This is question 9 from exercise 6.5.1 in Smith and Cusbert's Logic: The Drill.
It wants a translation and test of validity for the following:
Catch Billy a fish, and you will feed him for a day. Teach him to fish,
and you'll feed him for life. So either you won't feed Billy for life,
or you will teach him to fish.
My informal translation:
(1) Catch Billy a fish, and you will feed him for a day.
(2) Teach him to fish, and you will feed him for life.
Therefore,
(3) Either you will not feed Billy for life, or you will teach him to fish.
My formal translation with propositions and connectives:
(1) $C \wedge D$
(2) $T \wedge L$
Therefore,
(3) $\neg L \vee T$
I originally did semantic tableaux (truth trees) to test for validity (it came out valid), but I'm unsure how to do them math.SE. So, I will give a brief semantic proof. Suppose for contradiction that the premises (1) and (2) are true, and the conclusion (3) false. Hence $\neg L$ is false, and $T$ is false. Hence (2) is false as one of the conjuncts ($T$) is false. But this contradicts the supposition that all the premises are true, and conclusion false. Hence the argument is not invalid, hence valid.
The solution in the book (page 127) says my translation of (1) and (2) are incorrect (and that the argument is actually invalid).
It states the interpretation as:
(1) $C \rightarrow D$
(2) $T \rightarrow L$
Therefore,
(3) $\neg L \vee T$
Here is a screenshot of the question with translation and solution:
Which is correct?
I do not have access to the companion logic book Logic: The Laws of Truth, only the exercise and solutions book that the authors kindly put online here (PDF warning). So I am unsure if I have translated the conditional/conjunctions incorrectly based on some quirk that the authors talked about in the companion text. I cannot seem to find an errata on the website either.
Best Answer
A complement to the comments above.
It's worth mentioning that the meaning of the propositional connectives $\neg$, $\wedge$, $\vee$, $\to$ should not be regarded as a mere symbolic translation of the meaning of their English counterparts "not", "and", "or", "if ... then" respectively. See Hedman's A First Course in Logic (2004), p.1-2:
This is the case in the sentences above:
Note that the "and" here should not be interpreted as "$\wedge$". We have many similar cases:
We intuitively know that this sentence actually means:
Hence the argument is stated this way:
$\therefore \neg L \vee T$
Which is not valid (note that the conclusion says '$L \to T$').