# "Don't satisfy both" and "Don't satisfy either"

Are these two statements equivalent?

If you don't satisfy either of the conditions A and B

If you don't satisfy both of the conditions A and B

I interpret the "either" case as:

If you fail to meet at least one of A or B.

But the "both" case has me confused when a non-native English speaker questioned me on it. (I'm a native English speaker).

To use the self-service application to download and install Office 2016, you’ll need two things:

• Mac OS X 10.10 (Yosemite)
• The self-service software application included on L&S-managed Macs

If you don't meet either of those requirements

In the second stage of screening if you meet both of the following requirements you don’t need to do any further assessment of that substance. You’ll need to do detailed modelling of emissions that don’t meet both of the following requirements

My opinion is that using "both" to mean "either" is sloppy and confusing, because it seems like "both" should mean:

you fail to meet A and you fail to meet B

In other words, I think that the "both" case doesn't apply to you if you've satisfied at least one.

Is there a "standard" way to read the "both" statement?

• Both A and B = A + B; Not both A + B = A alone or B alone or neither (none) Jan 9 '17 at 20:07
• When you have a negative (don't or fail) and a conjunction (or, and) in the same clause, you are very likely to get ambiguity, because in language negatives, modals, quantifiers, and conjunctions do not always obey De Morgan's laws. Jan 9 '17 at 20:33
• @JohnLawler I'm not so sure it's an issue with applying De Morgans laws. The problem seems to be more foundational - how to construct the original expression, i.e. whether the negation should initially distribute to the 2 terms (`not(A) & not(B)`), or apply to the quantity (`not(A & B)`). Jan 9 '17 at 20:39
• That's one form of De Morgan's Laws; and that's why it's ambiguous -- it's impossible to construct the original expression. Information is always lost with deletion rules, and conjunction reduction has deleted enough structure to lose the trail here. Jan 9 '17 at 20:48