Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. It only takes a minute to sign up.
Sign up to join this community
I don't get it. Assuming there exists an individual Caesar, we can look at the set of prime numbers and not-(prime numbers), and Caesar will be in one of them.
I just don't see, even though it may be a bit silly to ask, why he rejects that this has meaning.
(A rejection at the time in his life when he wrote 'Elimination of Metaphysics' which I'm currently reading, at least.)
In the sections leading up to that claim Carnap discusses a first class of what he calls 'pseudo-statements', which are all sentences characterized by having in them some 'meaningless' word. The sentence "kjdfho is great" is a pseudo-statement of that first class, because it includes the presumably meaningless expression 'kjdfho'. Then, in §4 Carnap turns his attention to a second class of 'pseudo-statements', which includes sentences that while grammatically meaningful, are still not acceptable. As examples he considers the following ungrammatical sentence (1) and the grammatical sentence (2):
(1) Caesar is and;
(2) Caesar is a prime number.
The problem with (1) is obvious: it's not a well-formed (or grammatical) sentence. The assumption here is that 'and' is an operator on sentences, so its placement in (1), in an NP position, takes us out of the set of grammatical sentences of English. From the logico-semantical point of view, (1) is as ungrammatical as the aforementioned "kjdfho" sentence, and thus belongs to the first class of 'pseudo-sentences'. The problem with (2), however, is not that it's not well-formed. 'Caesar' is a grammatically acceptable name of an individual, and 'is a prime number' is an acceptable predicate expression. Nevertheless, Carnap claims that (2) is meaningless, and gives the following explanation:
"Prime number" is a predicate of numbers; it can be neither affirmed nor denied of a person (p. 68).
Although he doesn't state it explicitly in those terms, the idea here is that while sentence (2) is a well-formed sentence, it is not a well-typed one. If we were working with a formalized fragment of English, we would define an alphabet of the usual latin letters and then specify grammatical rules for generating a subset Wff of the well-formed sentences of English. Among those sentences we could find (2) and we wouldn't find (1). Of course, all these matters would depend on the expressive power of our formalized fragment. But suppose it's powerful enough to grant (2) the status of a well-formed sentence. Then, we would specify an extra layer of typing rules for generating the subset Wtf ⊆ Wff of well-typed sentences of our fragment. The point of typing rules is to ensure that the functional expressions of our language such as predicate expressions ('is prime') and functors ('father') are combined with expressions of a type compatible with the domains of definition of those functional expressions. To see how (2) is not well-typed, let's look at the domain of definition of 'is prime':
(T) prime : Nat → Bool.
The function prime, which is the referent of 'is prime' or 'is a prime number' is defined only for natural numbers and according to the known rule, maps natural numbers to true or false, depending on their primality. Consider the expression:
(3) π is a prime number.
Is (3) well-typed? Since π is not a natural number, (3) is not a well-typed expression. It may be a well-formed numerical expression according to a rule that says that a unary numerical function (e.g. prime) applied to a numerical expression (e.g. π) yields another numerical expression (e.g. 3). But it's nevertheless not well-typed because the domain of definition of prime doesn't include non-natural numbers in it (see T above).
After the realization that (3) is not well-typed, the non-well-typedness of (2) shouldn't be surprizing. 'Caesar' is not a numerical expression, so it's certainly not an expression that has a value in natural numbers. Now, that's a reason to regard (2) as not well-typed. The general question that arises here is whether we should consider not well-typed sentences as 'pseudo-sentences', as Carnap there does. Type-restrictions can be a helpful device for checking the correctness all sorts of mathematical constructions, so well-typedness is certainly an incredibly useful notion, but whether it should be grounds for partitioning the set of well-formed sentences of a given language into acceptable and unacceptable ones might be an interesting topic for another discussion.
References
Ayer, A.J. (1959) Logical Positivism.
Carnap, R. (1953) "The Elimination of Metaphysics Through Logical Analysis of Language", Ayer 1959.
Cumming, S. (2014) λ–Calculus and Type Theory, Lecture Course (Winter), UCLA.
Your second sentence might not be true, depending on what interpretation you take. If you mean "the set of prime numbers and the set of [not-prime] numbers," it's false because Caeser is not a number and therefore not part of either set. If you mean "the set of prime numbers and the set of not-[prime numbers]," that doesn't make sense because it's not really possible to have a set of everything (or a set of everything except primes).
The analysis I've seen of such statements used a different example:
(1) The present king of France is bald.
This was brought up in the context of wanting to formalize the meaning of statements and applying formal logic to them.
Frege and lots of mathematicians after him regard statements as predicates: given a context of interpretation, they are supposed to be either true or false.
Statement (1) certainly isn't true, so it must be false, right? And the negation of a false statement is true, right? Let's see:
(2) The present king of France is not bald.
This isn't true any more than statement (1), and for the same reason: both statements imply the truth of another (namely that there is a present king of France), and it is this implied statement which is false. Hence, regular predicate logic breaks down when applied to such statements: they are not just false, they are "not even wrong", and so are their negations.
The same can be said for Carnap's example: neither of
(3) Caesar is a prime number.
(4) Caesar is not a prime number.
is true, and this is because a statement the truth of which they both imply (namely, that Caesar is a number) is false.
