Learn Portuguese (Portugal): Não acho que o argumento dele seja válido. - I do not think that his argument is valid.

The verb achar (to think/believe) is tricky! When used affirmatively (acho que...), it introduces a strong opinion, so you use the indicative (é). But when it is negative (não acho que...), it expresses doubt or denial. This triggers the subjunctive, so we use seja, the present subjunctive of ser.

If the sentence were affirmative, how would it change?

It would become Acho que o argumento dele é válido. Because the sentence is affirmative, it expresses an opinion you hold to be true, so it takes the normal indicative verb é instead of the subjunctive seja.

Why do we say o argumento dele instead of using the possessive word seu?

While seu translates to his, it can also mean your, hers, or theirs. To avoid confusion, European Portuguese speakers frequently use dele (of him) right after the noun. So, o argumento dele literally means the argument of him.

Could I use acreditar instead of achar in this sentence?

Yes, you can say Não acredito que o argumento dele seja válido. Like achar, the verb acreditar (to believe) triggers the subjunctive when used in the negative, because you are denying a belief or expressing doubt.