"Neither Ana nor Bob can do every exercise but each can do some."

Question

I’m a bit confused as to how I should translate the following sentence:

• Neither Ana nor Bob can do every exercise but each can do some.

I've identified the atomic sentences $A$ = Ana can do every exercise and $B$ = Bob can do every exercise, and managed to translate the first part into $(\sim A \;\& \sim B),$ but I don't know how to go about "each can do some".

Answer 1

You are supposed to use quantifiers. Hence, you might want to define $A(x)$ as "Anna can do exercise x" and $B(x)$ as "Bob can do exercise x".

The second part of your sentence can now be translated as $\exists x A(x) \wedge \exists y B(y).$ I will let you translate the first part.

Answer 2

• Neither Ana nor Bob can do every exercise but each can do some.

It's instructive to sequentially deconstruct the given sentence. The following are all equivalent to the above:

1. Observe that 'but' and 'and' have the same logical meaning.

   • Neither Ana nor Bob can do every exercise, and each can do some.

2. 'Neither M nor N passed' means 'both M and N failed'.

   • Ana cannot do every exercise and Bob cannot do every exercise and each can do some.

3. Be extra careful: it is tempting to mistranslate the first conjunct as "for every exercise $x,$ Ana cannot do $x$", which actually corresponds to 'can't eat anything', which has a different meaning from 'can't eat everything'.

   • (It is not that Ana can do every exercise) and (it is not that Bob can do every exercise) and each can do some.

   These two pairs of parentheses are crucial delimiters, and mustn't be omitted.

4. Denote "Ana can do exercise $x$" and "Bob can do exercise $x$" by $Ax$ and $Bx,$ respectively. Notice that with this legend, the word 'each' in the above sentences will not be translated as 'for each'.

   • (It is not that Ana can do every exercise) and (it is not that Bob can do every exercise) and Ana can do some exercise and Bob can do some exercise.
   • (It is not that $∀x\,Ax$) and (it is not that $∀x\,Bx$) and $∃x\,Ax$ and $∃x\,Bx.$

   Symbolically rewrite the last sentence, optionally dropping all parentheses since quantification and negation have a higher precedence than conjunction & disjunction.

P.S. In step 2, 'neither M nor N passed' can alternatively be translated as 'it's not that (M or N passed)', noting that in logic 'or' is inclusive rather than exclusive.

P.P.S. In step 3, the difference is between $(∀x\,\lnot Ax)$ and $(\lnot∀x\,Ax).$