**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.

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