# Confusion about definition for a valid argument?

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?

• "If the premises are true and the conclusion is true, then the argument is valid" No. One can make invalid arguments for correct conclusions from correct premises. Argument: because I say so. – Daniel Fischer Sep 20 '13 at 18:17
• @DanielFischer So I'm guessing a better alternative would be to say, "If the premises are true and the conclusion is true, then the argument is sometimes valid." – user93971 Sep 20 '13 at 18:25

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.

• I want to understand what we mean when we use the term argument. When we use the term argument, are talking about all possible scenarios? In addition, I can have a case where the premises are all false and the conclusion is true, yet the argument is still valid? – user93971 Sep 20 '13 at 18:40
• Yes, you can: you might have a valid argument form in which one or more of the premises is false, and the conclusion is true. See the added link for clarification. – Namaste Sep 20 '13 at 18:43
• Yes. Here's a valid argument with false premises: (P1) The moon is made of green cheese. (P2) I have eight legs. (C) 1=1. It is valid because the premises cannot be true and the conclusion false; it is never the case that $1 \not = 1$. The sense of argument here is very abstract; just a set of sentences as premises and a sentence as a conclusion. – Neil Barton Sep 20 '13 at 18:47
• "The only scenario that makes an argument NOT valid is when the premises are all true, but the conclusion is false. In all other scenarios, the argument is valid." Not so. Validity requires it to be NECESSARILY the case that if the premisses are true the conclusion is too. – Peter Smith Sep 20 '13 at 18:49
• @PeterSmith Yes, thank you. But I had already revised the post, as you can see. – Namaste Sep 20 '13 at 18:50