Special cases of Hypotheses

1. Unfalsifiable hypothesis:

   $$ P(\mathcal{H}|\mathcal{O}) = 1, $$ where \(\mathcal{O}\) is the set of all possible observations.

2. Impossible hypothesis:

   $$ P(\mathcal{H}|\mathcal{O}) = 0. $$

3. Untestable hypothesis:

   $$ P(\mathcal{O}|H) = P(\mathcal{O}) $$

The unfalsifiable hypothesis is designed such that no matter what observation we make, the hypothesis cannot be rejected. For example, "whatever happens, I made that happen". Go ahead, prove it wrong. Equivalently, such hypotheses are true by definition. They can only function as axioms.

Impossible hypotheses are usually those which are logically inconsistent, or in other words, self contradictory.

Untestable hypotheses are those for which it is impossible to determine how they might affect any particular outcome or experiment. Needless to say, these are also useless, unless we have underestimated the size of \(\mathcal{O}\).