Goodman's Paradox

Paradox
All emeralds thus far observed have been green.

By philosophical induction, we conclude that all emeralds are green.

Let $t$ be some arbitrary time in the future.

Define the predicate:
 * '$x$ is grue'

to be true
 * '$x$ is green and was first observed before $t$'

or
 * '$x$ is blue'

All emeralds thus far observed have been grue, as they have been green and have been observed before $t$.

By philosophical induction, we conclude that all emeralds are grue.

But this implies that any emeralds observed after time $t$ will be blue.

We can extend the above paradox to arbitrary properties.

Let $P$ and $Q$ be predicates and define the gruesome predicate as follows:
 * '$x$ is $grue \paren { P, Q, t }$'


 * '$x$ is $P$ and was first observed before $t$'

or
 * '$x$ is $Q$'

Choose some class $C$ of objects that have all been observed to have some property $P'$.

All $x \in C$ thus far observed have been $grue \paren { P', Q, t }$, as they have been $P'$ and have been observed before $t$.

By philosophical induction, we conclude that all $x \in C$ are $grue \paren { P, Q, t }$.

But this implies that any $x \in C$ observed after time $t$ will have some arbitrary property $Q$.

Resolution
There is no agreed upon resolution to the paradox of gruesome predicates.

Two things are clear:
 * philosophical induction works for some predicates

and:
 * there is something about gruesome predicates that make them incompatible with philosophical induction

The resolution of this paradox must therefore identify precisely what about gruesome predicates makes them incompatible with philosophical induction.