This is not a
valid induction. Induction is defined on sequences.
Sequences are typically interpreted in terms of
set theory or
functions (In fact, if functions and sequences are given a set-theoretic
interpretation, sequences ARE functions int->alpha, where alpha is some type). Sets exist outside of
time, so
mathematical induction is not the
right tool to deal with time-dependent entities. Set-theoretic arguments, and most arguments grounded in
classical logic and classic
mathematical logic are only useful for describing things that are
eternal, or for describing an
instant in time, unless we specifically take account of
change in the world.
The above is also not a valid induction because it does not have a concept of 'successor': there is no order, not even partial order, defined over emeralds. As such, there is no "successor axiom" relating item n to item n+1.
I claim that the above example is in fact a deductive argument based on the following:
- Axiom 1: The Grue Axiom
- grue(A), for all A is true if:
- A was FIRST examined before January 1, 2003 AND is green. OR:
- A was not examined before January 1, 2003 AND is blue.
Anything that has ever been grue is always grue.
- Axiom 2:Premise of Grue's time-independance
- This is the axiom that grue(A) is a predicate that is time-independent.
- Axiom 3:Premise of Universalisability of predicates over emeralds
- This is the premise that all emeralds for are alike, so we may take a representative sample.
Clearly, the first and second axioms are
incompatible. The final axiom is interesting, because it interacts with the
definition of an emerald. If we require that all emeralds be grue, then nothing examined after 01/01/2003 is an emerald. (We may similarly require that emeralds be examined before 01/01/2003). If we do not require emeralds to be grue, we must resort to the third axiom to decide that all emeralds are grue.
This third axiom is what Mr. Frog would refer to as the inductive step. Clearly, it makes no sense in this context to apply it? Why? Because we consider emeralds to have a temporally-defined property (Time of examination), yet we also require all emeralds to be the same. Clearly, all emeralds in our sample will have the same time of examination, yet we do not conclude that all emeralds are examined at that time (Because, for there to be any problem with believing all emeralds grue, we admit emeralds not examined at this earlier time). If the logic is to be sound, we must refuse to believe in the existence of emeralds examined at another time. Using this argument to show that "induction" is worthless is like my using the fact that my car doesn't drive well on lakes to show that cars are useless.
See false mathematical proofs.