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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).

  • I thunk that you cannot find a single example of a "proof" (that satisfy the rigorous definition of math proof) in philosophy... – Mauro ALLEGRANZA Apr 19 '17 at 18:51
  • nevertheless, people apparently speak of "analytical proofs" in philosophy. So I am interested to see them. – user56834 Apr 19 '17 at 19:05
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    Please provide a citation/link to the article you mention. The word "proof" existed before mathematicians appropriated it, and continues to function colloquially outside of mathematics, so philosophers are under no obligation to follow mathematical strictures (Spinoza did not). This said, most analytic philosophers prefer the term "argument" (for or against) to "proof". Sometimes arguments are so formalized that they become deductive, but this means eliminating vagueness by explicitly stating all premises and axiomatizing all terms in them. – Conifold Apr 19 '17 at 19:40
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    See Spinoza's "proofs" here. Formalizations in analytic philosophy (of religious arguments in particular) are discussed by Silvestre. – Conifold Apr 19 '17 at 19:48
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    there is a very big difference in practice between mathematical proofs and formal proofs. there are very very few formal proofs of mathematical results. what counts as proof among mathematicians may be highly disciplined but it is just as informal as philosophical proof - which btw is just as disciplined as mathematical proof. – user20153 Apr 19 '17 at 21:05
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As mentioned in the comments, philosophers don't use the word proof as rigorously as mathematicians do. More often they speak of arguments, not proofs.

Some examples:

  • Gödel's ontological proof for God's existence (although I don't know if Gödel's proof counts as canonical).
  • Most of Wittgenstein's Tractatus; In fact Wittgenstein was a major forbearer of what later became known as Analytic Philosophy and his style of arguing in the Tractatus was significant influence on that school.
  • Quine's Two Dogmas of Empiricism and in particular sections I-IV of the paper, where he dissolves the analytic/synthetic distinction is a very good example of an analytic argument, even if doesn't use a formal notation like Gödel or Wittgenstein.
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    But there is nothing in Tractatus that looks like a "reasonable" example of a proof... :-) – Mauro ALLEGRANZA Apr 20 '17 at 7:15

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