A conversation a while ago with a friend sparked some reflection upon the sorts of arguments, or (more generally) philosophical "moves," that I find inherently intriguing. Not persuasive, but fascinating -- the kind that make me want to pick them up, turn them around, see how they work. Many -- most -- turn out to be what I would call grammatical arguments: they hinge upon taking a word's meaning seriously, as if with a kind of immanent critique. a few others introduce a crucial distinction. And one or two are the sort of cautionary warning that serves as a more elaborate version of a "rule for thinking."
Wittgenstein: distinction between saying / showing. The limits of language about language; a "performative" moment at the very beginning of "Analytic" philosophy.
Anselm: "ontological proof." The "prayer of the intellect," Simone Weil called it. Much as with Zeno's refereeing of the race between Achilles and the tortoise, generation after generation of philosophers has felt called upon to refute Anselm. When that many philosophers (who otherwise agree on so little) all feel you are wrong, you are certainly doing something right.
C.S. Lewis: foundering of naturalism ("cause" vs "reason"). This has been elaborated and made more sophisticated by Plantinga and others, but I still think that Lewis' presentation in Miracles is, while flawed, the best short summary. If all "reasons" reduce to "causes," then one can have, by definition, no reason to believe this.
Descartes: cogito. Do I even need to justify this? This was probably my first foray into the canon of philosophy, in a conversation with my next-door neighbor when I was 10 or so. There is much to object to in Cartesianism, but the beauty and simplicity of this grammatical moment is too often overlooked by Descartes' critics.
Frege: against psychologism. The locus classicus here is his review of Husserl's first book, but the import extends far beyond. At issue is, what do we mean by "valid"? Not just "convincing"! Not even really really really convincing.
Brentano: structure of intentionality. "Every mental phenomenon includes something as object within itself, although they do not do so in the same way. In presentation, something is presented, in judgment something is affirmed or denied, in love loved, in hate hated, in desire desired and so on." (Psychology From an Empirical Standpoint) A way of saying the obvious that suddenly shows that a great deal follows from the obvious.
Hume: is / ought non-transition. This one is taking a beating in some quarters these days via a kind of argument-by-poll. "Show me a counter-example!" is Sam Harris' refrain. This is a different argument.
Buber / Levinas: what encounter means. Otherness. This is the core of my debate with Monism, the claim that there is really One Big Thing. What I mean by "encounter" is not a sleight-of-perspective trick by which one part of a system forgets and then remembers another part.
Berkeley: The inconceivability of the inconceivable. Well, duh! Despite being lampooned by David Stove and many others as an argument "so bad it is hard to conceive of anyone being swayed by it," or words to that effect, it also has its defenders, notably (of late) Meillassoux. The beautiful thing about this defense is that Meillassoux doesn't believe the argument -- he just thinks it is strong. I.e., you don't have to be persuaded by an argument to admire it and think it is well worth, not just pausing over in the museum of Great Moments in Western Phil., but really being challenged by it.
Wilber: Pre-/Trans- Fallacy. This is a crucial distinction, and if Wilber is remembered for nothing else in fifty years, the elegance of this formulation will last. One thing that's often overlooked in considering it, is that those prepositions modify a noun: rationality.
Cantor / Gödel: knowing may exceed axiomatics. Like most of my generation I first bumped into this argument in Douglas Hofstadter's masterpiece. But I had the funny experience of being more and more persuaded by an argument Hofstadter was concerned to refute, by John Lucas, that (to put it briefly) human thinking was not reducible to the unfolding of algorithms. This application is only the beginning, though. It's not just that you can deduce something, it's what this capacity to deduce means.
Darwin, et al: time + randomness + stochastic mechanism + "anthropic" perspective can get you pretty much anything. Dennett's crucial modification in terms of "cranes" vs "skyhooks" (an elegant distinction) does much to expound the argument, which seems more or less irrefutable. So the pertinent question must be, is it -- or, in what sense is it -- interesting?
Great doubt, great enlightenment. Ok, this is not an "argument", it's a description of ascetic experience.
And, finally, the argument from Moral Realism. What do we mean by "Good"? And if we take this meaning seriously -- what follows? Or, if we think we "have no right" to this meaning -- then, (1) can we dispense with it? (and I don't mean "practically", I mean, coherently); and (2), if not, then... (This is very close to St Anselm's proof, above).
Looking over the above list, one could note that the verb "means" -- in italics, no less -- recurs frequently. Well, I warned you. One philosopher's Achilles' heel is another's Archimedean pivot. The trick is, to make it both.