Disinduction and the Heap
Consider mathematical induction:
Given: 1 has property P;
And: For any positive integer n, if n has property P then so does n+1;
Deduce: Any positive integer N has property P.
The middle term says that P is ‘inductive’; which means an infinite conjunction of implications: (P(1) implies P(2)) and (P(2) implies P(3)) and (P(3) implies P(4)) and … Therefore mathematical induction is a sorites of infinite length.
Denying the conclusion creates a trilemma:
1 has property P;
For any positive integer n, if n has property P then so does n+1;
Some positive integer N does not have property P.
This is the Disinduction Trilemma. It is also known as the Paradox of the Heap: for let P = ‘does not make a heap of sand grains’. Surely 1 grain of sand is not a heap; and surely adding one single grain of sand to a non-heap will not make it a heap, so non-heapness is inductive; yet surely there is some number N, denumerating a sand heap!
Disinduction can be expanded into this troika:
Moe: 1 has property P, P is inductive, and every N has property P.
Larry: P is inductive, some N does not have property P, and neither does 1.
Curly: 1 has property P, some N does not, and P is not inductive.
It also can be unpacked into these three deduction rules:
If 1 has property P, and P is inductive, then every N has property P.
If P is inductive, and some N does not have property P, then neither does 1.
If 1 has property P, and some N does not, then P is not inductive.