Question About The Validity of An Argument

logicpropositional-calculus

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.

Best Answer

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.)

Related Solutions

[Math] Confusion about definition for a valid argument

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.

[Math] Translate the following argument into propositional logic, and then assess it for validity.

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 '$L \to T$').

Reference

Best Answer

Related Solutions

[Math] Confusion about definition for a valid argument

[Math] Translate the following argument into propositional logic, and then assess it for validity.

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