# What is the difference between a sound argument and a valid argument?

In my notes, these are the definitions of a valid argument

An argument form is valid if and only if whenever the premises are all true, then conclusion is true. An argument is valid if its argument form is valid.

For a sound argument,

An argument is sound if and only if it is valid and all its premises are true.

Okay so to me, both definitions pretty much says the same thing to me. On a philosophy forum, I see that they distinguish the two by saying a valid argument is such that the truth value of the premises necessarily imply the truth values of the conclusion.

For example, the "Elimination" method say

$p \vee q$

$\sim q$

$\therefore p$

So the premises are $p \vee q$ and $\sim q$

Now if I were to substitute $p$ and $q$ for $p$ := "Jesse is my husband" and q:= "I am Jesse's wife" (assume p is true and q is true)

Then we have

Either "Jesse is my husband" or "I am Jesse's wife"

"I am not Jesse's wife"

Therefore, "Jesse is my husband"

So is this technically still valid or sound? (can't tell the difference) Both premises are true, but the conclusion is false? It should invalid right? Yet the method of elimination is said to be valid?

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A sound argument is necessarily valid, but a valid argument need not be sound. The argument form that derives every $A$ is a $C$ from the premises every $A$ is a $B$ and every $B$ is a $C$, is valid, so every instance of it is a valid argument. Now take $A$ to be prime number, $B$ to be multiple of $4$, and $C$ to be even number. The argument is:

If every prime number is a multiple of $4$, and every multiple of $4$ is an even number, then every prime number is even.

This argument is valid: it’s an instance of the valid argument form given above. It is not sound, however, because the first premise is false.

Your example is not a sound argument: $q$ is true, so the premise $\sim q$ is false. It is a valid argument, however, because for any $p$ and $q$, if $p\lor q$ and $\sim q$ are both true, then $p$ must indeed be true.

Note that an unsound argument may have a true or a false conclusion. Your unsound argument has a true conclusion, $p$ (Jesse is my husband); mine above has a false conclusion (every prime number is even).

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The punchline is that I took "I am not Jesse's wife" as true. if I did assume this, I am employing contradiction rather than elimination no? – Hawk Jan 18 '13 at 5:42
@sizz: That argument form is elimination. If you assume that I am not Jesse’s wife is true, then $p\lor q$ is false, so it’s another unsound but valid argument. I don’t see any reasonable way to construe it as an argument by contradiction. – Brian M. Scott Jan 18 '13 at 15:08
In your example, the premises "every prime is a multiple of 4 and every multiple of 4 is an even number" are all false, and the conclusion "prime is an even" is also false. So does the definition I stated in the OP incomplete? – Hawk Jan 20 '13 at 4:24
@sizz: Only one of the premises is false: it is true that every multiple of $4$ is an even number. But the truth or falsity of the premises has nothing to do with validity: the argument is valid because it follows a valid argument form. The fact that one of the premises is false means that it’s not sound. If both premises were false, it would of course still be unsound. There is nothing wrong with the definitions given in the original question. – Brian M. Scott Jan 20 '13 at 7:46
1. Eating cheese makes your nose longer.
2. Wild animals love the sound of the zither.
3. If I lived on the moon, I would eat green cheese all the time.
4. Long-nosed people always play the zither beautifully.
5. The moon is made of green cheese.
6. I am the king of the moon.

Because I am the king of the moon, I eat a lot of green cheese; as a result my nose is very long, and because of that I can soothe savage beasts with my exceptionally lovely zither playing.

This is a valid but unsound argument. The logic is valid, but it is based on false premises. (Can you identify them?)

You meet these arguments in the real world all the time, where the logic is straightforward but the premises are wrong. A recent valid but unsound argument that you might have been aware of is:

1. Arming school janitors with guns is the only way to keep our children safe.
2. We must keep our children safe.

Therefore, we should make sure the janitors have guns.

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In fact you argument is not sound. Soundness is a semantic quality of an argument. It is merely valid. The premises in a sound argument have to be true. – Erik G. Jan 18 '13 at 5:36
Thanks; I had switched "sound" and "valid". – MJD Jan 18 '13 at 5:40

Perhaps a valid argument is one that causes the listener to agree to some form of truth about a subject. A sound argument causes the listener to admit that there is more than one truth about the subject. It has the ability to raise the question of doubt. The difference lies in the listeners interpretation of the information provided. Arguments may be sound, but not necessarily valid. And valid arguments are not always sound. A good example of this is our judicial system....defense attorneys argument, prosecution attorneys argument and the jury or judges decision on the matter.

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The point of the question is to discuss the theory of valid arguments. Valid arguments are based on variables that can only be either true or false. The validity of an argument doesn't question the content of statements that may or may not be true when they are claimed to be. This is where soundness comes into play. Your answer is about the day to day use and misuse of logic by people who may or may not be well versed in the academic theory of arguments and not so much about the theory of its correct application itself. – Geoff Pointer Jan 15 '14 at 0:19