The fundamental equivalence between deduction and induction states that an inductive learning system’s output can be reproduced by a deductive system if we make the inductive bias explicit.

Formally, if L is a learning algorithm and B is its inductive bias, then for any new instance x’:

This means the classification produced by learning algorithm L on new data x’ after training on dataset D is equivalent to what would be logically deduced from the conjunction of the training data D, the inductive bias B, and the new instance x’.

Where:

  • is a learning algorithm
  • is a new instance from the instance space
  • is the training data set
  • is the set of assumptions that constitute the inductive bias of
  • represents logical equivalence (“if and only if”)
  • represents logical conjunction (“and”)

Example

Induction (Machine Learning)

  • You see examples: All swans I’ve seen are white
  • You guess a rule: All swans are white
  • You predict: The next swan will be white The Fundamental equivalence would then imply the following
  • Inductive learning: “I learned from data that swans are white”
  • Equivalent deductive system: “IF I assume the data represents all swans (assumption) AND I saw only white swans (data), THEN I can logically conclude all swans are white”