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For instance, when discussing "what is Justice", one of Rawls's key argument for "justice has to be a universal concept" is that we do not talk about anything that is "just for person A but not for person B".
In other words, Rawls is relying on our laymen's intuition about justice to narrow down the possible definition.
This is not an isolated example. Philosophers often rely heavily on common intuition to guide/justify their reasons. What is the justification for this?
I find it peculiar that this question associates 'universality' with 'intuition'; the two concepts are fundamentally in tension with each other, not in alignment.
Let's go back to that age-old debate about whether any two people are referring to the same color when they use the word 'Red'. Perception of color is internal and subjective. If I say 'Red' I may be thinking of something around the 640nm wavelength range while you may imagine something more like 720nm, and there would be no way for us to know we were imagining different colors without some serious testing equipment. Worse, if we ran across an alien species with a different visual apparatus (or a mere human with a genetic anomaly) we might consistently agree that a given object is 'Red' while experiencing entirely different internal states. But the point is that we can consistently agree that these kinds of things are 'Red' while those kinds of things are not. That constitutes a universal, the subjective, 'intuitive' aspect of it notwithstanding.
An intuition is something that lies within an individual person; a universal is something that lies across a collection of people. This distinction is difficult for a lot of people, because by default people are trapped in a solipsistic worldview in which it is extremely difficult to 'see' that which lies across them. They interpret everything as though it lies within them, and so universals come across as intuitions. But in actuality, abstract concepts like Justice and Liberty are a lot like the concept 'Red'. Different people will have different internal experiences of the concept, but the concept only exists (only has solidity) because it serves a common utility across a collective. We cannot even argue about whether a particular thing is 'Just' unless we are oriented to an abstraction that lies across all of us.
Of course, we have to take into account that some concepts are more contentious than others. I mean, I could dispute color assignments if I wanted too, saying that (say) all cherries are yellow except the yellow ones, which are blue. You couldn't really tell me I'm wrong (since you don't see my perceptions): more likely you'd say I was being obnoxious for some reason of my own, and roll your eyes in annoyance. But there isn't really much sense in someone being obnoxious that way (aside from a twisted sense of humor). With concepts like justice and fairness, by contrast, all sorts of social phenomena come into play: emotions, self-worth, wealth, power, and all the other ego-driven aspects of the human mind. There is far more reason to contest the boundaries (if not the very essence) of these concepts: to try to beak up that across-the-community universality into mere, personal intuitions. That is the essence of moral relativism. But the fact that there are distinct advantages individuals can achieve by breaking the universality of such concepts does not deny the existence of those universals, any more than (say) burning books denies the ideas that were written in them.
Because intuition works.
Our brain is a powerful tool that processes countless variables simultaneously on the background. When intuition says "A", that's the result of many calculations. See for instance a baby that learns to avoid falling from an edge way before it has learned to walk.
Or see how the main Daoist canon, the Dao De Jing, answers the question: how do Laozi knows that the world works that way? "Intuitively" or "just so" or "by this (the world itself)". And it gives this same answer three times (chapters 21, 54 and 57), in what is perhaps the smallest and most dense text in human history. So it shouldn't be understated.
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.
