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
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.