assigned already to this case. The doubt is in what that halakhah is. In this situation, we do not invoke rules like rov (majority), and every doubt is treated identically to an equal one.

The case of kol deparish is one where the reality was never established; this item never before stood out from the rest of the set. Therefore, we are assigning a halakhah to a case where the physical reality is in doubt. Here, majority is allowed.

See the posts in the above links for more detail. Here I want to add a mathematical analysis.

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	true	true

Aside from “OR”, it defines other operators, like “AND” (true only when both clauses are true) …

AND	false	true
false	false	false
true	false	true

… “NOR” …

NOR	false	true
false	true	false
true	false	false

… “NOT” …

	NOT
false	true
true	false

… 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)

This sounds more complicated than it is. 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).

Or, in English: Saying, “I am either missing work or missing my dinner” is the same as “I am not both attending work and having my dinner.”

Unlike boolean algebra, Qavu’ah Logic (hereafter QL) has three states — mutar, mechtza, asur (when speaking of prohibitions; patur, mechtzah, chayav when speaking of obligations). Kol qavu’ah kemectzah al mechtzah dami — all doubts about something that once had an established metzius is like half-vs-half, and we therefore respond to each half Safeiq Logic (SL) has 5, because it adds mi’ut and rov.

The case of sefeiq sefeiqa is much like a symbolic logic OR operator. You have two questions. If either were resolved “mutar” the result would be “mutar“. If both are assur, than the resulting ruling is assur.

	assur	mutar
assur	assur	mutar
mutar	mutar	mutar

Under QL, we add the case of unknown and therefore, a sefeiq sefeiqa would yield this truth table:

QL	assur	mechtza	mutar
assur	assur	mechtza	mutar
mechtza	mechtza	mechtza	mutar
mutar	mutar	mutar	mutar

Once we say that safeiq 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 sefeiq sefeiqa. In a sefeiq sefeiqa, 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. Sefeiq sefeiqa is much like the Boolean logic notion of OR.

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 bemaqom safeiq – a minority in a situation where there is already a doubt, keman deleisi dami – is as though it does not exist”, and sefeiq sefeiqa.

SL	assur	mi’ut	mechtza	rov	mutar
assur	assur	mi’ut	mechtza	rov	mutar
mi’ut	mi’ut	mi’ut	mechtza	rov	mutar
mechtza	mechtza	mechtza	rov	mutar	mutar
rov	rov	rov	mutar	mutar	mutar
mutar	mutar	mutar	mutar	mutar	mutar

(This table also presumes the opinion of the Rashba, the Sheiv Shemaatsa, et al, who hold that sefeiq sefeiqa is a kind of rov — thus the entry at the center of the table:

mechtza OR mechtza = rov

The R’ Shimon Shkop holds that sefeiq sefeiqa works because one safeiq reduces the question to a derabbanan, and the second is a safeiq derabbanan lequlah. See item #2 in this entry. In which case, that cell of the table should read “kemechtza derabbanan” a new value that violates this whole symbolic logic notion. However, I didn’t know of this machloqes back in 1994. So, let’s continue aliba deSheiv Shemaatsa…)

Negation (NOT) is defined intuitively, the gemara assumes a majority indicating A is equivalent to a minority indicating not-A.

	NOT
assur	mutar
mi’ut	rov
mechtza	mechtza
rov	mi’ut
mutar	assur

In parallel to sefeiq sefeiqa toward leniency is a sefeiq sefeiqa as grounds for stringency. In that case, something is permissible only if both criteria are found to be in the permissive possibility. It seems to be the direct reflection of the sefeiq sefeiqa 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)

But what if we

Let A = B = C = mechtzah

The left describes two sefeiq sefeiqos lechumerah. The right is A “AND” a sefeiq sefeiqa lequlah.

(mechtza AND mechtza) OR (mechtza AND mechtza)  ≠  mechtza AND (mechtza OR mechtza)
                assur OR assur                  ≠  mechtza AND mutar
		    assur                       ≠        metchtza