I am used to the proof methods of mathematics, and I have formally studied formal logic and mathematical logic. I like philosophy but have never followed a rigorous course in analytical philosophy.
Mathematical proofs are air-tight (as far as it can get) deductive arguments that definitively prove (assuming no mistakes were made) that some very precisely formulated proposition is correct.
Mathematics, formal logic, and mathematical logic are generally about very simple and extremely "reductionist" (for lack of a better term) concepts, such as numbers, sets, functions, algorithms, etc.
On the other hand, philosophy tends to be about concepts that have a much larger degree of complexity, and indefinitness, such as materialism, causality, holism/individualism, and much more.
I once came across an article however, that literally claimed to provide an "analytical proof" that a particular type of "methodological holism" did not logically imply another type of holism. I was surprised that the term "proof" was used in the context of what seems to me to be e much more indefinite concept than those used in mathematics and formal logic. unforunately I've since lost the article.
This leads me to be very curious about the idea of using analytical proofs in philosophy. What are some simple canonical examples of analytical proofs in philosophy?
NOTE: I am already very familiar with formal propositional logic, set theory, predicate logic, mathematical logic. I am only referring to the application of "analytical proofs" to traditionally more vague concepts.
EDIT: I have found the paper I was referring to. Here is an excerpt from the online appendix:
The following, informal argument for our central proposition and its corollary can in principle be formalized. This means that, under an appropriate formalization of the different variants of individualism and holism, it could be turned into a proof (in the technical sense). Since formal philosophy is not our concern here, however, we confine ourselves with giving an expositionally simpler informal argument (broadly in line with Stoljar 2009).