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206 STRÖSSNER
4.1 Carnap’s Logical Probability
In “Logical Foundations of Probability” (Carnap 1962) Carnap fleshes out the concept of
logical probabilities. In his theory probability is used as a measure of the degree of
confirmation of a hypothesis by given evidence. In a first step, probabilities are assigned to
state descriptions, which give a complete characterization of a possible state of affairs. The
probability measure m of a sentence is the addition of the probability of all state descriptions
to which the sentence belongs. Finally, the confirmation c(e/h) of hypothesis h by evidence e
is defined as conditional probability m(e∧h)/m(e).
Carnap’s theory of probabilistic confirmation is twofold. One part is reasoning from the entire
population to a smaller sample: the direct inference. Another part is the reasoning from one
sample to the population or to another sample. We will restrict our considerations to the
direct inference. Let c be a confirmation function for the evidence e and the hypothesis h.
Confirmation functions for direct inferences need to be regular and symmetric (Cf. Carnap
1962: VIII): 7
Def. 5: For every regular and symmetric confirmation function c(h/e) it holds:
– Regularity: c(h/e) = 1 iff e |= h. The best possible confirmation is entailment.
– Symmetry: c(h/e) is defined as conditional probability m(e∧h)/m(e) on a measure
that gives the same value to state descriptions which do not differ in the number of
individuals that are in the extensions of all possible predicates.
Now, let us assume r exclusive and exhaustive predicates M 1, M 2, ..., M r and the following
statistical distribution for the population of n individuals: n 1 individuals are M 1, n 2 individuals
are M 2, ..., and n r individuals are M r. This is our evidence. Our hypothesis for a sample of s
individuals is that s 1 individuals will be M 1, s 2 are M 2, ... and s n individuals are M n. Then, as
shown by Carnap, every regular and symmetric confirmation satisfies the following equation:
c(h/e)=
n-s!
× n 1!n 2 !…n r !
(n 1 -s 1 )!(n 2 -s 2 )!…(n r -s r )! n!
. (Carnap 1962: 495)
If we restrict ourselves to two exclusive and exhaustive predicates (e.g. being right-handed
and not being right-handed) and a sample of only one individual (e.g. Plato), the equation
above can be simplified:
c(h/e)=
n-1!
× n 1!(n-n 1 )!
(n 1 -1)!(n-n 1 )! n!
= n 1
n .
This shows that, as one would expect, the confirmation of the hypothesis that an individual
will have a property is identical to the relative frequency of the property in the population.
4.2 MOST, MOSTLY and Probability
The semantics of “most” in “Most S are P” is expressible by the number of S which are P and
the number of S which are not P. In a finite universe “Most S are P” translates to a disjunction
of statistical distributions, which have the form “n 1 S are P and n − n 1 are not P”. The
disjunction contains all statistical distributions in which n 1 is greater than n − n 1. All of these
distributions confirm the proposition that some arbitrary individual S is P better than the
statement that this individual is not P. That means that the thesis that an individual which is
S (e.g. the man Plato) has a property (being right-handed) is always better confirmed by the
evidence that most individuals of that kind have this property than a contradictory
7
Note that this symmetry is related to individuals only. The rather problematic symmetry for Q-
predicates, predicates characterizing an individual completely, is not required for the results on direct
inference.