I will consider the objections that Kripke has to the Description Theories of Proper Names in general -Kripke thinks that Cluster theory is in this kind of theories too.
Kripke summarizes such theories in six theses. The theses roughly say:
“Theses 1: each proper name corresponds to family properties φ such as a speaker A believes that x satisfied φ A(φ x)
Theses 2: A believes that one of these properties φ or some of them considered conjointly pick a particular object, and only that particular object.
Theses 3: Most of that property is satisfaced by an object y which is the referent of x.
Theses 4: If the vote doesn´t pick out an object x then x doesn´t refer.
Theses 5: “If x exists then x has most of the properties φ” are known by A.
Theses 6: “If x exists then x has most of the properties φ” is a necessity truth.”
The principal condition that every theses has to satisfy is the principle of circularity.
The first argument that I´m going to consider is precisely such argument considering that if we accept the Descriptive theory falls into circularity as it violates the second theses, for intance a speaker A names “Newton” A refers to an particular object because A believes that “Newton is the discover of gravity´s law” if we ask A what is the gravity´s law A can´t answer “Newton´s law” because A falls into circularity but if A doen´t have more knowledge about gravity´s law then A is definitly in a vicious circle.
The next arguments disable the theses 2, 3 and 4 such arguments are from the speaker´s ignorance and speaker´s error.
The argument of Ignorance as the name suggests highlights the ignorance of the speaker. The speaker A only knows a property asociated with the name “Mozart” i.e. “He is a famous melodist”. However we can´t say that such property -the property of being a famous melodist- picks out one particular object. There are more than one famous melodist, if B asks who is Mozart? A answers a famous melodist. B could think that is Beethoven inasmuch as the name “Beethoven” satisfies the descripcion asociatted with the name. In this case Theses 2 is violated because of the properties associated with the name don´t pick out only one and the same object, the Theses 3 and 4 are violated too because “Beethoven” satisfaced the descriptions associated with the name “Mozart” but “Beethoven” is not “Mozart”. The most properties can be associated with the name Beethoven. Beethoven is not the referent of the name “Mozart” (T.3). “Beethoven” and “Mozart” aren´t a unique object (T.4).
The argument of ignorance has to do with the ability of the speaker using the name, the knowledge of the particular object which they want to refer. If the speaker doesn´t have enough knowledge of a proper name and doesn´t find a sufficient cluster description associated with the proper name which satisfies the particular object then the Descriptive theory would say that the name doesn´t refer but Kripke defends that even if there aren´t any properties the proper name designs the reference because determinating the reference of a proper name does not depend of the descriptions that a speaker is associating.
The error argument. The properties associated with a name correspond with a wrong person. If a speaker A using the property “The author of The concept of anxiety” to refer the object “Kafka”but if “Kafka” wouldn´t be the real author i.e imagine that the book was written for a person whose was Vigilius Haufniensis Everybody thinks that is a seudonim of Kierkegaard but in fact a different person other than Kierkegaard called Vigilius wrote the book. A want to design the object “Kierkegaard” but in fact A is not talking about Kierkegaard but another person namely Vigilius. Kripke defends that that is not true, even
if the description doesn´t correspond with Kierkegaard A is not talking about Vigilius, A is still talking about Kierkegaard. That argument cancels out the theses 3 and 4.
The last potential objection against the Descriptive Theory is the modal argument. This argument is against the theses 6, i.e. Attacks the notion of necessity. It is not necessary that most of the descriptions associated with a proper name apply to a particular object because we can imagine a different world where the object doesn´t satisfy any properties and we will still assert that we are talking about the same object because according to Kripke the proper name has nothing to do with the descriptions .e.g. We can imagine a world where “Kafka” is not the author of Metamorphosis, where he doesn´t die yonger, where he is married... etc. Even so we know that we are referring the name “Kafka” to the particular object Kafka even though he doesn´t satisfy any properties associated with the name.
Finally Kripke holds that theses 5 is violated because the theses 3 and 4 aren´t a priori but accidents so the a priori known by the speaker doesn´t occur.