We describe our outcomes using a language with one predicate, G$,$ and six names denoting individuals: a$1,$ a$2,$ a$3,$ a$4,$ a$5,$ a$6.$ So$,$ the atomic sentences of this language are Ga$1,$ Ga$2,$ Ga$3,$ Ga$4,$ Ga$5,$ and Ga$6$; we can form more sentences by using our connectives $(,,$etc$.).$ Assume our initial credence function is Carnap's C$*,$ which divides credence equally among structure descriptions, and also divides credence equally among state descriptions that are consistent with a given structure description.

$($Recall:$$ A state description is a conjunction that includes, for every atomic sentence in the language, either that sentence or its negation.$$ A structure description is a statement that says how many individuals are correctly described by each predicate.$$ Most structure descriptions are consistent with many different state descriptions.$)$

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$1.$ What is C$*($Ga$6)?$

$2.$ What is C$*($Ga$6|$Ga$1)?$

$3.$ What is C$*($Ga$6|$Ga$1$Ga$2)?$

$4.$ What is C$*($Ga$6|$Ga$1$Ga$2$Ga$3)?$

$5.$ Do your answers to $1-4$ suggest anything interesting about how ideally rational agents whose prior credences are given by C$*$ will reason?

Now, let's introduce a grue$-$like predicate, Q$,$ which we can define as follows:

Qa$1$ is true if and only if Ga$1$ is true

Qa$2$ is true if and only if Ga$2$ is true

Qa$3$ is true if and only if Ga$3$ is true

Qa$4$ is true if and only if Ga$4$ is true

Qa$5$ is true if and only if Ga$5$ is true

Qa$6$ is true if and only if Ga$6$ is not true

$($Here$\text{'}$s the thought: Q is like "grue": You have property Q iff you are either G $("$green$")$ and first observed before a certain time, or else you are not G and first observed after that time. And a$1-$a$5$ are the objects that get observed before that time; a$6$ gets observed after that time.$)$

So: We can rewrite $"$Qa$1"$ as $"$Ga$1,"$ and we can rewrite $"$Qa$6"$ as $"$Ga$6".$

$6.$ What is C$*($Qa$6)?$

$7.$ What is C$*($Qa$6|$Qa$1)?$

$8.$ What is C$*($Qa$6|$Qa$1$Qa$2)?$

$9.$ What is C$*($Qa$6|$Qa$1$Qa$2$Qa$3)?$

$10.$ Do your answers to $5-9$ suggest anything interesting about how ideally rational agents whose prior credences are given by C$*$ will reason?