Rudolf Carnap an Cooper Harold Langford, 26. August 1935 August 1935

Thank you very much for your kind and very interesting letter. Please accept my apologies for not having it answerd earlier. I always had the intention to write not merely a few lines but a real reply to your argument. And because of this I have postponed my answer several time.

Your question concerning vicius regressions (= \(v.r.\)) is very interesting. I agree with you in the opinion that the occurance of a \(v.r.\) makes nonsense. But I think that, if a sentence asserts something about itself, then there is not always – though indeed sometimes – a \(v.r.\) involved. It may be that my concept of \(v.r.\) is a narrower one than yours; it seems to me that we have to reject the \(v.r.\) in this narrow sense only. I should speak of \(v.r.\) if we had only explicit definition of such a kind that the defined symbol occurs in the defeniens. (This is the simplest case; the more complicated ones consist in a chain of definitions of such a kind that the symbol defined by the last definition of the chain occurs in the definiens of a preceding definition of the chain; for our purpose perhaps the consideration of the simplest case will suffice.) Thus e. g. the following definitions are \(v.r.\):

\((D\textsubscript{1})\)\(A\textsubscript{1}\) = Df \(A\textsubscript{1}\) is true.

\((D\textsubscript{2})\)\(A\textsubscript{2}\) = Df \(A\textsubscript{2}\) is false.

\((D\textsubscript{3})\)\(A\textsubscript{3}\) = Df \(A\textsubscript{3}\) is not refutable.

\((D\textsubscript{4})\)\(A\textsubscript{4}\) = Df \(A\textsubscript{4}\) is not provable.

If we construct a language \(L\) we have to state the formative rules for definitions in \(L\). Then all \(v.r.\) are excluded from \(L\) automatically so to speak. That means, if we wish to know wether or not a certain sentence of \(L\) involves a \(v.r.\) we need not understand its meaning. All we have to do is only to take care that all definitions which we state in \(L\) are in accordance with the formative rules for the definitions of \(L\), and especially with the following rule:

\((R\textsubscript{1})\) A defined symbol must not occur in the definiens of its definition.

And this is a purely formal consideration, because \(R\textsubscript{1}\) is a formal rule. A non-formal consideration is made once for all when we decide to state \(R\textsubscript{1}\); this non-formal consideration is however not involved in the rules of \(L\), but has a motivational character only.

In constructing the languages \(I\) and \(II\) (in the book Syntax) \(I\) stated \(R\textsubscript{1}\) among the formative rules for explicite definitions (p. 22, 80) – as is commonly done in logistical systems. Therefore I believe that no \(v.r.\) can occur in these languages. On the other hand, \(II\) contains the sentence \(G\) (p. 92) corresponding to Gödels 🕮 sentence, and \(I\) contains an analogous sentence \(G\textsubscript{I}\) (p. 95). Each of these two sentences is an arithmetical one which, when interpreted as a sentence of arithmetised syntax means that it itself is not provable. The decisive question question now is wether these sentences involve a \(v.r.\) I Think, they do not. I agree with you in the following: if the symbol ‘\(G\)‘ were introduced like ‘\(A\textsubscript{4}\)‘ by ‘\(D\textsubscript{4}\)‘ above, the forms \(D\textsubscript{1}\)-\(D\textsubscript{4}\) are excluded from \(I\) and \(II\) by \(R\textsubscript{1}\). In fact, ‘\(G\)‘ is introduced by the following definition:

\((D\textsubscript{5})\)\(G = DF^{subst(b\textsubscript{2}, 3, str(b\textsubscript{2}))}\)

here \(‘b_2‘\) is a certain numeral (\(b_2\) is the series-number of the sentential function \(‘\sim BewSatzII(r, subst(x, 3, str(x))‘)\); ‘subst‘ and ‘str‘ are numerical functors; hence ´\(G\)‘is a numerical (designating a certain number which is by far too great to be actually written down in the decimal-system, but which is nevertheless defined unambiguously by the definitions stated). Now the number \(G\) can be construed as the series-number of a certain sentence. And when we build up this sentence (it is ‘\(\sim BewSatzII(r, subst(0“‚3‚str(0“))\)als Null, nicht Buchstabe O transkribiert where after the 0 both places is to be written a very long series consisting of \(b_2\) accents), we find that it asserts that a sentence having a certain series-number \(G\) and which itself is then simply called \(G)\) asserts that it itself cannot be proved.

I shall try to explain the situation without referring to the arithmetised syntax. This explanation is perhaps more easily intelligible, but of course less exact. We defin ‘\(G\)‘ as a syntactical name of a sentence as follows:

\((D_5)\)\(G\) = \(Df\)Formatierung? that sentence which arises from that and that expression \(E\) by that and that substitution \(Sb\). When we now build up the sentence \(G\) according to the description given in the definiens of \(D_5‘\), i. e. carry out the prescribed substitution \(Sb\) on the expression \(E\), we find (in this special case,

\((S_1)\) “There is no demonstration in \(II\) for a sentence arising from \(E\) by the substitution \(Sb“\).

\((S_2)\) “No sentence of the form \(G\) is provable in \(II\)“. Thus \(G\) means that \(G\) is not provable in \(II\). The decisive point is, that ‘\(G\)‘ is not defined trough this property – as was ‘\(A_4\)‘. \(G\) is rather defined in a quite regular way. Only after analysing the definition of ‘\(G\)‘ we find that ‘\(G\)‘ is the name of a sentence of a certain form \(F_1\) attributing a certain quality (namely improvability) to sentences of a certain form \(F_2\), and then it turns out that \(F_1\) and \(F_2\) are the same form. And this form can actually be described withaut any ambiguity, withaout any circle. 🕮

The result is: the definition of ‘\(G‘\) has not the slightest similarity to \(D_4\). \(G\) is an actually constructed sentence-form. Only after its construction we see that is speaks about this form itself.

As regards HumePHume, David, 1711–1776, brit. Philosoph: I think you are right that his formulations are not all on the same line, some of them being of a rather sceptical character, regarding the answer to metaphysical questions as being unknowable, and some regarding these questions as meaningless. And I must confess I am not sure wether he means here ‘meaningless‘ in the same strict sense as we do. I think with you that there is a radical difference between the two views. But it seems to me that even up to the 19th century this difference was not sufficiently noticed by some antimetaphysicians.

I am afraid that my letter will not be a satisfactory explanation of my view on the point in question. An oral discussion woul certainly be more fruitful. As I shall come to U.S. next year I hope to have an opportunity of meeting you. I am invited to deliver lecturesB at Chicago UniversityIUniversity of Chicago, Chicago IL during the winter-quarter (January 3rd-March 5 1936). Perhaps I shall attend the Congress of the Eastern Section of the Philos[ophical] Ass[ociation]IMeeting der Eastern Division der American Philosophical Association, Baltimore, 1935 in Dec. 1935, if I should arrive early enough in U.S. For the time from April to June I have no definite plans so far. In July and August 1936 I shall give lecture-courses at the Harvard Summer SchoolIHarvard Summer School. For September 1936 I have been invited to the Harvard Tercentenary and to the ConferenceISymposium zum 300. Jubiläum Harvard, September 1936 preceding it. After that I hope to be able to stay a few more weeks in U.S.