A Note on a Parameterized Version of the Well-Founded Induction Pricible

The well-known and powerful proof principle by well-founded induction says that for verifying 8x : P(x) for some property P it suuces to show 8x : 8y < x : P(y)] =) P(x)] , provided < is a well-founded partial ordering on the domain of interest. Here we investigate a more general formulation of this proof principle which allows for a kind of parameterized partial orderings < x which naturally arises in some cases. More precisely, we develop conditions under which the parameterized proof principle 8x : 8y < x x : P(y)] =) P(x)] is sound in the sense that 8x : 8y < x x : P(y)] =) P(x)] =) 8x : P(x) holds, and give counterexamples demonstrating that these conditions are indeed essential.