Editing New riddle of induction (section)

===Carnap===
[[Rudolf Carnap]] responded{{sfn|Carnap|1947|p=139}} to Goodman's 1946 article. Carnap's approach to inductive logic is based on the notion of ''degree of confirmation'' ''c''(''h'',''e'') of a given hypothesis ''h'' by a given evidence ''e''.{{efn|he uses another variant, ''c''<sup>*</sup>(''h'',''e''), for which he gives a formula to compute actual values;{{sfn|Carnap|1947|p=138, 143f}} different from Laplace's [[Rule of Succession]]. See Carnap's book ''Studies in inductive logic and probability'', Vol.1. University of California Press, 1971, for more details, in particular sect.IV.16 for ''c'', and app.A.1 for ''c''<sup>*</sup>.}} Both ''h'' and ''e'' are logical formulas expressed in a simple language ''L'' which allows for
* multiple quantification ("for every ''x'' there is a ''y'' such that ..."),
* unary and binary predicate symbols (properties and relations), and
* an equality relation "=".
The [[universe of discourse]] consists of denumerably many individuals, each of which is designated by its own constant symbol; such individuals are meant to be regarded as positions ("like space-time points in our actual world") rather than extended physical bodies.{{sfn|Carnap|1947|p=134}} A state description is a (usually infinite) conjunction containing every possible ground atomic sentence, either negated or unnegated; such a conjunction describes a possible state of the whole universe.<ref>This might be seen as corresponding to [[Wittgenstein]]'s [[Tractatus Logico-Philosophicus#Proposition 1|Tractatus]], Nr.1.11.</ref> Carnap requires the following semantic properties:
* Atomic sentences must be logically independent of each other.<ref>cf. Tractatus Nr.1.21</ref> In particular, different constant symbols must designate different and entirely separate individuals.{{efn|For example, if ''a'' and ''b'' had a part in common, then "''a'' is warm and ''b'' is not warm" would be an impossible combination.}} Moreover, different predicates must be logically independent.{{efn|For example, "is a raven" and "is a bird" cannot both be admitted predicates, since the former would exclude the negation of the latter. As another example, "is warm" and "is warmer than" cannot both be predicates, since "''a'' is warm and ''b'' is warmer than ''a'' and ''b'' is not warm" is an impossible combination.}}{{efn|Carnap argues{{sfn|Carnap|1947|p=135}} that logical independence is required for deductive logic as well, in order for the set of [[analytic-synthetic distinction#Frege and Carnap revise the Kantian definition|analytical sentences]] to be decidable.}}
* The qualities and relations designated by the predicates must be simple, i.e. they must not be analyzable into simpler components.{{sfn|Carnap|1947|p=136}} Apparently, Carnap had in mind an [[irreflexive]], [[partial order|partial]], and [[well-founded]]{{sfn|Carnap|1947|loc=p. 137: "... carry the analysis [of complex predicates into simpler components] to the end"}} [[order theory|order]]{{efn|Carnap doesn't consider predicates that are mutually definable by each other, leading to a [[preorder]].}} ''is simpler than''.
* The set of primitive predicates in ''L'' must be complete, i.e. every respect in which two positions in the universe may be found to differ by direct observation, must be expressible in ''L''.{{sfn|Carnap|1947|p=138}}
Carnap distinguishes three kinds of properties:
# Purely qualitative properties; that is, properties expressible without using individual constants, but not without primitive predicates,
# Purely positional properties; that is, properties expressible without primitive predicates, and
# Mixed properties; that is, all remaining expressible properties.
To illuminate this taxonomy, let ''x'' be a variable and ''a'' a constant symbol; then an example of 1. could be "''x'' is blue or ''x'' is non-warm", an example of 2. "''x'' = ''a''", and an example of 3. "''x'' is red and not ''x'' = ''a''".

Based on his theory of inductive logic sketched above, Carnap formalizes Goodman's notion of projectibility of a property ''W'' as follows: the higher the relative frequency of ''W'' in an observed sample, the higher is the probability that a non-observed individual has the property ''W''. Carnap suggests "as a tentative answer" to Goodman, that all purely qualitative properties are projectible, all purely positional properties are non-projectible, and mixed properties require further investigation.{{sfn|Carnap|1947|p=146}}