
There's a very simple philosophical method that we learn at an extremely early age, and is then beaten out of us by our parents because it's just too darn powerful in the hands of a small child. You could call it "Infinite Regression Questioning", but only if you like sounding pompous, because it consists of asking "Why?" indefinitely (or until the other person gets fed up).
Logical systems start with some assumptions/axioms and then combine them in various ways (according to the rules of logic) until conclusions are reached. To choose a simple example - the two axioms "Andrew is a man." and "All men are evil." lead very swiftly to the conclusion "Andrew is evil."
Asking "Why?" allows you to work back up that chain to previous statements, and unwind things until you reach the axioms. "Andrew is evil." Why? "Because Andrew is a man, and all men are evil". At which point the obvious thing to do is to ask "Why?" again. I mean, you don't _know_ if those are assumptions or if they were produced by logical steps from some pointer higher up the chain.
There are two ways that this can progress. You can reach a statement that didn't come from anywhere - it's an assumption/axiom which isn't proved, just believed. This is _fine_ - so long as you're comfortable that your beliefs spring from inside you in the first place, discovering the point at which they appear isn't a bad thing, it just tells you something about yourself.
Alternatively you can find a causal loop where the next step up is also a point you've already reached. A very obvious example of this is "We know God says so because it's in The Bible." "We know the Bible is true because God says so." - at which point you can either decide that one of the points is something you have belief in without any cause (turning this into a case of the first kind) or you can accept that this logical chain never actually grounds itself anywhere, and floats freely around as a little bubble.
Of course, if you don't care about your morality being logical then you can just treat all of your beliefs as axioms - you feel that way because that's the way you feel, and you don't need to justify any of it. Which leads us handily back round to the start, and the answer parents give when asking "Why?" gets on their nerves - "It just is, ok?"
Of course, when dealing with people who aren't prepared to just believe you, you might need to be a touch more presuasive than that...