Logic: Do fictional characters exist?

Recently, I was confronted with a question that can be briefly phrased as follows-:

--------Question
how to deal with things like Santa and purple cows in logic. If they do not exist, then how are we supposed to represent them in predicate logic.
--------end of Question

The following is an attempt at an answer.

There would be no problem here if we did not insist upon the "existence" of all entities for them to be true, etc. When I say that "the number 2461 exists", the existence implied is different from that which is implied by my saying that "in my pocket, a pencil exists". For the latter existence, the verification has to be done empirically in the spacetime of my pocket. But the existence of the number 2461 or any other is a matter of pure logic. That is, if 2461 ia an member of the class of numbers, then one might say it exists. This kind of usage is uncommon but I have introduced here with a purpose. Consider the following:

---------ILLUSTRATION 1:

X is an entity satisfying the following:
1. X belongs to the class A.
2. X belongs to the class B.

Then, does X exist? The answer can be given by looking for a member of the class A which is also a member of the class B. If such a member can be found then X exists, else it does not.

---------end of ILLUSTRATION 1

If you agree with the above illustration, then we can come to the question of the entity called Santa. To know what is Santa, we need to know the complete list of those attributes which when satisfied by an entity, the entity can be said to be Santa. Call this list of properties/attributes P. Then, to verify whether Santa exists or not one just has to look for an object that satisfies all those properties. If one finds any, Santa exists; otherwise, he does not.

The problem arises from the fact that when listing the attributes in P, one generally leaves a lot to be assumed. Suppose P included the attribute f: "Santa belongs to the class of characters in fictions" along with the other attributes, then depending upon what those other attributes are Santa may , or may not, exist. But if P included the attribute f': "Santa is not a fictional character" then one would probably assume that Santa's verification would require to look for him in the physical world.

To summarize the above, from the point of view of logic it is unnecessary to maintain a difference between entities depending upon whether they satisfy the attribute f ("is a fictional character") or its opposite f' ("is not a fictional character"). Therefore, there appears to be no reason why the fictional characters (that do not exist in the physical world) should be treated any differently from others. Also, they would not require a different representation in predicate logic unless the representation is unnecessarily made dependent upon the attribute f.

Fictional characters exist on the same footing as the concepts, ideas, etc. exist. The difference between fictional and physical begins to show when one moves from the world of pure logic to that of verification of existence. The verification could end up being entirely within logic as in the case of contradictions and tautologies, or it could cross the boundary of logic into the empirical sciences.