I was wondering of examples where authors examine their own framework using their framework. I'm especially interested in 'larger' examples where an author may put their reasoning for creating a metaphysic through their system of metaphysics or where theorists examine their own theory using that theory, but I would also appreciate 'smaller' examples, like for example "all categorical syllogisms are constructed of words, all words are constructed of letters, therefore all categorical syllogisms are constructed of letters"
This is what Wikipedia says about Stephen David Ross
He began in American pragmatism, reading it to question itself fundamentally and to entail the inexhaustibility of nature and reason, the mysteriousness of things.
But I guess the most famous example would be Jesus :
If a man slap you on one cheek turn the other as well
If a man take your shirt give your coat as well
Treating the gospel account as factual the manner of his death seems to have authenticated his commandments.
He claimed that unlike others similar to him who had some claim to be "wise men" – the sophists – he not only was interested in wisdom (sophia) he actually loved it (her?) – philo-sophia.
And so like Jesus, when he was murdered
- he refused to back down from his claims/beliefs
- he refused to run away (and stop making trouble)
- most strikingly his (recursive) love in his sophia-love stood him in such good stead that he died cheerfully and fearlessly.
Brings me to the confusion that your question seems to be causing: you've phrased it in terms of fixpoints. It would be better stated in terms of recursion. The former expresses the latter via the fix point combinator so it's not wrong. But it is confusing!
This question on performative contradictions is in the opposite direction to your question. This direction can also be interesting for the egregious example of failure to "practise what you preach".
I particularly commend the comment by @conifold