Is rigour always related to definitions and axioms?
Whatever be the case, can a set of philosophical analyses be rated in terms of rigour?
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There is definitely such a thing as philosophical rigour although it is often an ideal rather than something achieved. Language makes rigour very difficult in philosophy but it can be maintained if one is aware of the ptitfalls.
Philosophical analysis can and must be rated in terms of rigour. Once rigour is lost philosophy becomes a muddle of words and concepts. Lack of rigour is public enemy number one in philosophy just as in mathematics.
Rigour in itself does not make for good philosophy but in its absence philosophy cannot be good.
Yes, because while all reason may rest on the unprovable, no one can surpass these starting points, except by encompassing the whole universe and developing a new synthesis.
So, don't give up on reason. It will hone your mind until one of two things happen: you see with greater clarity or (and I have witnessed this phenomenon) someone will resort to using the name of YHVH and put your mind in a baffled state.
The most common way I've seen the latter used is for one side to state (matter of factly) "That is YOUR opinion.". Such statement uses the holy name of YHVH twice (the word "opinion" speaks the name after the "n") and corrupts your mind, so that you cannot argue back, when in fact, it may not have been an opinion at all, but the Truth inside your own genetic history.
Rather there is such a thing as "lack of philosophical rigour", such as not reading up on prior writings on a subject, not using formalisms to produce correct, consistent and complete arguments, falling prey to common fallacies, and not responding to quality rebuttals.
Most of religion can be seen as lacking philosophical rigor, which is why religion is not part of philosophy.
But it is hard to define and measure philosophical rigor in a positive way (rather than counting all mistakes made).