Can I intuit the validity of the statement and be justified in my belief in said statement in abscence of a defeater? Is it rational to believe as such? I don't think so. I myself would attack the notion of rationality itself; what is and isn't rational, I believe, is more a matter of practical than it is of determining wihch holds more value in an epistemic sense. If one could present to me an argument which demonstrates the existence some sort of bar by which one can measure the rationality of a particular statement, I would be quite thrilled. Without such proof of the existence of such a metric, however, I would then contest whether or not any belief can be justified beyond being consistent within its own axiomatic system. That is to say, statements can only be evaluated with presumed axioms, with no other axiomatic system of any greater or lesser epistemic value than the other---as far as one can tell.
ah, establishing validity. to begin, i want to discard this notion of rationality. i will not address it. and you are correct in assuming that the absence of a defeater is not justification for a belief. furthermore, let us situate the discussion properly, as you have been using words such as justification, belief, and epistemic. this is an epistemological activity. we are trying to come to terms with why we "know" what we do.
what we hold as knowledge is belief justified through evidence. in your explorations of axioms, you tentatively hold the existence of a logical system based on axioms that is self consistent to be justified and or valid. the justification in this instance comes from the absence of contradiction.
you are on the right path in looking for other justifications such as intuition. i am sure we both recognize that a logical system could be less than accurate, and in the extreme absolutely false. at this point, we shall say that a logical system is necessary but not sufficient. therefore, intution becomes necessary.
in our epistemological exercise, we must guard against looking for justifications outside of epistemology. many people have a bad habit of looking for evidence and justifications outside of the thing in question, instead of allowing the thing to disclose itself. i am sure everyone has witnessed, or experienced this tail chasing spectacle. essentially, what i am saying is only epistemology can answer the question that has been raised.
so to the question of justifying a belief through evidence given through "intuition". intuition is a word that i will not use on this forum in this discussion. it brings too much other stuff with it. i will however use the term consciousness. so, i seek to show how a belief is justified through evidence given through consciousness.
i will also say that consciousness exists in intensities. everyone can recognize that they are less conscious when they wake up in the morning than when they have had a double espresso with extra sugar. in the same manner, we can recognize more or less consciousness in the world around us. furthermore, those flashes of scientific or artistic inspiration are always associated with a mental lucidity.
here is your metric then: consciousness is the evidence by which we justify our beliefs.
we now can say that as there is varying degrees of consciousness, we have varying degrees of validity; whereby the ultimate validity is pure consciousness. we are justified in our beliefs through the evidence of consciousness, as consciousness is the ultimate axiom. this is the one metric by which we are all familiar and all can identify.
in closing, i will also link this: http://www.the16types.info/forums/viewtopic.php?t=2489 note the last line of bertrand russels "value of philosophy", and einsteins articulation of the most important function of art and science.