Question About The Validity of An Argument I have been reading and doing exercises form the book How to Prove It: A Structured Approach until I have reached section 1.2.
I have come across the definition of a valid argument:
An argument is a valid argument if the premises cannot all be true without the conclusion being true as well.
This can be translated as:
"For a valid argument, if all the premises are true, then the conclusion must be true."
The contra-positive of the statement is:
"For a valid argument, if the conclusion is false, then some of the premises is false."
On page 26 exercise no. 18:
Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument? What if the conclusion is a contradiction? What if one of the premises is either a tautology or a contradiction?
I noticed from the implication and the contra-positive of the definition for a valid argument that the form of an argument is the same as:
\begin{align*}
(P_1 \wedge P_2 \wedge P_3 \wedge ... \wedge P_n) \rightarrow C
\end{align*}
where each $P_1,...,P_n$ represents each of the premises respectively and $C$ represents the conclusion.
If the form I have written is correct, then the only case where the argument is invalid is when the conclusion is a contradiction and all the premises are true.
I want to know whether the form I have expressed is correct or not.
Reference
Velleman, D. J. (2019). How to Prove It: A Structured Approach. Cambridge University Press,
3. edition.
 A: 
the form of an argument is the same as:
\begin{align*}
(P_1 \wedge P_2 \wedge P_3 \wedge ... \wedge P_n) \rightarrow C
\end{align*}
If the form I have written is correct,

Yes, that's correct.

then the only case where the argument is invalid is when the conclusion is a contradiction and all the premises are true.

This is not so: consider the argument with atomic premise $P$ and atomic conclusion $R,$ and note that its conclusion is not a contradiction.


Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument?


This argument is valid, because any premise logically entails a tautology.

What if the conclusion is a contradiction?

This argument may not may not be valid. (Why?)

What if one of the premises is either a tautology or a contradiction?

This argument may not may not be valid. (Why?)
However, if one of the argument's premises is a contradiction, then the argument is valid, because a contradiction logically entails any conclusion.
(Read more here and here.)
