Validity of Fundamental Equivalence between Induction and Deduction

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 $x^{\prime }$:

$$\forall x^{\prime }\in X:L\left(x^{\prime },D\right)\iff \left(B\wedge D\wedge x^{\prime }\right)$$

This means the classification produced by learning algorithm $L$ on new data $x^{\prime }$ 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 $x^{\prime }$.

Where:

• $L$ is a learning algorithm
• $x^{\prime }$ is a new instance from the instance space $X$
• $D$ is the training data set
• $B$ is the set of assumptions that constitute the inductive bias of $L$
• $\iff$ represents logical equivalence (“if and only if”)
• $\wedge$ 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”