It is alleged by the noncognitivist that talk of abstract objects as existing things is literally meaningless. What does it mean to say, for example, that the law of non-contradiction exists, or that the number 7 exists? Can such statements be defended from the noncogntivist's attack?
Notice that one needn't hold the belief that abstract objects actually exist in order to consistently maintain that it's meaningful to talk as if they exist. The nominalist might say that the number 7 does not exist, but that "the number 7 exists" is a meaningful, coherent proposition.
One may advance an argument based on the indispensability of abstract object-talk:
1. If a proposition is meaningful, each of its referents must possibly exist. (Premise)
For example, if I were to say, "unicorns are magical horses with a horn," I am expressing a proposition in which each of the words in the sentence at least possibly exists, even though this particular entity does not, in fact, exist.
2. Some propositions include abstract objects as referents. (Premise)
"The Prime Minister is not a prime number," includes an abstract object - prime number (e.g. 7).
3. Some of the propositions in (2) are meaningful. (Premise)
"The Prime Minister is not a prime number" expresses not only a truth, but a necessary truth. Prime Ministers just aren't the type of entities that can be prime numbers. Yet, a proposition can only be true if it is meaningful. From this it follows that:
4. Therefore, abstract objects are meaningful referents. (From 1 - 3)
As stated before, the meaningfulness of abstract objects does not necessarily entail that abstract objects exist. However, one may very well advance an additional argument that states that indispensable truths must exist. It is self-refuting to reject the laws of logic, for example, which entails their indispensability, and it's their indispensability that entails their existence as abstract objects. (I'm thinking of logic as abstract for fairly obvious reasons. The law of non-contradiction doesn't do anything; it doesn't stand in any causal relations, so it cannot be concrete.)
Logic, then, is necessary at all times and all places. This is a viable starting point for an argument of natural theology. Logic is either the concept of a mind, or else it is mind-independent. If it is conceptual, then it cannot be the concept of just any mind (you and I possibly fail to exist at various times), but must be the concept of a necessary mind, God.