I have the following sentences:
I won't go the library unless I need a book
p: I will go the library
q: I need a book
I replaced unless with if not as follows:
I won't go the library if I don't need a book
Then: $\lnot q \rightarrow p$ is my translation correct here?
And what if I paraphrased the sentence to the following:
If I won't go the library then I (don't) need a book.
$\lnot p \rightarrow \lnot q $
Would it still be correct?
$\endgroup$ 12 Answers
$\begingroup$"$A$ unless $B$" is usually read in English as
$A$, if not $B$.
Thus, for I won't go the library unless I need a book, will be:
I won't go the library, if I do not need a book.
With:
$p$: I will go the library
$q$: I need a book
will be:
$\lnot q \to \lnot p$
that is the same as:
$p \to q$.
$\lnot q \to \lnot p$
is not equivalent to:
$\lnot p \to \lnot q$,
and this is consistent with the fact that:
If I won't go the library, then I don't need a book
is not the same as the previous:
I won't go the library, if I do not need a book.
Trough the truth-functional equivalence between "if $B$, then $A$" and "not $B$ or $A$", we have that :
$\endgroup$ 7 $\begingroup$"$A$ unless $B$" is equivalent to "$B$ or $A$".
Let's think about your sentence carefully:
I won't go the library unless I need a book.
This sounds like you hate the library, and that as a general rule of thumb, you should not be expected to be caught dead there, except for one extenuating circumstance (you need a book). Let's evaluate your proposed translations.
Translation 1 ($\neg q \to p$): If I don't need a book, then I'll go to the library.
This definitely doesn't make sense. After all, you hate libraries. And you don't even need a book!
Translation 2 ($\neg p \to \neg q$): If I won't go to the library, then I don't need a book.
This isn't quite as bad of a translation, but it doesn't necessarily follow from the original sentence. Perhaps the library is merely your last resort for getting a book, and so it's possible that there are other alternatives (like borrowing your friend's old copy, for example). Thus, you are able to successfully avoid a trip to the dreaded library and yet simultaneously satisfy your need for a book.
Correct Translation ($p \to q$): If I go to the library, then I need a book.
This works. There are only three possible combinations and one impossible combination:
- Possible #1: You are not at the library and you don't need a book. (happens all the time)
- Possible #2: You are not at the library and you need a book. (hopefully there are alternatives)
- Possible #3: You are at the library and you need a book. (rarely happens, but you had no other choice)
- Impossible: You are at the library and you don't need a book. (...then why are you there?)
