It turns out that the argument above is "valid". Arguments can have many other structurally valid forms, such as "modus ponens" or "modus tolens". "Valid", however, doesn't mean that an argument is true. Consider this:preg_replace('/<\/?p( [^>]*)?>[ ]*/', ' ', preg_replace('/
[ ]*/', ' ', '
All happy people are made of cheese.'))
Tanya is a happy person.
Therefore, Tanya is made of cheese.
This is a valid argument. Why? Because, and tattoo this into your brain: validity is structural; it has a valid structure. The further step is to determine whether the argument is "sound".
An argument is "sound" when all of the premises are true, or colloquially, make some sense. The first example about Socrates: it has the structure of a valid argument, and all of the premises are true, so we can call it "sound". The second example: it has the structure of a valid argument, but not all of the premises are true, so we cannot call it "sound". That's where journalism often fails: often, journalists do not investigate how reasonable the premises are, and so only recognize validity (at best).
That's one notion of argument. You might have guessed that there are lots of way to think of what counts as an argument. In the discussion above, you have to know what counts as a valid structure, and you have to be able to identify which statements play which roles in that structure. That's pretty subtle stuff.
Another way to think about arguments is "Argument to the best explanation". The rough idea is that you're presented with some data that stands in need of explanation - let's assume that you're able to evaluate what that data is. For example, you're standing in front of a burning building, you smell gasoline, and you see someone running away with a gas can in his hand. Your task is to explain this data. Your explanation will count as an argument, because your explanation plays the role of a "conclusion" to the "premises" that are the data. That's a helpful way to start thinking of the structure of an argument to the best explanation.