A) How does Frege's distinction between Sense (sinn) and Nominatum (Bedeutung/ Reference) help us solve the puzzle of (true) informative identity statements such as : "Mount Everest is Chomolungma"?

The puzzle says that “a=b” must provide new information given that a and b are the same, even though the relationship an entity has with itself is a tautology. The first approach, disapproved by Frege, to decoding the statement “Mount Everest is Chomolungma” is that the objects referred to by “Mount Everest” and “Chomolungma” are the same. Then, “Mount Everest is Mount Everest” would mean the same thing as the previous statement, which is false, because the first proposition is a posteriori and the second is a priori. Therefore, equality is not between objects. To avoid the paradox, he realized that the two sentences must state different propositions. Frege comes up with the theory that the informative identity is related to the names. Now, we say the name “Mount Everest” denotes the same thing as “Mount Everest” and “Chomolungma”. However, saying names are equivalent means only referring to the linguistic property of denotation, making no claims about reality, and not expressing any ‘proper knowledge’. Neither referents nor names can be considered equal because then we don’t learn anything about the world. Frege solves this by introducing the mode of presentation: how an object is presented to us perceptually. If “Mount Everest is Chomolungma” is true, it tells us the two ways that we designate the same referent. The referent is the same but the senses (containing mode of presentation) are different. Sense represents an aspect of an object, and there can be many senses of the same object, given many aspects. Due to this, we always gain new insights through informative identity statements, because it is impossible to know every sense of each object beforehand.

B) What were Russell's major objections against Frege's theory of Sense as an ingredient of the meaning of a word, distinct from its reference/ nominatum or denotation?

According to Russell, we can’t refer to the sense of an expression without leading back to the referent. Further, the recursion when an expression refers to a sense was an objection: what sense would the expression have then? These highlight the baseline argument that we can’t capture what sense is - though not argued concretely by Russell. The meta nature of trying to understand what ‘sense’ is, means we must continue to give denotations to it and try to capture the sense of those denotations of the sense and so on. Through this confusion of the relationship between sense and reference, comes the main point of contention for why they must be separate. This is highlighted if we have a descriptor

without a referent, such as “the King of France”. When there is no referent, we say there is no truth value to the sentence, and that the sentence is rendered meaningless (neither true nor false). This is a contradiction because Frege would say that there is a sense, and therefore meaning. Russell also found this bizarre because clearly, we know the statement is false. Therefore, if we have a distinction between sense and reference, we arrive at a corner case where a sentence without a reference is both meaningless and meaningful at the same time. Therefore, Russell adopts the system where he removes sense altogether, and directly relates meaning to referents.

C) How does Russell's theory of definite descriptions help us solve the problem of true denials of existence, e.g. "The Gyphxon does not exist"?

Russell believes if a name has no reference, it has no meaning. Since Frege regards definite descriptors as proper names, we assume “The Gyphxon” is a name. “The Gyphxon” doesn’t refer to anything, so it is an empty descriptor. Given the above premises, we conclude that “The Gyphxon does not exist" is a meaningless statement, since there is no referent. Therefore, the problem of true denials of existence is that the given sentence cannot be true since it is meaningless, but we know it is true. How can we say something doesn’t exist without using a descriptor, which would imply the contradictory existence of said object? Russell says that definite descriptors are not names, but propositional functions with no reference. They have no meaning of their own, and gain it when they are assigned an instance. Now, in the denial of existence statement, we don’t mention a particular entity but instead, talk about a propositional function having no instances. We say ‘It is not the case that X is a Gyphxon and X exists’. This paraphrasing removes the need for a reference and allows us to talk about nonexistent things. Without Russell’s system, Frege would say there is sense but no reference, and since the sentence has no reference, it would have no truth value. According to Frege, the statement would be neither true nor false, which is incorrect, as the statement is clearly true. Therefore, in Russell’s system, we do not presuppose the existence (or non-existence) of a referent. We can conclude whether the statement is true or false, instead of rendering it as meaningless.