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Irrationalists quite rationally believe that there is no rationality. Rationalists irrationally believe that it exists, because it is doubtful that it is possible to create some closed, self-justifying justification for the existence of rationality.
For myself, I solve this contradiction by saying that rationality is such a convention. That is, if enough people agree that something is rational, it is rational. But our agreement is not based on nothing. We have a common experience – an experience that the other person can understand and, if necessary, reproduce with sufficient accuracy. There is a world around us and this world is noticeably not chaotic, but ordered. We have the ability to make judgments about this world and distinguish false judgments from true ones. For this conversation, it does not matter where we get this ability and why the world is knowable-someone believes that this is evidence of the goodness of God, someone just says ” the anthropic principle! if the world were chaotic, we wouldn't be able to exist in it!”, and the ability to distinguish lies from truth is attributed to evolution. It is on these two pillars that the belief in the existence of rationality is based. If it didn't exist, how could we possibly understand each other?
To be honest, it seems strange to apply the criterion “is this word rational or irrational” to words, but if you do apply it, the word is rational to the extent that you are not the only one who understands it this way.
In addition, just yesterday in the Telegram channel “Hungry Philosophers” there was an entry on a related topic, I will quote a piece from there:
That is, I am not the only one who connects rationality and the ability to communicate, this is an idea that is floating in the air.