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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.)

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    Seems pretty well explained on Wiki. en.wikipedia.org/wiki/Vienna_Circle There's a lengthy discussion of this point. – user4894 Jun 9 '14 at 21:48
  • I read his essay. What he says here "The problem is that "prime number" is a predicate of numbers, not a predicate of human beings." .......... doesn't make sense to me. Seems like I can say "Joe is a book" just as well, and just as meaningfully, even if it's false. Not sure why he's restricting how we can relate individuals to predicates. – Casey Jun 9 '14 at 21:53
  • Well I agree with you, that "Caesar is prime" is FALSE, since Caesar is not a natural number. But the question was what Carnap thinks; and it's clear that he considers the proposition meaningless. This raises the question of whether every proposition is either true or false, or whether "meaningless" is a third truth value that can be assigned to some propositions. – user4894 Jun 9 '14 at 21:56
  • @Casey: this may not be what Carnap intends, but consider questions that contain a false assumption. "Is your wife tall?" presupposes that you have a wife, and cannot be answered "yes" or "no" if you don't have one. The alternative "do you have a wife who is tall?" can. Carnap might simply be saying that to evaluate "Caesar is a prime number" we must presuppose that Caesar is a number, and has factors, and we must test a property of the factors he doesn't have. The first rule of logical positivism is, you don't talk about <s>logical positivism</s> non-existent things ;-) – Steve Jessop Jun 10 '14 at 10:39
  • ... now I'm not sure I agree with that parse of the sentence, but that's a dispute with Carnap over English-language common usage and whether or not "is a prime number" can only be evaluated for numbers. It's not a dispute over the issue of whether we can meaningfully assert inappropriate properties of Caesar, just over whether his example is suitable to illustrate an attempt to do so. – Steve Jessop Jun 10 '14 at 10:45
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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.

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  • Fascinating; Gary Bernhardt just tweeted something about using option types in thinking about society -- curiously enough, in the context of something about category errors in confusing the absence and negation of a property (so amorality becomes immorality, incompetence is perceived as malice, etc.) Anyway, great answer!! – Joseph Weissman♦ Jun 10 '14 at 0:39
  • @JosephWeissman Thanks! I've been a pretty dormant programmer for the last five years, so I hadn't heard of Gary Bernhardt. Option types and society sounds great. I'll check his stuff out on github. – Hunan Rostomyan Jun 10 '14 at 0:53
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    You can recall also Frege's Julius Caesar problem (and I think that Carnap's example is not casual; he assisted Frege's lectures in Jena). For Frege, the universe of discourse included absolutely all; thus he struggled trying to prove that : not (Julius Caesar = n), for every n, because without a working definition of the objects which are numbers, there are no syntactical rules in his Begriffsschrifft preventing the above well-formed identity statement. – Mauro ALLEGRANZA Jun 10 '14 at 9:59
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    Great answer. I was also in Sam's class Winter quarter. I think he'd approve :) – David Titarenco Jun 10 '14 at 17:21
  • @DavidTitarenco David! Good to see you here. And thank you :) – Hunan Rostomyan Jun 10 '14 at 17:32
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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).

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    Exactly. I'm glad you mentioned the fact that there is no 'universe'. – Hunan Rostomyan Jun 10 '14 at 2:28
  • @David Young Yeah I meant not-(prime numbers). I edited my OP. – Casey Jun 10 '14 at 19:18
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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.

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  • Speaking English, I would say that "Caesar is not a prime number" doesn't imply that Caesar is a number. In the same way that "I am not a fat slob" doesn't imply that I'm a slob. However, saying "I am not a fat slob" may indeed be to concede that I'm a slob. So I think the basic semantics of the sentence are in dispute here between Carnap and the questioner, before we even get to assessing what implication the semantics have on the meaningfulness of the sentence. – Steve Jessop Jun 10 '14 at 10:50
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    NO The famous Russell's example regards a definite description. "The present king of France is bald" must be formalized as "exists x such that (King of France(x) and bald(x))". Thus the negation of the above is: "does not exists x such that (King of France(x) and bald(x))" wich is "for all x (not King of France(x) or not bald(x))" which is perfectly true when, as in present time, there is no King of France. In Carnap's example, we assume that "Caesar" is a name and not a description; thus, if we assume that it is "syntactically" correct (i.e.it does not violate types restrictions)... – Mauro ALLEGRANZA Jun 10 '14 at 12:38
  • ... to "speak of" Roman consuls and numbers in the same "discourse" we are allowed to ask if Caesar is a prime, exactly as to ask if 4 is a prime. Carnap's argument is aimed at "excluding" meaningful contexts where we "predicate" the same properties of "kinds" of entities so different as numbers and Roman consuls. – Mauro ALLEGRANZA Jun 10 '14 at 12:41
  • @Steve Jessop: this was discussed in class. Indeed, it was argued that "Caesar is not a prime number" could in fact be uttered, but only with a special stress on 'not' to indicate that a special kind of negation was going on. – reinierpost Jun 10 '14 at 13:51
  • @ Mauro ALLEGRANZA: Yes, and I think the professor who explained this to me would agree with you, but this depends on their distinction between definite descriptions and names, which I don't really believe in. – reinierpost Jun 10 '14 at 13:55

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