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 if and only if:

$x$ 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 $\map {\operatorname {grue} } {P, Q, t}$
$x$ 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 $\map {\operatorname {grue} } {P, Q, t}$, as they have been $P$ and have been observed before $t$.

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

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

## Resolution

Upon first glance, this paradox may not seem like a mathematical issue, as philosophical induction is not a mathematical proof technique.

While it is used in everyday life, the notion that one can assume that the future will be like the past is an unjustified assumption.

This observation is called Hume's problem of induction, and is a philosophical - not mathematical - problem.

However, this paradox is much stronger than Hume's problem, as it shows that a naive formulation of induction is not just unjustified, but mathematically inconsistent.

Consider the following loose principle:

Suppose that all objects of some class $C$ have been observed to have some property $P$.
Then we can accordingly adjust the probability that all objects of class $C$ have property $P$.

The phrase "accordingly adjust" is ambiguous, and can be replaced with any precise description of such an adjustment.

If one uses any principle like this to conclude that the objects of $C$ have some property $P$ with some probability, one can also conclude that they have all properties of the form $\map {\operatorname {grue} } {P, Q, t}$, where $Q$ is arbitrary, with the same probability.

Thus, one can conclude that the objects of class $C$ observed after $t$ will have all properties with equal probabilities.

Thus, any induction principle that indiscriminately allows predicate to be inducted upon is not just unjustified, but mathematically inconsistent.

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

There is clearly something about gruesome predicates that make them incompatible with philosophical induction, so any consistent induction principle must disallow them.

However, it is not clear what this incompatibility is.

It is tempting to assert that the issue with gruesome predicates is that they are time dependent - that is, that they are of the form $\map {\operatorname {grue} } {P, Q, t}$.

However, suppose there were some language in which the predicate grue meant $\map {\operatorname {grue} } {\operatorname {green}, \operatorname {blue}, t}$, and the predicate bleen meant $\map {\operatorname {grue} } {\operatorname {blue}, \operatorname {green}, t}$.

In this language, the English predicate green is of the form $\map {\operatorname {grue} } {\operatorname {grue}, \operatorname {bleen}, t}$.

We can similarly translate any predicate into an equivalent gruesome form.

Thus, it cannot be the form $\map {\operatorname {grue} } {P, Q, t}$ itself that is problematic.

The resolution of this paradox must therefore identify precisely what about gruesome predicates is problematic so that consistent induction principles can disallow them.