I've just stumbled upon this rather seemingly simple problem:
Translate this sentence into propositional logic:
You can go to the fair if you are not sick, but otherwise you must stay home and rest. Use
f : you can go to the fair;
s : you are sick;
h : you stay at home;
r : you rest;
My first answer to solve this was $\neg(\neg s \implies f) \implies (h \wedge r)$. After a few observations, this statement is equivalent to $(\neg s \wedge \neg f) \implies (h\wedge r)$. The trouble is both $\neg s$ and $\neg f$ cannot happen at the same time (I think), thus either $(h \wedge r)$ true or false will make this statement valid. This leads me to confusion.
Second approach: $(h \wedge r) \implies (s \wedge \neg f)$. The reasoning behind this was if I must stay home and rest, then I must be sick and I must not in the fair, therefore I can't go to the fair. But, again, after a few observation, this is equivalent to $(h \wedge r) \implies \neg (s \implies f)$ which is contradiction because if I'm sick then I can go to the fair. Then I've been completely stuck until now.
Hopefully someone can point out where is the error, or a solution to this problem. Any help will be veeerrryyy appreciated. Please note that English is not my first language.
Cheers
"But" is another form of "and", used before or after a (perhaps implicitly) negated or negative statement. "Fred went to the party but Sally didn't", "John is nice but stingy", "I'm tired but I'm happy". "Otherwise" here means "if not (you are not sick)" i.e. "if you're sick", a bad/negative thing. Thus the second half means "if you are sick then you must stay at home and rest".
Really, you don't even need to break up that last statement into two statements, because you have no further need of the individual parts. But in case it's required to represent them separately I'll preserve that:
So, you can represent the entire statement in this way: $$ (\neg s \to f) \wedge (s \to (h \wedge r)) $$