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ISSN-L: 2223-9553, ISSN: 2223-9944

Academic Research International Vol. 3 No. 1 July 2012




Meiji University




This paper takes two tasks. The one is elaborating on the relationship of inductive logic with decision theory to which later Carnap planned to apply his system (§§1-7);

this is a surveying side of this article. The other is revealing the property of our prediction of the future, subjectivity (§§8-11); this is its philosophical aspect. They are both discussed under the name of belief in causation. Belief in causation is a kind of “degree of belief” born about the causal effect of the action. As such, it admits of the analysis by inductive logic.

Keywords: Carnap’s inductive logic, decision theory, belief in causation, subjectivity


When we intentionally act, we take future affairs into account. Suppose, e.g. one parks his car on some street; he will then wonder whether his parking will cause a traffic jam later. Let us call this mindset belief in causation. Sometimes the agent may care about a causal effect of his action in the future. It is the object of study below.

For its analysis, we dare to use Carnap’s inductive logic. Some may think it miscasting, since Carnap’s system is so formalized that it appears inapplicable to ordinary situations like the above. Nonetheless, in his later years, Carnap actually planned such an application (Carnap, 1962, p.vii). He noticed: the main field for his logic is decision theory rather than natural science; his logic takes on the pure, theoretical, logical part of normative decision theory (Carnap, 1971, p.26).

Thus, our investigation also directs itself toward decision theory, so that we shall deal with the relationship between inductive logic and decision theory in the first half of this paper, which includes the overview of decision theory (§2), the foundation of mathematical expectation (§3), and the actual application of inductive logic (§§5-7).

In the second half (§§8-11), the results of our investigation are examined; we shall particularly recognize the subjectivity of inductive logic.


In his later years, Carnap planned to apply his inductive logic to decision theory. But what on earth is the decision theory? To begin with, we must clarify it.

1 This paper was read at “Philosophy of Science Colloquium” in Institute Vienna Circle (the

–  –  –

We may, for this purpose, refer to R. C. Jeffrey’s book (Jeffrey, 1983) 2. To introduce his

explanation, let us imagine the following scenario:

(1) An owner of a certain bookshop is wondering whether or not he should order a certain book, though its amount is only one. Reflecting on his past experience, the possibility of a customer coming to buy the book is 0.3. But once the book is sold, his gain will be 100 euro, where, of course, the charge for the ordering, 40 euro, is subtracted. Now, should he order the book? 3 According to Jeffrey, this scenario is decomposed into three factors: Act, Condition, and Consequence (Jeffrey, 1983, pp.1f.). Act is the option that the agent can choose: in (1), the agent is the owner, acts are “ordering the book” and “not ordering the book.” Condition is the possible affair that has much to do with the acts: in (1), conditions are “a customer coming to buy the book” and “nobody coming to buy the book.” Consequence is the possible affair that results from the chosen act together with a specific condition: in (1), consequences are “ordering the book and a customer coming to buy it,” “ordering the book but nobody coming to buy it,” “not ordering the book but a customer coming to buy one,” and “not ordering the book and nobody coming to buy one.” We can name these four consequences in terms of money, stating in order: “+100 euro,” “-40 euro,” “0 euro,” and “0 euro.” These are called numerical desirability.

These factors are put into the desirability matrix and the probability matrix respectively.

(2) The Desirability Matrix for Situation (1)

–  –  –

By reference to these matrices, we can calculate mathematical expectation.

(4) The mathematical expectation of ordering the book = (+100) × 0.3 + (-40) × 0.7 = +2 (euro) (5) The mathematical expectation of not ordering the book = 0 × 0.3 + 0 × 0.7 = 0 (euro) These calculations are explained as follows: firstly, we multiply each correspondent entry between a desirability matrix and a probability matrix, and secondly, we add entries on the same row (because each row shows each act). And by their comparison, we can decide the best act; in the present case, since +2 0, the owner should order the book.

–  –  –


This is an overview of decision theory, to which Carnap planned to apply his logic. But in the explanation above, it was not yet clarified how inductive logic works in that theory. So we must elucidate it further.

First, let us review the use of mathematical expectation, in terms of which optional acts were compared. But, on what ground? Calculations like (4) and (5) may possibly seem easy amalgams.

–  –  –

The weighted mean is a well-known concept in the circles of investment, for example.

Consider closing prices of a certain stock, supposing that it was 80 euro three months ago, 50 euro two months ago, and 20 euro last month. How can we calculate the average or the mean

of this stock price? Presumably, at first, we use a simple average:

80+50+20 = 50 (6) 1 +1 + 1

–  –  –

Carnap’s strategy was deriving mathematical expectation from this formula. While we have so far decided the weight (wp) in terms of time (i.e. the latest datum is more highly weighted than earlier ones), Carnap took the probability of a hypothesis predicting a certain magnitude or a consequence to which the magnitude is assigned, instead. This idea is developed in the following way. Let hp be such a hypothesis that predicts magnitude mp itself or the

consequence to which mp is assigned. Then, the probability of hp is symbolized as follows:

4 Why is the denominator of the left side not “1+1+1” but “1+2+3”? Of course, “1” corresponds to the weight of

the datum “80”, “2” to “50”, and “3” to “20”. The reason is known, transforming (7) into this:

80 +(50 + 50) +(20 +20 +20) (*) 1+(1+1)+(1+1+1)

–  –  –

*(hp, e) (9) Here we may already find a piece of inductive logic as well. The function “*” is peculiar to inductive logic; it is called -function, which assigns probability to each hypothesis “hp” with regard to evidence “e” (Carnap. 1962, pp.293f.). Carnap’s proposal was to replace the values

of this function with the preceding time-values at wp in (8). The result is as follows:

∗ =1 [ × (ℎ , ) ] (10) ∗ =1 (ℎ , )

–  –  –


In this way, Carnap justified the caluculation of mathematical expectation. His argument also tells us where inductive logic works; that is, “*(hp, e)” in formula (11).

From this angle, let us review the scenario of a bookshop owner, adjusting (11) to its calculation, (4) and (5) 7. As just stated, inductive logic is concerned with probability assignment. How about the case of the owner, then? On checking over his probability matrix (=3)8, we know: he assigns probability independently of the chosen acts. The probability assignment to each condition (“a customer coming to buy the book,” “nobody coming to buy the book”) in each column in (3) are not changed whichever act (“ordering the book,” “not ordering the book”) might be chosen.

However, we often experience the opposite cases in real lives. As an example, let us take the case of nuclear deterrence (cf. Jeffrey, 1983, pp.2f., pp.8f.). Suppose a certain country deliberates on whether or not it should arm itself with a nuclear weapon; then, acts are “nuclear armament” and “nuclear disarmament,” and conditions are “war” and “no war.” In this situation, the country surely cannot think about the probability of each condition without taking chosen acts into account. In other words, it cannot think about the probability of war and of no war, independently of chosen acts. This is because the probability of war in the case of nuclear armament is obviously different from that in the case of nuclear disarmament.

This is why the probability matrix in this case is formed in a different way from that of the bookshop owner (cf. Jeffrey, 1983, p.9); that is, (12) The Probability Matrix for Nuclear Armament 5

Let sentences h1,…, hs be exhaustive and exclusive. Then the following holds:

–  –  –

The influence of the acts on the conditions is clear in this matrix. Focus on, e.g. the lower left entry: if the country (agent) does nothing, in other words, chooses nuclear disarmament (act), then the probability of war (condition) becomes high (=0.8). On the contrary, see the upper left entry: if the country chooses nuclear armament, the probability of war gets lower (=0.1).

In this way, the same condition has different probabilities depending on the chosen acts.


This relationship between the act and the condition can be classified into causation9. So now, we have returned to our original interest, belief in causation. To discuss it further, then, let us take up the following scenario on the basis of the one mentioned at the beginning of this paper (§1)10.

(13) X is about to park his car on C street. But he does not want to cause a traffic jam. So he looks back to his past experience, and tries to predict the probability of a traffic jam in his parking.

How does he make this prediction?

We apply the preceding analyses of nuclear deterrence to this situation as well. It is, firstly, decomposed into parts: acts and conditions. Acts are “parking” and “not parking,” conditions are “traffic jam” and “no traffic jam.” We may put aside consequences or numerical desirability at present, because the pressing problem for X is not choosing an act but the probability of the conditions caused by his act. His interest is how probable the causal relationship between his parking and a later traffic jam is. We can then formulate it as


* ((ε causes a traffic jam), e) (14) Here, “ε” stands for an act11. “e” is the past data that X looks back to in situation (13). Like this, we think the chosen act (=ε) influences not the probability of a condition but the condition itself (=traffic jam), and assign probability to causation as a whole. This is the way of thinking developed in the following 12.


Now then, let us scrutinize how we can apply inductive logic to our formulation (14).

9 Jeffrey also mentioned “causal influence” in decision theory (Jeffrey, 1983, p.24). However, his interest was rather in the problem of “evidential significance” found in Prisoner’s Dilemma (Jeffrey, 1983, pp.15f.).

Although he tried to deny those relations (Jeffrey, 1983, p.9), we may disregard those negative arguments. For Jeffrey merely made an extra convention to refuse them (ibid.).

10 We might deal with the same situation, nuclear deterrence, here consecutively. However, for that, we need much more detailed preparations. So in this paper, we stick to the instance of parking.

11 In this paper, I use “ɛ” as an individual constant for an event, and “e” as a variable for an event. In these notations, I follow Davidson’s idea of logic of events (Davidson, 1980, Essay6). I ask the readers to make a distinction between “e” (=event) and “e” (= evidence).

12 This treatment was presented initially in Kaneko, 2012a, where I elaborated on its technical matters as well.

The relationship of inductive logic with causation was treated also in Uchii, 1972 and Uchii, 1974; but the main focus there was causal modality, not causation itself. Rather, as the work close to our argument, we may refer to Köhler, 2011. Again, as for so-called probabilistic causality (Salmon, 1980 and Suppes, 1970), I elaborated on why I do not favor their doctrines in Kaneko, 2012a, §2.

Copyright © 2012 SAVAP International www.journals.savap.org.pk www.savap.org.pk 15 ISSN-L: 2223-9553, ISSN: 2223-9944 Academic Research International Vol. 3 No. 1 July 2012 Review (13) first. We can name each event X has in mind “ε1,” “ε2,” “ε3,” and “ε4.” Therein, “ε3” expresses that X parks his car on C street within a certain period of time; “ε1” and “ε2” are the past events similar to X’s act, which X looks back to in (13); for example, “ε1” expresses that Y parked his car on A street, and “ε2” expresses that Z parked his car on B street; “ε4” expresses something different from parking, not meaning a negative event like X’s not parking; at present, we regard it as the future event that there is a rush of cars on C street. Again, “ε1,” “ε2” and “ε3” are arranged in the order of time, but we do not decide which precedes, “ε3” or “ε4,” since we suppose that “ε3” and “ε4” have not been observed yet.

These “ε1”~“ε4” are regarded as individual constants. We put them in one class:

Const. = {ε1, ε2, ε3, ε4} (15) This class is taken to be the expression of the population that X has in mind (cf. Carnap, 1962,

p.207, pp.493f.). In the present situation, we divide it into two subclasses, further:

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