\brief{Rudolf Carnap an Cooper Harold Langford, 26. August 1935}{August 1935}

\anrede{Dear Professor Langford,}

\haupttext{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 \uline{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.

\noindent{}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.

\noindent{}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 \neueseite{} 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: \uline{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:

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

\noindent{}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 `$\sim BewSatzII(r, subst(0``,3,str(0``))$\blockade{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:

$(D_5)$ $G$ = $Df$\blockade{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 $II$``.
\noindent{}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. \neueseite{}

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 \uline{Hume}\IN{\hume}: 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 lectures\IC{} at Chicago University\II{\universitychicago} 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]\II{\kongressbaltimore} 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 School\II{\harvardss}. For September 1936 I have been invited to the Harvard Tercentenary and to the Conference\II{\tercentenary} preceding it. After that I hope to be able to stay a few more weeks in U.S.}

\grussformel{Very sincerly yours,\\
R.C.}


\ebericht{Brief, msl., 3 Seiten, \href{https://doi.org/10.48666/854847}{WQ}, Briefkopf: hsl. \original{In Tyrol}, msl. \original{August\,26, 1935}.}