Why might moral reasoning be a problem for moral non-cognitivism?

Moral non-cognitivism is the view that moral claims (e.g. 'Suffering is evil', 'Kindness is a virtue', 'Stealing from others is wrong') are not true or false, indeed, do not aspire to be true or false. Rather, they serve other purposes. One prominent version of non-cognitivism that I will focus on is expressivism. Expressivism is the view that moral claims are, despite appearances, just expressions of feeling or emotion. When I see you steal from your friend, and I tell you, 'Stealing from your friend is wrong!', what I am really doing is expressing my feeling of disapproval. Critics have labelled this the 'Boo-Hurrah theory', since moral claims seem to amount to no more than jeering and cheering, rather than telling us anything informative about the moral sphere. One crucial problem for moral non-cognitivism is that we often engage in moral reasoning. Having seen you steal from your friend, I might present you with the following argument to show you the wrongness of your actions: (1) stealing from others is wrong; therefore, (2) stealing from your friend is wrong. In this piece of reasoning, it seems like (2) follows from (1). It seems, that is, that these moral claims, (1) and (2), are subject to logical laws. But logical laws only apply to claims which have a truth-value - that is, claims which are true or false. To make this clearer, let’s take any two claims - sentences - P and Q, and assume that P follows from Q. Now, P can only follow from Q if there is no way that Q could be true at the same time as P is false. Applying this to the stealing case, it seems (2) could only follow from (1) if there is no way that (1) could be true at the same time as (2) is false. Think about it: if it’s wrong to steal from others, then doesn’t it follow - necessarily - that it’s wrong to steal from your friend? But it seems that the moral non-cognitivist has to deny this chain of reasoning: after all, if moral claims amount to no more than mere expressions of emotion and the like - that they are really just cheers and jeers - then they are simply not the sorts of things that can follow from one another. Think about other expressions of emotion: nothing follows from my crying out in pain, shaking my fist at a horse at the races or celebrating the victory of my home team. Logic just doesn’t reach such expressions. It is very intuitively plausible that moral reasoning is a legitimate and important part of our everyday lives; and so moral non-cognitivism, and specifically expressivism, seems tied down to a very implausible position.