The Problem of Apparent Reference to Nonexistents

    Let S be the sentence, ‘The present King of France is bald.’  The following set of statements is inconsistent:

        1.  S is meaningful (significant, not meaningless).
        2.  S is a subject-predicate sentence.
        3.   A meaningful subject-predicate sentence is meaningful (only) in virtue of its picking out some individual thing and ascribing some property to that thing.
        4.  S’s subject term fails to pick out or denote anything that exists.
        5.   If S is meaningful only in virtue of picking out a thing and ascribing a property to that thing (1,2,3), and if S’s subject term fails to pick out anything that exists (4), then either S is not meaningful after all (contrary to 1) or S picks out a thing that does not exist.  But:
        6.   There is no such thing as a ‘nonexistent thing’.
 

The Problem of Negative Existentials

    This is a special case of the preceding puzzle, but an aggravated one.  Let P be the sentence, ‘The difference between A and B does not exist’.  Now, P seems to be true and seems to be about the difference between A and B.  But if P is true, P can’t be about the difference between A and B, for there is no such thing as the difference between A and B for it to be about.  Likewise, if P is about the difference between A and B, then P is false.  (For a less stilted example, try ‘The round square’, ‘The faster-than-light drive’ or ‘The free lunch’.)
 

Frege’s Puzzle About Identity

    An identity-statement such as ‘Scott = the author of Waverley’ contains two singular terms, both of which pick out or denote the same person.  It seems, then, that what the statement says is simply that that person is identical with that person, i.e., that that person is identical with himself.  If so, then the statement is trivial.  Yet the statement seems nontrivial, in each of two ways:  (i) It is informative, in that someone might learn something new upon reading it.  (ii) It is ‘contingent,’ as philosophers say; the fact it states is one that could easily have been otherwise.
 

The Problem of Substitutivity

    The function of a singular term is to pick out an individual thing and introduce that thing into discourse.  It is in virtue of this denoting role that singular terms are meaningful at all.  Therefore, we would expect that any two singular terms that denote one and the same thing would be semantically equivalent:  we could take any sentence containing one of the terms and substitute the other of the two for the first term, without changing the meaning or at least without changing the truth-value of the sentence.  But consider the sentence

        (R)  George IV wished to know whether Scott was the author of Waverley

and suppose R is true.  Now, Sir Walter Scott was the author of Waverley; so ‘Scott’ and ‘the author of Waverley’ denote the same thing.  Therefore we should be able to substitute the latter for the former in R without changing R’s truth-value.  But we cannot; the result is a false sentence, since George IV did not wish to know whether Scott was Scott.
 

The Problem of Excluded Middle

    Let S again be ‘The present King of France is bald’, and let Not-S be ‘The present King of France is not bald’.  The following set of statements is inconsistent:

    1.   Every meaningful sentence is either true or false.
    2.   S is meaningful.
    3.   S is not true, since the class of bald things contains no such entity as the present King of France.
    4.   If S is false, then Not-S is true.
    5.   Not-S is not true, since the class of non-bald things contains no such entity as the present King of France.