# Translate the following argument into propositional logic, and then assess it for validity.

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.

• Unfortunately this boils down to a English reading exercise. If you interpret it as it is literally written, then you'd be right. However, the intended meaning behind "Catch Billy a fish, and you will feed him for a day" seems to be "If you catch Billy a fish, then you will feed him for a day". – Mathmo123 Oct 27 '14 at 8:45
• So there are cases in which "and" is a connective in English, but the translation to $\wedge$ doesn't work? In this case, in (1) $C$ has to occur before $D$ occurs, thus making it not truth functional in the $\wedge$-sense, as both conjuncts would have to be true, whether the authors are saying that the truth of $C$ comes before the truth of $D$, so a $\rightarrow$ translation would be more appropriate. Is that correct? – jsmith95 Oct 27 '14 at 9:33
• @jsmith95 - you have find the point ! $\land$ is symmetric, while $\rightarrow$ is not. Thus, if we reverse: "Catch Billy a fish, and you will feed him for a day", we get : "You will feed Billy for a day, and you will catch him a fish". The two have not the same meaning. – Mauro ALLEGRANZA Oct 27 '14 at 9:53
• Thanks Mauro, that makes a lot of sense. – jsmith95 Oct 27 '14 at 10:02

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:

Unlike their English counterparts, these symbols represent concepts that are precise and invariable. The meaning of an English word, on the other hand, always depends on the context. For example, ∧ represents a concept that is similar but not identical to “and.” For atomic formulas A and B, A ∧ B always means the same as B∧A. This is not always true of the word “and.” The sentence

She became violently sick and she went to the doctor.

does not have the same meaning as

She went to the doctor and she became violently sick.

Likewise ∨ differs from “or.” Conversationally, the use of “A or B” often precludes the possibility of both A and B. In propositional logic A∨B always means either A or B or both A and B.

This is the case in the sentences above:

(1) Catch Billy a fish, and you will feed him for a day.

(2) Teach him to fish, and you'll feed him for life.

Note that the "and" here should not be interpreted as "$\wedge$". We have many similar cases:

(i) Jump and you die

We intuitively know that this sentence actually means:

(i') If you jump then you will die

Hence the argument is stated this way:

1. $C \to D$
2. $T \to L$

$\therefore \neg L \vee T$

Which is not valid (note that the conclusion says '$L \to T$').