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:

mechtzaORmechtza=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*.

(mechtzaANDmechtza) OR (mechtzaANDmechtza) ≠mechtzaAND (mechtzaORmechtza)assurORassur≠mechtzaANDmutarassur≠metchtza