On the issue of homosexuality, they merely express repugnance at the idea, and this forms the basis of the view that gay marriage should be illegal. I have never heard a logical argument for why gay marriage is wrong, rather it all stems from emotion.
On the issue of Islam, they express anger and disapproval towards the teachings of the Quran, and again, they form their anti-Islamic views merely on this emotional basis. Yet there is no logical argument for how or why the teachings of Islam are factually wrong.
On the issue of abortion, they again merely express an extreme dislike at the idea of infants getting murdered. Yet again, they are completely unable to offer a logical argument to prove that infanticide is actually wrong...again they rely purely on their feelings to say that abortion is morally wrong.
Now obviously the most bizarre facet of the illogic of conservatives, is that they seem to project this lack of logic onto their political opponents. You've heard them claim that liberals are illogical who can't think rationally. When a conservative make this claim, what he is actually doing is spewing forth psychological projection, which is a sign of weakness, insecurity, and dishonesty. Because as I've just shown above, they in fact have no rational arguments to support their own views.
Are conservatives incapable of rational argumentation?