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:
- $C \to D$
- $T \to L$
$\therefore \neg L \vee T$
Which is not valid (note that the conclusion says '$L \to T$').
There are many 'laws of deduction': there are many different systems of deduction, each of which with their own set of laws or rules ... so it would be good to know which rules you are allowed to use.
Nevertheless, here is a proof using fairly commonly used rules:
Best Answer
(2)~Q&(3)Pand not~QΛP?Because assuming $ A \land B $ is equivalent to assuming $ A $ and assuming $ B $, it's in almost any set of rules of inferences, hard to imagine that rule of inference not being present. The author just didn't make this step explicit because it's not important.
The validity of a proof and the validity of it's conclusion are 2 different things. The validity of a proof depends on every step being correct; the final statement being true is not sufficient for a proof to be correct.
A conclusion is invalid if it's negation is consistent with your assumptions. In the above example, the negation of the conclusion is that you didn't solve the test questions. Does that contract any assumptions? No, you could have gotten extra credit for being the teacher's special friend. No one said that the only way to get extra credit was the 2 possibilities listed.
Modus ponens is the statement that, if you assume $A \implies B$ and $A$, then you can conclude $B$.
The above proof starts with the following 3 assumptions.
(1) $(Q \land R) \implies P$
(2) $\lnot Q$
(3) $P$
The proof then tries to apply modus ponens to (1) and (3). However that would only work if (1) was $(Q \land R) \leftarrow P$. Since (1) isn't a reverse implication, the application of modus ponens is incorrect.
This is a common error to make in English. If you said "You will get extra credit if you write a paper or if you solve the test questions", how many students do you think would assume that those are the only way? If your mother said "if you don't eat your vegetables, then you can't have dinner", wouldn't you assume that you can have dinner if you eat your vegetables? This is logically incorrect, and it's the error the author is trying to present.