quick "help!" question

Sean McGrew mcgrew at DOLPHIN.UPENN.EDU
Fri Mar 26 14:28:33 UTC 2004

```Ok, here's my contribution--pretty hard to do natural language in
math/logic terms at all, much less SIMPLY!

I also feel that

*3. Both Bob and Joe didn't study

Or at least that it's ambiguous and awkward. I myself would either say:

3a) bob and joe didn't both study. (=2)
or
3b) bob and joe both didn't study. (=1)

As for the mathish way to write it, maybe this:

Say Both X & Y [predicate] = X [predicate] and Y [predicate]
Then the difference is in the scope of the negation, in 3a the whole
sentence is negated, while in 3b only the predicate is negated.

positive version: Both studied

3a) not (both study) = not [ (Bob studied) AND (Joe studied) ]
vs.
3b) both (not study) = (Bob didn't study) AND (Joe didn't study)

your 3, if it's acceptable, is ambiguous because the scope of the
negation is unclear.

back to either and neither:

Neither X nor Y predicate = not (X predicate) AND not (Y predicate)

Either X or Y predicate = (X predicate) *OR (Y predicate)

*where OR = exclusive, i.e. either, but NOT both. Logicians and
computer programmers typically have distinct symbols for inclusive and
exclusive OR.

Then you just need the rule that

not (X predicate) = X not predicate

so finally,
Neither Bob nor Joe studied = not (Bob studied) AND not (Joe studied)
=(Bob didn't study) and (Joe didn't study)

Either Bob or Joe didn't study = (Bob didn't study) OR (Joe didn't
study)

Sean McGrew