Question About The Validity of An Argument

Question

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.

Answer 1

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.)