When philosophy can actually "prove" unintuitive conclusions, such as that the world is in motion, even on windless days, it becomes science. The concepts are validated by prediction and "experience," which is itself a kind of "common sense."
When philosophy deals with ideas and values such as justice, the reasoning can only be compared with "common sense" in the end. For the bad sophists in Plato's dialogues it is enough to merely refute common sense as a kind of trick with words, while Socrates attempts to clarify the perplexities arrived at in ordinary ways of speaking. He strives to earn common sense agreement at every turn in a possibly surprising conclusion.
For Wittgenstein, similarly, the task of the philosopher is to provide "therapy" for the problems in common sense speech or to "let the fly out of the bottle," as he put it. And, most famously, for Kant the idea that his "categorical imperative" turned out to be a logically clarified form of the golden rule was, he felt, very much in its favor.
As it branched away from philosophy, science produced ways to validate surprising ideas. But if you are reasoning about unpredictable matters outside of that highly structured context, how would you "refute" common intuitions without some ultimate recourse to common intuitions? This legacy of guiding or "drawing out" the layman's intuitions is demonstrated, if somewhat unconvincingly, in Socrates' instruction of the slave boy in the Meno.
Philosophy attempts to carry us further along in thought while "hugging the shore" of our practical reasoning and ways of discussing things, even where the language gets necessarily obscure or tedious. Science, with its compelling demonstrations, can refute common intuitions, as with Galileo's moon or Darwinian evolution, but at some risk.
Philosophers such as Husserl and Wilfrid Sellers have rightly pointed to the kind of social crisis that can evolve if science becomes at once our authoritative basis of knowledge and utterly divorced from the common sense understanding of the majority of people. I admit I feel this way myself at times when trying to read about the interiors of black holes or multiple worlds.