You can also have a valid argument when one or more of the premises is false. In that case, the truth or falsity of the conclusion changes nothing about the validity of the argument.
Hence, it is correct to say $$(P\land \lnot C) \implies \lnot A.$$
See Wikipedia for more on when an argument is valid, which starts:
In logic, an argument is valid if and only if its conclusion is logically entailed by its premises. A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.
Note that the linked entry above makes a distinction between a valid argument and a sound argument:
Validity and soundness
Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:
All animals live on Mars.
All humans are animals.
Therefore, all humans live on Mars.
The problem with the argument is that it is not sound. In order for a deductive argument to be sound, the deduction must be valid and all the premises true.
- why the You don't write a paper,however you get an extra credit. sentence has been split into two propositional sentences
(2)~Q & (3)P and 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.
- There is an explanation given for the solution which I understood. Now, does that explanation can be summed up saying I wasn't able to apply any propositional laws to the sentences to derive the conclusion and hence it is invalid? I mean is that how you generally solve the valid/invalid question? by trying to apply the laws to the sentences and see if it gives you the conclusion?
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.
- and if possible can someone explain about converse and inverse error in detail when it comes to prove the validity?
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.
Best Answer
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: