It is sense #2 that drives philosophers much of the time. (not all of the time)
Then, the first distinction is between VALID arguments and SOUND arguments.
Try this: a sound argument is a valid argument with true premises.
Example: imagine a conversation 100 years ago between father and daughter:
D: Soon I will be 21 and able to vote!
F: Well no, you won’t.
D: Why?
F: Only persons 21 and over can vote. And, females are not persons; so, you cannot vote.
D: I see that’s a valid argument, but the premises suck!
How should the daughter proceed to convince her father?
“Yesterday was a great day for the women of Saskatchewan,” the Feb. 15, 1916, issue of the Regina Morning Leader proclaimed.
At the time, the Dominion Elections Act, which dictated how provincial elections were run, stipulated “No woman, idiot, lunatic or criminal shall vote.” On Valentine’s Day 100 years ago, the government decided it was time that change.
Good discussion here.

English: All 2-ary logical connectives, expressed with the Sheffer operator NAND. The operation NAND (argument 1, argument 2) is displayed as a black circle, including both arguments. Deutsch: Alle zweistelligen Junktoren, dargestellt mit dem Sheffer-Operator NAND. Die Operation NAND (Argument 1, Argument 2) ist graphisch dargestellt durch einen schwarzen Kreis, der die beiden Argumente enthält. (Photo credit: Wikipedia)
More here.
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And it’s very unfortunate that the two meanings share the same word! For the particularly defensive person, defensive because they are one sweet argument away from having their convictions shattered, to misinterpret an inquiry for an attack is the end of that…
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But the question is, “How should the daughter proceed to convince her father?”!
I’ll try:
Dad, you may have a valid argument, but do you really think the premises are true? Do you think that I am not a person? If not, what am I? I realize ‘person’ is ambiguous and that you are using the legal sense of the word. But, think about it, please. Long ago only male Greek landowners were persons! But that has changed. Don’t I have a brain? Human biology? Do you really believe women are like idiots?
Please, Dad, work with me to change this out of date law.
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Dad: But don’t you see, daughter, it’s the law of the land. You don’t want me to be an outlaw do you? And, we all know that women are too emotional to vote rationally.
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