The validity of the definition of a valid argument

Question

I have some serious problems understanding what counts as a valid argument and what does not. I have read some different definitions of what a valid argument is:

An argument is valid if

(1.) The premises cannot all be true without the conclusion being true as well

(2.) The truth of the premises guarantees the truth of the conclusion.

(3.) It is impossible for the premises all to be true and the conclusion not be true.

Ok so I've included a bunch of them just because I think it is good to hear it in different wordings. Personally, I would like to say that:

(4.) If the premises are all true then the conclusion is always true.

My confusion began when I was supposed to answer the following question:

What can you say about the validity of an argument if:

a) The conclusion is a tautology

b) The conclusion is a contradiction.

c) One premise is a contradiction

My thoughts on a)
It is not possible to say anything because one cannot know if it is possible for all premises to be true and therefore one cannot check if the argument is valid or not valid based on the definition for a valid argument, since it just says that if the premises are all true then the conclusion is always true. That is, since the condition "if the premises is all true" can never be met in the first place, there is no way to use the definition. I would like to say that the argument is undefined.

My thoughts on b) Reasoning same as in a)

My thoughts on c) Kind of the same thing, the definition cannot be used because all premises can never be true.

I read somewhere that the answer to c) should be that the argument is always valid since there is no way that all premises are true and at the same time, the conclusion false. Now, that kind of makes sense, there is certainly no scenario where all premises are true and the conclusion is false, so by (3.) it holds since it was the exact thing that cannot be possible if an argument is valid. If, however, I think about definitions (1.), (2.) and (4.), this does not make sense, I again think that there is no possible way to use these definitions to say anything about the argument, because the premises are not all true from the first place.

So, now I have come to the point where I think the definitions say different things. I feel like (3.) is about the possibility of a scenario and the other definitions are about implication. At the same time I feel like (3.) implies the other definitions and the other definitions implies (3.), but I have a hard time to see the equivalence.

One final thing that adds to my puzzlement is that I seem to be able to construct an example that supports my claim that one can not say anything about question c) :

John is happy and John is not happy
Alice is happy or George is happy
George is X
Therefore Alice is Y

Edit:

For example if X = "unhappy", and Y = "unhappy", then the argument does not make sense by it's structure, so how can it be valid? And if it is still valid, what is the point about an argument being valid if they clearly do not have to make sense by their structure anyway?

Sorry for the excessively long post, but I am desperate for someone to help me make this
illogical logic logical.

Answer 1

Reading through your question, it's a common worry that many people share. I think the problem often stems from being confused about the role validity plays in logic.

defining validity

(there are at least two other definitions of validity that work differently than the answer I'm going to give you (but the answer below reflects what you're probably learning):

1. Model theory - an argument is valid if and only if you can construct a system of the premises. This is called model theory).
2. Validity via inference - an argument is valid if each premise proceeds either from an assumption or a valid rule of inference (2 actually works out to be the same as the answer below for at least tautologies )

Using the following definition of validity,

an argument is valid if and only if it can never have all of its premises be true and the conclusion be false.

We can first look at the definitions you suggest. Truth-preservation (your (2)) is a consequence of validity rather than the definition of validity. In fact, nearly all of your wordings are things we can say about valid arguments rather than efficient ways to test validity.

On this definition, two types of arguments meet the test of validity:

(1) Arguments with conclusions that are always true, either trivially so or because the premises when true always yield a true conclusion.

(2) Arguments where there is always at least one false premise.

You then want to consider three cases:

(a) The conclusion is a tautology

In your consideration of (a), you seem to be working from a slightly altered definition of validity that hinges on the truth of the premises. For the standard definition of validity, we never need to consider the premises if the conclusion cannot be false. But regardless, there's no reason an argument whose conclusion is a tautology should not be valid. A tautology means the conclusion takes the form A v ~A. And that's always going to be true, because it is merely a restatement of the law of the excluded middle.

Thus, arguments of type (a) can never have a false conclusion and is always valid.

(b) The conclusion is a contradiction.

For (b), you are correct. A contradiction is false, because it is A & ~A. But that does not tell us that the premises can all be true at the same time. Thus, we cannot determine the validity of an argument merely by knowing its conclusion is a contradiction.

If we could know all the premises are true, then the argument would be invalid.

(c) One premise is a contradiction

For (c), if one premise is a contradiction, then the premise is false. Thus, the argument is valid. Because by corollary (2) of the definition, we know that the argument can never have all true premises and is thus incapable of having all true premises and a false conclusion.

I believe (c) is the case where things diverge the most between models of formal logic, but assuming the sort of logic taught in many first or second year undergraduate courses, the above account holds.

Answer 2

Let's take your version 2 of validity "(2.) The truth of the premises guarantees the truth of the conclusion" and a tautology as the conclusion. Being a tautology, the conclusion is always true. You may have looked at the word "guarantees" which makes it seem as if the premises do something actively to make the conclusion true, but that isn't necessary for the argument to be valid.

All that matters for the argument to be valid is that if the premises are true, then the conclusion is true. And yes, that is the case. The conclusion is always true. So if the premises are true, the conclusion is true. If the premises are false, the conclusion is true. If easter and christmas fall on the same day, the conclusion is true. If the premises cannot possibly be true, the conclusion is still true. But all that counts is: IF the premises are true then the conclusion is true, and that is the case. And therefore the argument is valid.

The second case was that the conclusion is a contradiction. So the conclusion is never true. If the premises are true, then we have true premises and a false conclusion. So if the premises are true, the argument is not valid.

However, it is possible that the premises are not true. If the premises are not true, then the argument is still valid. The requirement is "IF the premises are true then the conclusion is true". Nothing is said, and there are no requirements, for the case that the premises are true. "If Easter and Christmas fall on the same day then Santa Claus delivers easter eggs". As long as Easter and Christmas don't fall on the same day, this is a valid argument. So if the premises are not true, then the argument stays valid even if the conclusion is a contradiction.

For the last case that one of the premises is a contradiction, let's look at 3 "(3.) It is impossible for the premises all to be true and the conclusion not be true."

If one of the premises is a contradiction, then it is impossible for that one premise to be true. It is even more impossible for all premises to be true since we already know that one premise is not true. And it is even more impossible for the contradiction premise to be true, the other premises to be true, AND the conclusion to be not true. Therefore the argument is valid.

What can be confusing is that an argument can be valid, but rather useless and pointless at the same time. "If Easter and Christmas fall on the same day then Santa Claus delivers easter eggs" - nobody will claim that this is a particularly useful statement, but nevertheless it is valid.