[Math] Confusion about definition for a valid argument

logicpropositional-calculus

In my book it defines "an argument is valid if the premises cannot all be true without the conclusion being true as well."

I had trouble understanding what the author meant ,so I did my best -out of curiosity- to see if I could write that definition in logical form.

This how I tried paraphrase the author's definition of a valid argument:

If the premises are true and the conclusion is true, then the argument is valid, and if the premises are true and conclusion is false, the argument is invalid.

Now here is how I wrote my paraphrase of the author's definition in logical form:

Let P stand for the statement "The premises are true " , C stand for the statement "The conclusion is true", and A stand for the "Argument is valid."

This what my paraphrase in logical form is written:

$[(P\land C)\to A]\land[(P\land\neg C)\to\neg A]$

I do not know if the logical form I created is an accurate representation of what the author wrote as the definition of a valid argument. Do you think it's correct my paraphrase?

Best Answer

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.

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