Intuitions are not usually appealed to at the beginning of an argument, because then there wouldn’t be an argument at all. Instead there would simply be the assertion: intuitively things must be as follows …, which might result in the shortest paper in the history of philosophy. Rather intuitions are often appealed to when arguing against an opponent’s position, by showing that their position implies something deeply unintuitive, or something that is intuitively wrong.
One special case of intuition is physical intuitions. A physical intuition is an intuition that a system in a certain environment will work a certain way, or that if certain physical conditions hold then a given result will follow. For example, I “intuitively” know that if I throw something into the air it will come backs down, unless something interferes with it. Or, more complexly, I “intuitively” know that a room that is painted in inverted colors will look normal if I am wearing contacts that also invert colors. These intuitions are not really intuitions at all, because they aren’t appeals to the way I feel the world works, but rather appeals to what I think will happen given that I have an understanding of science. These intuitions could in principle be checked, rigorously, in a scientific fashion, either by appeal to theory or by doing an experiment. Thus they are more like a convenient way to appeal to scientific truths without going into scientific detail.
But not all appeals to intuition are like this. For example, a philosopher might attempt to refute an opposing theory by showing that it labeled some situation as just that is intuitively unjust. There is, I think, a problem with such appeals to intuition. Our physical intuitions were justified because they were based on our experience of the way things physically work, and on the scientific knowledge we have picked up more formally. But what justifies an intuition about justice? We don’t experience or see justice in the world. It is true that when we come upon certain situations we may, without reflection, seem them as just or unjust, and we may base our judgments about other circumstances on these experiences. But then our intuitions aren’t about objective justice, rather they are about our concept of justice. And concepts can vary between people, and so if we are really only studying a concept then we are studying something that is purely a matter of convention, which in turn means that there are no real, objective, facts to be uncovered in this area. And if that is the case then what does it matter if your opponent isn’t describing your concept of justice, or even the majority opinion about justice; maybe he or she is describing their own concept of justice, and since we are working with a matter of convention that doesn’t make them wrong.
This might seem to make philosophy impossible. After all most of the subjects dealt with by philosophy do not seem to be captured by our knowledge of science. Philosophy deals with abstract things like ethics and meaning and truth; since we can’t point to them how are we to study them at all? Obviously intuitions will play a role, but instead of playing a specific role, in refuting or supporting specific conclusions I think that they are better suited to playing a definitional role. For example, if we are studying ethics we should start with an understanding of what ethics is. Now this shouldn’t be a detailed definition, since that would be to build the conclusion into the premises. Instead it should be more like a functional definition, meaning that we should start from a definition of what ethics does. For example, ethics governs interpersonal relations. Furthermore, ethics is a self-imposed regulation of these interactions, which is what distinguishes it from economic and legal rules. And then we can ask ourselves why self-imposed rules of this kind should exist at all. Are they imposed by genetics, imposed by society, or chosen freely? If it is the first then they are to the benefit of the species. If it is the second then they are to the benefit of society. And if it is the third then they are to the benefit of the individual. From there we can choose one or a combination, and then go on to define in detail what the ethical rules are, how they are beneficial, and what they dictate in specific situations. Now it is true that someone may disagree with the initial definition, but if this is the case it doesn’t mean that the object of our investigations is in some way conventional, it means that we are simply using the same word to signify different things; meaning that we have a disagreement about language, not a disagreement about content.
The important issue here, however, is not about the method of a successful philosophical investigation. Rather it is to point out that whenever we start appealing to intuitions we may inadvertently be derailing our investigations to be about our concepts. And since our concepts are generally conventional we won’t be making any real headway on the issue at all. Arguments about ethics and epistemology tend to have this flavor, with positions being “refuted” by demonstrating that some consequence is unintuitive. Obviously such arguments refute nothing, but they do show that the matters being discussed have become largely conventional. And I don’t think that ethics and epistemology are matters of convention, and so any approach that treats them as such has necessarily strayed from its intended topic and is instead discussing something else, such as our ethical or epistemic intuitions.