Defining Name-R

I have said that the relation Belief_r(S,t,R,x,y,z) may hold even though there is no a such that Symbol-R(S,x,a,t). You may use a name even though it lacks reference. You may have as much evidence that it has a reference as you do for other symbols, but you are wrong. We may define Name_r as

In special symbols:

Name_r(x) iff ($S)($R)($y)($z)$t)Belief-R(S,t,R,t,x,y,z)

Thus we may say such a x is a name but not a symbol. We can tell from our beliefs that some objects are names, but, if they fail reference they are not symbols. Generally

if belief_r(S,t,R,x,y,z)

then

belief_r(S,t,($a)symbol_0r,S,t,x,a)

In plain symbols:

Name_r(x) iff (ES)(ER)(Ey)(Ez)Et)Belief-R(S,t,R,t,x,y,z)

Thus we may say such a x is a name but not a symbol. We can tell from our beliefs that some objects are names, but, if they fail reference they are not symbols. Generally

if belief_r(S,t,R,x,y,z)

then

belief_r(S,t,(Ea)symbol_0r,S,t,x,a)

What does such a name (that is not a symbol) mean?

Such names will have a practice of use similar to some other names which do have a reference. This use would provide some evidence to the person using the name that it is a symbol, i.e. there is an object it stands for. This evidence need not be conclusive. If it turns out wrong, the person will generally be able to explain what he meant by the name with a definite description. The practice of the use of the name will be sufficient to indicate the definite description, although the choice between some alternatives maybe arbitrary.

What about impossible objects?
For impossible objects, such as the round square, their names will always be in terms of definite descriptions. This may be true of objects which are not impossible as well. The golden mountain is understood by a description, even though it is not logically impossible. I am using definite descriptions here as defined by Russell. The examples I gave are really definite descriptions, not names. ‘Pegasus’ might do for the name of an unlikely, but not impossible, object. Some might argue that ‘God’ is the name of an impossible object; others might not.