In his famous paper “On Denoting,” Bertrand Russell argues that, just as experiments are used to test the merit of scientific theories, logical theories either sink or swim based upon their ability to “deal” with puzzles. For his present purposes, Russell gives us three puzzles “which a theory as to denoting ought to be able to solve” (36):

The Problem of Substitution

Russell’s first puzzle involves substitution (36). If a=b, then whatever we can say about a must also hold true for b, for they are identical to one another. This is often called the “indiscernibility of identicals” or sometimes “Leibniz’s Law,” after the philosopher who first formulated it. Given that a and b are identical, we can substitute one for the other in a given proposition without altering the proposition’s truth or falsehood.

But problems arise when we apply this principle in everyday life. For example, consider the following proposition (I’ve updated Russell’s example to deal with an author that I personally enjoy):

     “Daniel wished to know whether Mark Twain was the author of Huckleberry Finn.”  

As many know, it is true that Mark Twain was the author of Huckleberry Finn. Thus, according to the indiscernibility of identicals, we can substitute “Mark Twain” for “the author of Huckleberry Finn” without changing the truth of the proposition. But after doing so, we end up with this sentence:

     “Daniel wished to know whether Mark Twain was Mark Twain.”  

Surely, something has gone wrong here—it is not the case that I am interested in Mark Twain’s identity to himself.

The Problem of the Excluded Middle

Russell’s second puzzle involves applying the well-accepted law of the excluded middle to what Meinong would have referred to as “nonexistent entities” (36). We all know the law of the excluded middle: p or not-p. Thus, it is either the case that “A is B” or “A is not B.” But what if we replace A with a “nonexistent entity”?

     It is either the case that: “the present King of France is bald” or “the present King of France is not bald.”

Now, if we, in an act somewhat similar to Santa Claus, make a list of all the things that are bald and a second list of all the things that are not bald, we will find that “the present King of France” has failed to make it onto either of our lists—-even if we check it twice. Of course, the reason is simple: there is no present King of France. But, nevertheless, the law of the excluded middle seems to have failed us somehow. The question is: why?

The Problem of Difference

Russell’s final puzzle involves the problem of making a non-entity the subject of a proposition (36). If we can say truthfully that there is a difference between A and B, then we can express this fact by saying that “the difference between A and B subsists.” Accordingly, if there is no difference between A and B, we should be able to say that “the difference between A and B does not subsist.” But in this sentence, we find ourselves dealing with a non-entity as a subject—a non-subsisting difference—and, hence, seem to be contradicting ourselves. We are left with two uncomfortable choices. We can either toast our glasses with Meinong and affirm the being of non-entities, or we can deny the being of non-entities and stand in apparent self-contradiction. How can we get out of this trap?

How To Make Sense of Propositions with “Non-Entities” In Them

Russell thinks that we run into trouble when we try to analyze denoting phrases in isolation from the propositions in which they occur (34). For example, when we isolate “the present King of France” from the propositions that contains this denoting phrase, we become stumped while trying to find its denotation. In order to avoid this puzzle, Russell suggests that we translate the entire proposition into a particular logical form. Thus: “the present King of France is bald” becomes something like:

     “There is an x such that x is the present King of France, nothing other than x is the present King of France, and x is bald.”

It is now easy to see how the problematic sentence “the present King of France is bald” really presents no problem at all. The sentence is not meaningless, as Frege would suggest, and it does not rely on the nonexisting entities of Meinong’s theory to derive its truth value. Rather, the sentence is simply false given Russell’s translation.

Primary & Secondary Occurrences: Quine and the Shortest Spy

Russell makes a distinction between “primary” and “secondary” occurrences for denoting phrases (38). This distinction is often blurred in ordinary language, but can be easily distinguished in logic. The usefulness of this distinction is that by attending to this distinction, we can solve the three puzzles raised earlier. I believe it will be easier to explain this distinction if we just see it in action. Consider the following sentence:

     “Quine believed that Ralph Ortcutt was the shortest spy.”

For Russell, this sentence disguises at least two different propositions, one in which “the shortest spy” occupies a primary occurrence and one in which it occupies a secondary occurrence. Here they are respectively:

1. (Primary) “One and only one man was the shortest spy, and Quine believed Ralph Ortcutt was that man.”
2. (Secondary) “Quine believed that one and only one man was the shortest spy and Ralph Ortcutt was that man.”

What has occurred in each of these translations? In the original sentence, there is a main clause and a subordinate clause. The subordinate clause is “Ralph Ortcutt was the shortest spy” and the main clause is “Quine believed that Ralph Ortcutt was the shortest spy.” As we have already seen, the subordinate clause contains a denoting phrase: “the shortest spy.” Now, whether that denoting phrase is primary or secondary seems to hinge upon how we separate it from the sentence. If we separate the denoting phrase from the subordinate clause in which it occurs, we end up with another subordinate clause: “one and only one man was the shortest spy and Ralph Ortcutt was that man.” This yields a secondary occurrence for Russell, because the denoting phrase is still part of the subordinate clause and is subordinate (or secondary) to the main clause—it is subordinate to Quine’s belief.

However, if we separate the denoting phrase from not only the subordinate clause, but the main clause as well, we end up with a primary occurrence: “One and only one man was the shortest spy, and Quine believed that Ralph Ortcutt was that man.” Notice that the denoting phrase is no longer subordinate to the original main clause—it is no longer subordinate to Quine’s belief. Thus, we have a translation in which the denoting phrase occupies a primary occurrence.

I’m not quite sure how to answer the final question: could Quine have believed either of these propositions without knowing that anyone is named Ralph Ortcutt? I don’t see why not. In both sentences, “Ralph Ortcutt” is subordinate to Quine’s belief. Thus, if Quine truly believes that “Ralph Ortcutt” is the man in question, I suppose he must believe that there is someone who is named “Ralph Ortcutt”—-that “there is some x, such that x is human and x is named ‘Ralph Ortcutt.’” Whether he knows or not seems to me like a separate question.

-- DanielS - 12 Sep 2005