## Thursday, June 27, 2013

### On Troikas: Disinduction and the Heap

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.