Russell, Frege, and the Law of the Excluded Middle

Foreword

This was my first paper for Philosophy 516: Language and Meaning. It talks about Russell's Theory of Descriptions, particularly from 'On Denoting'. We discuss the problems his theories solve, and why we should prefer his theory over Frege's.

Body

Bertrand Russell tries to push forward some new ideas for understanding denoting phrases in his “On Denoting”. He is interested in a few puzzles with previous understandings of how language operates (Frege). One of note deals with the law of the excluded middle, the idea that either “A is B” or “A is not B” must be true. The issue with language is that a statement like “The present King of France is bald” and “The present King of France is not bald”, prima facie seems false in both cases, considering there is no current King of France, and that an enumeration of both things that are and are not bald will not uncover ‘the present King of France’. This issue persists with anything denoting a sentence that seems grammatical, yet attempts to denote something that does not exist. We will discuss Russell’s arguments on denoting phrases, how his ideas solve such problems, and discuss how Frege would try to solve the same problem.

Russell needs to do a little bit of work to get his theory off the ground. He defines a denoting phrase, which roughly is a phrase that can be used as the subject of a full sentence, not including proper nouns (e.g. ‘a man’, ‘every man’, ‘the present King of France’, etc.). These are the types of phrases Russell is interested in. It is noteworthy that a denoting phrase may not necessarily denote something (‘the present King of France’), which is similar to Frege’s signs, specifically that a sign ought not necessarily have a referent. This is where the issue of the excluded middle comes about, something that does not exist clearly will not show up in the set of things that are or are not B.

Russell’s theory involves converting denoting phrases into a logical form, by changing the grammatical structure to a logical statement “C(x)” where x is a variable that could be anything in the material world (e.g. a point in the universe, my cat, the Eiffel Tower). We can thus convert any sentence into a logical form. For example, Russell states that “I met a man” asserts that “‘I met x, and x is a man’ is not always false.” Essentially stating that there is at least one thing in the universe in which the statement ‘I met x, and x is a man’ is true. We can also consider the philosophical classic “All men are mortal.” Russell translates this to “‘If x is a man, then x is mortal’ is always true.” Meaning there is no counterexample to the statement ‘If x is a man, then x is mortal’

Next, Russell goes into an interesting tangent about the word ‘the’ and its implications in a denoting phrase. Russell argues that ‘the’ implies uniqueness, that if a denoting phrase is preceded by ‘the’ then it is implied it is denoting a single thing (regardless of if that thing is actually denoted or not). Thus, a statement such as “The father of Charles II” is translated in logic to “It is not always false of x that x fathered Charles II and that x was executed, and that ‘if y fathered Charles II, y is identical to x’ is always true of y.” It seems wordy. But, the first half of the sentence says that there is some thing x that fathered Charles II and was also executed, which shouldn’t look unfamiliar. Followed by a statement that just ensures uniqueness, if anything is Charles II's father then that thing is x.

Consider again the law of the excluded middle. The issue is attributing a property to a denoting phrase that does not exist in the material world. Russell, cleverly, shows that the denial of a property existing on something has multiple interpretations, namely:

(1) The present King of France is not bald.

(1p) There is now an entity that is a King of France and it is not bald.

(1s) It is false that there is an entity that is the King of France and it is bald.

(1) represents the grammatical sentence being spoken, while (1p) is the rephrasing into a pseudo-logical format where ‘the King of France’ is primary, while (1s) is where it is secondary. We evaluate (1p) to false, intuitively, but we call (1s) true, as it just negates that there is an entity that is the king of France. This preserves the law of the excluded middle, because now there is an interpretation of “C is not b”, where C is a denoting phrase that denotes something that does not exist, and b is any possible property, in which the statement evaluates to true.

Frege would likely have some greater trouble attempting to conclude this same thing. For context, Frege is interested in propositional phrases and proper nouns (e.g. “Mark Stanley”, “The evening star”, etc.). The sign of these phrases is the words themselves; the referent of the sign is the actual physical thing that a sign relates to. He then argues that there is a third thing, sense, which in a way is like the vibe of the phrase. He argues that something like “the morning star” and “the evening star” cannot be replaced by each other in sentences, and thus there must be some deeper meaning further than just referent, this is what he calls sense. This seems prima facie. Frege then becomes interested in the discussion of the referent of an entire sentence. He argues that the referent of a whole sentence is the sentence’s truth value.

The way Frege gets out of the issue of the law of the excluded middle seems like a cop-out. He argues that sentences that involve things that don’t have actual referents (or that don’t denote something) cannot be evaluated as a truth claim. Frege argues that because we are doubtful about whether a constituent of the sentence can have a referent, there is no reason to believe that the entire sentence can have a referent. This, in a way, begs the question, as it assumes that the truth value of the entire sentence is of the same type as what a sign refers to; there is no clear reason to call both ‘referents’. There is a reason to be able to cast doubt on the existence of the truth value without acknowledging them as referents, however that specific line of reasoning is weak. Fictional names also presents issues to this. When on makes the claim “Sherlock Holmes is a detective.” our first response is not “but Sherlock Holmes doesn’t exist.”

This alone is a reason to prefer Russell’s view over Frege’s. Frege may have some argument to solve this problem that mimics what Russell did, while keeping his definition, but Frege was so abstract in what he meant by “sense” that it is not clear that Russell’s view doesn’t simply encapsulate Frege’s, with a few modifications here and there.