While he thinks he understands what is meant by truth and logic, his hold on logic is shaky indeed.
He only identifies with truth and logic because he sees them as desirable things to have and therefore his right to have, even if he cannot apply them.
However, should the wheels of logic grind towards a conclusion he finds unpleasant, frightening or inconvenient, his emotions will take over and he will break his own rules. Whatever grasp on truth or logic he has goes out the window and he is only left with the reality of his emotions and someone else's truth and logic, which he will predictably reject.
Being an egotist, he will always reject anyone else's truth and logic however truthful and logical in favour of his own emotions however irrational, because his emotions are his alone and therefore special and better to anything else in the world.
In short, the nihilist believes only in himself, but he believes in and acts on no system of rules but his emotions, from moment to moment. To deal with the obvious problems this kind of mindset would create for any society, religion was created. With religion, there will be scripture and if everyone is said to subscribe to this religion, then at least the rules will be official and contained in scripture, even if in practice they will be broken. If the rules are law, then those who break them will be punished, to discourage undesirable behaviour.
Most people these days are nihilists, because that was the way they were brought up.
Nihilists are incapable of sustained political activism because they are infirm of purpose. A moral system is necessarily and logically a system of morality subscribed to by a group of people. This is impossible to explain to a nihilist who thinks falsely and illogically that what he thinks is morality is anything more than a list of rules that he only imperfectly follows. He will insist loudly and repeatedly that neither he nor his society needs a religion while in the next breath saying that the country is going to the dogs and he is getting out as soon as he can.
What he means - as an egotist - is that because he does not think he need or wants a religion, therefore no one else should need or want such a thing. The pathological egotist has not even the self-consciousness to know that the rest of the world is not him, and he is not the rest of the world. This means he will conflate what he thinks are his own best interests with what is the national interest when he is in fact not particularly adept at looking after what he will call his Number One, precisely because he so often rejects truth and logic in favour of his own emotions and appetites.