For instance: Sally can play the guitar and the piano. Martin can't play the guitar or the piano.
This could be seen as a matter of formal logic:
In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.
The rules can be expressed in English as:
The negation of a conjunction is the disjunction of the negations. The negation of a disjunction is the conjunction of the negations.
or informally as:
"not (A and B)" is the same as "(not A) or (not B)"
"not (A or B)" is the same as "(not A) and (not B)"
There is a logical rule, because in Boolean arithmetic:
¬(A ∧ B) ⇔ ¬A ∨ ¬B
In less formal language though, it doesn't always work that way.
We would generally not understand "Martin cannot play the guitar and the piano" as meaning "it is not true that Martin cannot play both the guitar and the piano, but I am telling you nothing about whether he can play one of those instruments". Rather we would understand it as meaning that he can play neither.
We might favour "Martin cannot play the guitar or the piano" because it better matches the formal logic while remaining understandable, but in terms of understanding what others mean, applying such a rule will not work.
The other answers discussion de Morgan's laws are relevant here, but I think it is instructive to look at all the possible combinations and what they mean. Let's say
A means that Sally can play the guitar and
B means that Sally can play the piano. There are 4 possible scenarios:
- Sally can't play the guitar. Sally can't play the piano.
- Sally can play the guitar. Sally can't play the piano.
- Sally can't play the guitar. Sally can play the piano.
- Sally can play the guitar. Sally can play the piano.
Sally can play the guitar and the piano
A ∧ B - This means that Sally can play both instruments. Only scenario 4 is possibly true.
Sally can play the guitar or the piano
A ∨ B - This could mean that it is known that Sally can play at least one of the two instruments. Perhaps we are investigating Sally's room and we find a copy of Easy Guitar and Piano Parts for Popular Show Tunes.
From a logical perspective this means scenario 2, 3 and 4 are all possible. The only impossible scenario is scenario 1.
It reality this situation is not commonly encountered. This particular phrasing might also be used to mean that in some arrangement of players to instrument, the options are to assign Sally to guitar or piano—"Jane can play drums, Alice can play bass, and Sally can play the guitar or the piano."
Sally can't play the guitar or the piano
¬(A ∨ B) - This means that Sally is unable to play either instrument. Sally can't play the guitar and Sally can't play the piano. The only possible scenario is scenario 1.
Sally can't play the guitar and the piano
¬(A ∧ B) - This means that Sally is unable to play both instruments, but leaves open the option that she might be able to play one of them. The possible scenarios are 1, 2, and, 3. The only impossible scenario is scenario 4. In common usage, it is unusual to make a statement like this and mean scenario 1—if she can play neither instrument, the you'd be explicit about it and say "Sally can't play the guitar or the piano". So, for practical purposes, in this case you can assume that only options 2 or 3 are possible (in logical terms, "exclusive or").