This is actually a fairly sophisticated question, so let's analyze a bit.
Consider the claim that a fully rational person does not believe both p and not-p. Very likely, as you considered it, that claim seemed true to you. Something similar probably happens when you consider the following propositions:
What is this saying? It's addressing the fact that most people intuitively accept the principle of bivalence:
In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.
Then, several types of assertions are presented. To wit:
[I1] If not-not-p, then p.
[I2] Torturing a sentient being for fun is wrong.
[I3] It is impossible for a square to have five sides.
[I4] A person would survive having their brain transplanted into a new body.
Note that the propositions come across domains:
- I1 is logical.
- I2 is ethical.
- I3 is geometrical.
- I4 is anatomical and physiological.
Something similar does not happen to me in the second statemen[sic] (or even the last).. Why are these intuitions considered similar?
So, let's address some facts about intuitions. As the article says:
[A]gents with sufficient experience in a given domain (e.g., neonatal nursing, fire-fighting, or chess) arrive at judgments and make decisions on the basis of a cognitive process other than conscious considerations of various options and the weighing of evidence and utilities.
So first, there are a range of intuitions which philosophers recognize might be seen as on a spectrum from naive (as in naive physics) to expert (as in the intuitions of Einstein's Gedankspiel that led to special and general relativity). What makes an intuition an intuition is that given a naturalistic approach to philosophy as the author(s) of the article use they are subconscious. The philosophical investigation of intuitions, therefore, is dependent upon psychological research. This sort of philosophy has become known as experimental philosophy.
In the last ten to fifteen years, there has been an explosion of interest in research involving the scientific or empirical investigation of intuitions of philosophical interest. Such projects are now frequently grouped under the rubric of “experimental philosophy.” (For general characterizations of experimental philosophy see Knobe and Nichols 2008, Appiah 2008, and Nadelhoffer and Nahmias 2007.) At least four broadly-individuated projects in experimental philosophy can be distinguished.
This is also the thesis, for instance, of Malcom Gladwell's Blink (GB). That is, psychology asserts that there's no fundamental distinction between a baby's intuitions regarding a visual cliff and a physicists intuitions on manifolds as a basis for understanding spacetime. What makes intuitions special in philosophy is that they are one of a handful of sources of epistemic knowledge, along with reason, revelation, and faith. Intuitions is the word philosophers accept and use when propositions are built and appear spontaneously to the conscious mind.
Now, you say the intuitions are different. Sure, for starters above, each of those is a source domain. Logic, ethics, geometry, and anatomy-physiology are well recognized disciplines. And if we accept that there are various levels of sophistication of intuition from baby to world's greatest expert, then it makes sense that not all people will experience intuitions in the same way in various domains. Consider for a moment that a person diagnosed with OCPD will show a marked preoccupation with conscientiousness which may present as self-righteousness. For such a person, I2 may seem more obvious than I1, I3, or I4. So, given that one can have various levels of intuitional sophistication in a domain, there are multiple domains of knowledge, and that individuals may have constitutions that favor domains, it should come as little surprise that individuals have vastly different intuitions. To anyone learned in philosophy, that's not a surprise. No professional worth his salt would believe that Rene Descartes had the same intuitions David Hume. Most philosophers, ultimately, appeal to intuitions when establishing their positions, and for every three philosophers, there are certainly four positions.
In fact, a sadist might actually want that. Why are these intuitions considered similar? I would argue that the very notion of a moral intuition seems nonsensical: it's just a desire that many people share. Desire != intuition.
You're intuitions are good on this account, if you consider that you are essentially espousing what Ayer espoused last century in a metaethical position known as non-cognitivism. From the article:
Non-cognitivism is the meta-ethical view that ethical sentences do not express propositions (i.e., statements) and thus cannot be true or false (they are not truth-apt). A noncognitivist denies the cognitivist claim that "moral judgments are capable of being objectively true, because they describe some feature of the world".1 If moral statements cannot be true, and if one cannot know something that is not true, noncognitivism implies that moral knowledge is impossible.
What do naturalistic philosophers make of this position? Well, psychologists, sociobiologists, and at least primatologist will tell you that pre-ethical moral impulses are what hold social animals together. In fact, there's a term for it: psychological altruism. Frans de Waal's Primates and Philosophers: How Morality Evolved (GB) presents the argument that essentially, ethics is nothing more than taking the intuitions nature provides us on treating our fellow human well, and adds the process of justification to it.
Are moral intuitions considered equally valid as other intuitions?
So naturalistically speaking, there is no grounds to conclude that our moral impulses are any less important than those of logical, mathematical, or scientific thoughts. In fact, excepting clinical psychopathy, I think a strong case can be made that peoples moral intuitions are every bit as a consistent as our logical and mathematical ones, and in the case of most people, probably even more reliable since the human brain has evolved to make psychological altruism a central pillar of its function in a way that logical and statistical make people struggle to understand.