Different definitions 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:

(Sorry if this post is missplaced, I did not know if this apply to mathematics or philosophy or both. I did, however, first encounter my problems in a book about mathematics)

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 this example :

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 = "not happy" and Y = "not happy", the argument does not makes sense structurally I think. And if it does not make sense structurally, what is the point of a valid argument in the first place?

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

Answer 1

Ok, lets try to untangle this.

Logical Consequence

You are pondering about logical consequence, the relation that holds between the premises and conclusion of a valid argument.

Your definitions (1)-(3) are the usual/historical conception of what logical consequence is, they are all equivalent. Also, let me emphasize their modal nature:

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.

Your suggestion (4) is also equivalent to 1.-3., if you take 'always' in a modal sense: Whenever the premises are true, the conclusion is necessarily true. But modal talk is vague. By e.g. 'impossible', do we mean metaphysically impossible/epistemically impossible, or just the absence of a counterexample? How can we put those definitions to use?

Luckily, Tarski came along and offered the now prevailing account of logical consequence. He neatly replaced the modal talk above with more transparent concepts: Formulas are interpreted in set-theoretic models. In a model (or, if you prefer, possible world) every formula is either true or false. Logical consequence is then defined as truth-preservation across interpretations: A formula $a$ is a logical consequence of a set of premises $B$, iff in every model in which all $B$ are true, $a$ is also true. In familiar formal terms: $$B\models a \iff \text{ for every model } \mathfrak M \text{ with } \mathfrak M\models B \text{, also } \mathfrak M\models a$$ (where $B\models a$ is read as ''$a$ is a logical consequence of $B$'', and $\mathfrak M\models a$ means ''$a$ is true in model $\mathfrak M$''.) The modal component of the first definitions was thus replaced by quantification over models. The Stanford Encyclopedia of Philosophy has a helpful entry on logical consequence.

But that's enough for now, let's apply the Tarskian definition of logical consequence:

Your Examples

a) The conclusion is a tautology.

This means that the conclusion is true in all models, so it is the logical consequence of any set of premises. The argument is trivially valid.

b) The conclusion is a contradiction.

This means that the conclusion is false in every model. So it won't be the logical consequence of any set of premises, unless that set of premises is already inconsistent. If the premises are contraditory themselves, then the argument is valid (because, vacuously, in every model in which the premises are true (none!) the conclusion is true).

c) One premise is a contradiction.

Any conclusion follows from contradictory premises. Contradictory premises aren't true in any model, so the definition of consequence is again vacuously satisfied. The argument is trivially valid.

(I couldn't follow your final concrete example of an invalid argument with contradictory premises, so I don't comment on that)

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Two more notes: (i) In addition to the Tarskian semantic account of consequence sketched above, there are also accounts starting from proof-theory. There, a sentence is said to follow logically if it can be formally derived from the premises using only valid rules of inference.

(ii) Logical consequence additionally depends on which rules of inference you consider valid. There is e.g. classical consequence and intuitionist consequence. But both notions can be captured in a Tarskian model-theoretic framework (intuitionist logic just requires more involved models).