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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.

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  • $\begingroup$ 1) If the conclusion is a tautology, then it is always TRUE. Thus, there is no case when some of the premises are TRUE and conclusion FALSE, that means that an argument with conclusion TAUT is always valid. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Apr 26, 2022 at 9:20
  • $\begingroup$ 2) If the conclusion is a contradiction, then it is always FALSE. Thus, the only case when the argument is valid is when also the premises are contradictory (because otherwise we have some cases with premises TRUE and conclusion FALSE). $\endgroup$
    – Mauro ALLEGRANZA
    Commented Apr 26, 2022 at 9:21
  • $\begingroup$ @MauroALLEGRANZA In case where the conclusion is false, isn't there exist a case where the premises are also false which leads to the argument being valid? $\endgroup$
    – Approxiz
    Commented Apr 26, 2022 at 9:30
  • $\begingroup$ @MauroALLEGRANZA To me, it seems that the definition for the valid argument is "For a valid argument, if all the premises are true, then the conclusion must be true." which is the rephrasing of "An argument is a valid argument if the premises cannot all be true without the conclusion being true as well." I might be wrong. Please correct me if I am wrong. $\endgroup$
    – Approxiz
    Commented Apr 26, 2022 at 9:32
  • $\begingroup$ @MauroALLEGRANZA If you draw a truth table for the statement, you can see that the only case where an argument is invalid is when the premises are true but the conclusion is false. Thus, in case where the conclusion is a contradiction, there is a case where the argument is valid which is when the premises are false. $\endgroup$
    – Approxiz
    Commented Apr 26, 2022 at 9:36

1 Answer 1

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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.)

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  • $\begingroup$ I found an article about similar topic. What do you think about it? $\endgroup$
    – Stats Cruncher
    Commented Feb 15, 2023 at 6:33
  • $\begingroup$ To follow up on why? questions: the argument is valid iff for every critical row there is a true conclusion. Therefore, the argument is invalid iff there is a critical row for which the conclusion is false. There might be a case when an argument has no critical rows, then such an argument is vacuously valid (why?). Now you'll need to determine the validity of an argument obeying the constraints that the exercise poses. $\endgroup$
    – Vlad Mikheenko
    Commented Jul 3 at 15:58

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