[Avodah] Halachah and De Morgan's Law

[Micha Berger via Avodah]

There are two laws in boolean algebra (and therefore also in propositional
logic) discovered by 19th cent British mathematician Augustus de Morgan.
Known, logically enough, as de Morgan's Laws.

One, which I am just including for completeness, is that
	not (A or B)
is the same thing as
	not-A and not-B
To clarify with an English example:
	John did not go to the store or the shop.
Means the same thing as:
	John did not go to the store, and he did not go to the shop.

The other law is the one I thing I stumbled across in the Arukh haShulchan.
This rule looks similar, but has to do with
	not (A and B)
is the same thing as
	not-A or not-B
To clarify with an English example:
	John did not go to both the store and the shop.
Means the same thing as:
	John did not go to the store, or he did not go to the shop.
	(And possibly neither.)

Here's how the AhS invokes it in YD 188:141. (That's not a typo, YD 188
has 158 se'ifim of AhS.) The SA se'if 40 discusses a case where a father
is mashbia his son not to borrow money unless Re'uvein and Shimon agree
to it. The son may take a loan if either Re'uvein *OR* Shim'on agree.
(This is based on the Rashba, and it means we hold like R' Yonasan
over R' Yoshiah in a recurring machloqes.)

The ra'ayah is from "aviv ve'imo qileil", where the issur is against
cursting either, not both. So you see "ve-" means or.

The BY is bothered -- yes that shows the vav hachibur could be used for
or, but who said it /has/ to mean "or"? Like "venasan lakohein hazeroa
vehaleichaim vehaqeiva" and numerous other examples where it means "and".
Vesafeiq nedarim lechamir!

So AhS se'if 140 proposes the possible case where Re'uvein died, and
we can assume that the father did not mean that borrowing was prohibited
if only one could be asked.

But se'if 141 draws a distinction between a chiyuv and an issur. In a
chiyuv, yes you do have to give the kohein all of the matanos listed.
But in an issur -- like meqalal aviv ve'imo -- "ve-" by default means
"or", that X is assur and Y is assur, ie it is assur to X or Y.

(Unless, as R Yonasan says, the pasuq says "yachdav" to be clear that
the issue is only when both.)

Similarly the "unless" of our neder.

De Morgan's law swapping "and" and "or",no?


Tir'u baTov!
-Micha

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