Value concepts (1958)

Abstract

Carnap wrote a continuation of his reply to Kaplan (§32 of Carnap’s replies in the 1963 Schilpp volume), which would, however, have made that reply, already by far the longest in the book, too long. So he set aside his projected notes for a continuation to serve as the basis for a separate paper, which he never got around to writing. It is transcribed here from his shorthand and translated into English, with some introductory notes to provide a little context.

Appendices

Value Concepts (a shorthand manuscript by Rudolf Carnap, transcribed and translated by A.W. Carus)

Value concepts and rational agent First written to supplement my reply to Kaplan in the Schilpp volume. But that would have got too long. So better as a basis for a later paper!

21 February 58

Value Concepts

Relatively to a value system

Let V be a value function (It is not assumed that there is a person whose value function is V.) This means that for every possible history of the world W, V(W) is a real number. Since only the differences among values of V matter, in the following definitions, two value functions V und V$^{\prime }$ that differ only by a constant (for every $\hbox {W},\hbox {V}^{\prime } (\hbox {W}) =\hbox {V}(\hbox {W}) + \hbox {A}$ with constant A) may be viewed as equivalent.

Let the proposition q apply only to a limited time interval $\hbox {t}_{\mathrm{q}}$ und a limited spatial region $\hbox {R}_{\mathrm{q}}$. Then V(q) is to be understood as follows, where ${{ W}}_{{{ T}}}$ is the true history:

$(\upalpha )$ :

(a) If q is actually the case, then $\hbox {V(q)} = \hbox {V}(\hbox {W}_{\mathrm{T}})$.

(b) If q is false, then $\hbox {V(q)} = \hbox {V}(\hbox {W}_{\mathrm{q}})$, where $\hbox {W}_{\mathrm{q}}$ is the possible history of the world that would occur if q were always the case.

In (b) a counterfactual conditional is used. The explication of these is still controversial. For our purposes the following indications should suffice, though they would need to be made more precise. In the present context, we will use only counterfactuals in which the condition q is limited in the above way and moreover in which q is consistent with the totality PL of the actual physical laws (in the sense of §. . ., so not in the sense of the laws currently recognized by scientists). $\hbox {W}_{\mathrm{q}}$ is therefore the history of the world that meets the following conditions: [2]

$(\upbeta )$ :

(a) $\hbox {W}_{\mathrm{q}}$ coincides with $\hbox {W}_{\mathrm{T}}$ over its entire range before the time interval $\hbox {t}_{\mathrm{q}}$,

(b) as well as during the interval $\hbox {t}_{\mathrm{q}}$ outside the region $\hbox {R}_{\mathrm{q}}$,

(c) within the space-time region $\{\hbox {t}_{\mathrm{q}},\hbox { R}_{\mathrm{q}}\}, \hbox { W}_{\mathrm{q}}$ coincides as far as possible with $\hbox {W}_{\mathrm{T}}$ and diverges from $\hbox {W}_{\mathrm{T}}$ only as far as is necessary to make q true;

(d) after the interval $\hbox {t}_{\mathrm{q}},\hbox { W}_{\mathrm{q}}$ coincides with $\hbox {W}_{\mathrm{T}}$ in all space-time regions not affected causally by the previous q, while they diverge from $\hbox {W}_{\mathrm{T}}$ in the regions affected by q as determined by q in conjunction with the laws PL. [3a]

$(\upgamma )$ :

p is better than q with respect to the value function $\hbox {V} =_{\mathrm{Df}}\hbox { V(p)} - \hbox {V(q)} > 0$.

$(\updelta )$ :

p is good with respect to the value function $\hbox {V} =_{\mathrm{Df}}$ p is better than not-p. [3b]

Assuming that an agent X has a choice among the possible actions of a set $\hbox {A}_{\mathrm{X}}$, we define:

$(\upvarepsilon )$ :

The possible action a in $\hbox {A}_{\mathrm{X}}$ is an optimum with respect to the value function $\hbox {V} =_{\mathrm{Df}}$ no action in $\hbox {A}_{\mathrm{X}}$ is better (in the sense of ($\upgamma$)) than a with respect to V. [3c]

(22 February)

A person X at a given time has not just a single value function, but a great many of them, representing different value aspects. If X, following the dietary advice of his doctor, says “It is better for me to avoid a certain kind of food”, he has a certain value function in mind, one that represents only health values, and only for himself. Other partial value aspects might be: his business profit, his aesthetic pleasure, his own well-being with respect to all aspects jointly, the well-being of a family, that of a large group, that of a nation, that of all humanity. But there is also a comprehensive value function of X that comprises all aspects, and in which the relative weight of each aspect in any possible overall situation finds expression—aspects that are sometimes in mutual conflict. Different things are meant by [the expression] “moral value judgement.” Perhaps it is best to use this term for the overall value judgement, in which the different aspects are included. [4]

The rational agent

$(\upzeta )$ :

Relative rationality With respect to a value function V, a credibility function Cred, a body of evidence E and a set A of possible actions, an action a in A is ${{rational}}=_{\mathrm{Df}}$ for no action a$^{\prime }$ in A different from a is V(W) using Cred on the basis of E and a$^{\prime }$ preferred to V(W) on the basis of E and a. (The degree to which V(W) is preferred with respect to a certain body of evidence is the sum over all possible W of the products of V(W) with the credibility of W on the basis of the evidence in question; see § [25(II)].) [5]

There are certain standards on the basis of which a Cred-function can be criticized as irrational; these have been discussed elsewhere (Kemeny’s essay §[III]; and my §[26(IV)] in this reply). It is the task of inductive logic to arrive at such standards.

Are there also standards of rationality for value functions? The above-mentioned standards of inductive logic are not applicable here. The acceptance of a value function is completely independent of factual questions, for what the value function primarily evaluates is not particular actions or processes but rather entire possible histories of the world. Considerations about the consequences to be expected from an action do not come into the picture, for in a W all consequences are already included and given. [For instance, take the case where] the function $\hbox {V}_{1}$ values $\hbox {W}_{1}$ more highly than $\hbox {W}_{2}$, while the function $\hbox {V}_{2}$ does the reverse:

• (a) $\hbox {V}_{1}(\hbox {W}_{1}) >\hbox { V}_{1}(\hbox {W}_{2})$

• (b) $\hbox {V}_{2}(\hbox {W}_{1}) <\hbox { V}_{2}(\hbox {W}_{2})$.

Assume that the agent $\hbox {X}_{1}$ accepts $\hbox {V}_{1}$ and $\hbox {X}_{2}$ accepts $\hbox {V}_{2}$. Assuming that $\hbox {X}_{1}$ and $\hbox {X}_{2}$ discuss their value functions and, in particular, the descriptive results (a) and (b). In their discussion they will consider only the two histories $\hbox {W}_{1}$ und $\hbox {W}_{2}$. $\hbox {X}_{1}$ may have different evidence values than $\hbox {X}_{2}$ for each of these two histories; but that is irrelevant for the question of choosing between $\hbox {V}_{1}$ and $\hbox {V}_{2}$. This [6] question concerns only whether one values $\hbox {W}_{1}$ more highly than $\hbox {W}_{2}$ or vice versa; [and] that has no bearing on the question whether $\hbox {W}_{1}$ will occur or has a higher probability [of occurring] than $\hbox {W}_{2}$.

Although all logic, including inductive logic, and factual knowledge are irrelevant, it nonetheless seems to me that there are other, purely valuational criteria by which to judge a value function as more or less rational than another. I am not going to attempt to set up fundamental standards for such judgements here. I only want to mention some considerations whose justification in such a judgement seems plausible and would likely be approved by most people, even if they diverge markedly in their valuations. First, it seems reasonable to require that a value function V(W) is derivable from general principles regarding the valuation of particular processes; specifically that the value of V(W) be an algebraic sum (or integral) of positive or negative values determined by some sort of principles governing certain very specific processes, while the remaining processes are irrelevant. (The relevant processes [7] arise e.g. from certain affective processes in humans, or from a more general kind of processes in beings that are animate or regarded as such; while the inorganic processes are of course irrelevant.) Then it should also be required that the principles have a general character, that they are expressible by mathematical functions of the relevant properties of the processes involved, specifically mathematical functions that are continuous and relatively smooth, rather than jumping up and down. These examples of requirements may be doubtful. I have not mentioned them to defend their validity, but only to indicate why I think that there are certain standards a value function must meet to be rational. The clarification of such standards I can’t attempt here. But it seems clear that if such standards were worked out, they would only exclude as irrational certain value functions, and still admit an infinite set of different value functions that are extremely different from each other, and among them would be many that would be considered by most people, perhaps by all, as completely wrong and immoral. So the standards I speak of do not at all have the function of excluding “immorality” [8] or of distinguishing between value judgments that occur psychologically in controversies about moral or political questions. In the following I will speak of “the standards of rationality for value functions” as if they had already been arrived at. [9]

Now we define:

The behavior of an agent X is perfectly rational during a certain time period $\Delta \hbox {t}$ when it meets the following conditions:

$(\upeta )$ :

(a) In deductive thought, which includes the whole of pure mathematics, he never makes any errors during $\Delta \hbox {t}$.

(b) During the period $\Delta \hbox {t}$ he uses a rational method in his inductive thought; specifically, there is a credibility function $\hbox {Cred}_{\mathrm{X}}$ for him that meets the criteria of rationality.

(c) His behavior during the period $\Delta \hbox {t}$ is governed (in the way to be described under (d)) by a value function $\hbox {V}_{\mathrm{X}}$ that meets all standards of rationality.

(d) Whenever X has a choice, at a time t within the period $\Delta \hbox {t}$, among different actions in a set $\hbox {A}_{\mathrm{X,t}}$, and if at t his total evidence is $\hbox {E}_{\mathrm{X,t}}$, then the action chosen by X has relative rationality (in the sense of $\upzeta$) with respect to $\hbox {V}_{\mathrm{X}},\hbox { Cred}_{\mathrm{X}},\hbox { E}_{\mathrm{X,t}}$, and $\hbox {A}_{\mathrm{X,t}}$. [10]

Assume X is perfectly rational at time t and chooses action a in $\hbox {A}_{\mathrm{X}}$. Then it is nonetheless still possible for a not to be an optimum with respect to $\hbox {V}_{\mathrm{X}}$. It could be that an action a’ is better than a with respect to $\hbox {V}_{\mathrm{X}}$, due to certain circumstances not known to X at the time of the action. It could even be that the objectively better, i.e. more successful action a’ would not be rational for X. As emphasized elsewhere (§[26.IV]), rationality is not to be determined by success.

No one is ever perfectly rational in the sense just defined. “More rational”, whether applied to different periods or to two possible behaviors of the same person in the same period, cannot very well be exactly defined. Roughly speaking, a behavior is more rational than another when it comes closer to perfectly rational behavior. But since deviations from perfectly rational behavior are possible in completely different ways, e.g. in the ways mentioned above $(\upeta )$ (a), (b), (c), (d), and within each of these once again in different ways, it is hardly possible to decide without an arbitrary convention under what conditions a deviation in one way should be considered equal to a deviation in another way.