Halakhah and Phenomenology – Symbolic Logic

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:

ORfalsetrue
falsefalsetrue
truetruetrue

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

ANDfalsetrue
falsefalsefalse
truefalsetrue

… “NOR” …

NORfalsetrue
falsetruefalse
truefalsefalse

… “NOT” …


NOT
falsetrue
truefalse

… 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.

assurmutar
assurassur mutar
mutarmutarmutar

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

QLassurmechtzamutar
assurassurmechtza mutar
mechtzamechtzamechtzamutar
mutarmutarmutarmutar

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.

SLassurmi’utmechtzarovmutar
assurassurmi’utmechtzarov mutar
mi’utmi’utmi’utmechtzarovmutar
mechtzamechtzamechtzarovmutarmutar
rovrovrovmutarmutarmutar
mutarmutarmutarmutarmutarmutar

(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
assurmutar
mi’utrov
mechtzamechtza
rovmi’ut
mutarassur

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
		    assurmetchtza

A Quantum of Time

There are two halachic indivisible units of time. When it comes to interruption, or for defining a single statement (e.g. when correcting oneself in davening) the unit is tokh kedei dibbur — within the time it takes to say ["Shalom eilekha Rebbe uMori"], a greeting of 4 words consisting of 10 syllables. For calendrical calculations, the unit is the cheileq, 3-1/3 seconds. My instinct is that the two are intended to be equal — it takes around 3-1/3 sec to say 10 syllables, and for someone who isn’t timing themselves with a stopwatch, the tokh kedei dibbur definition is more useful.

Here is a possible explanation for the halakhah:

Hugs Follow a 3-Second Rule
by Rebecca Kessler on 28 January 2011, 1:23 PM
Science Magazine

Ever wondered how long a hug lasts? The quick answer is about 3 seconds, according to a new study of the post-competition embraces of Olympic athletes. But the long answer is more profound. A hug lasts about as much time as many other human actions and neurological processes, which supports a hypothesis that we go through life perceiving the present in a series of 3-second windows.

Crosscultural studies dating back to 1911 have shown that people tend to operate in 3-second bursts. Goodbye waves, musical phrases, and infants’ bouts of babbling and gesturing all last about 3 seconds. Many basic physiological events, such as relaxed breathing and certain nervous system functions do, too….

© 2011 American Association for the Advancement of Science. All Rights Reserved.