## Wednesday, October 14, 2020

### Bennett's positive and negative instrumentality

Bennett offers this account of positive vs. negative instrumentality. If the volume of the space of possible bodily movements occupied by doing A is greater than that occupied by doing not-A, then doing A is a negative instrumentality; if it is less, then it is positive. Thus, raising one’s hand is positive: one can, for instance, raise, lower or keep one’s hand level, and raising occupies less volume of movement space than not-raising.

Here’s a curious consequence. Let M be the maximum speed at which I can move. Let A be moving at a velocity of magnitude greater than half of M. Then A occupies more of the space of possible bodily movements than not-A and hence counts as negative by Bennett’s criteria.

Why? Well, velocity is a vector: it has a magnitude and a dimension. The relevant action space (assuming the movement is two-dimensional—we can’t fly) is a disc of radius M. The subset occupied by not-A is the a (closed) disc of radius M/2. The area (the two-dimensional analogue of volume) occupied by not-A is (1/4)πM2. The area of the whole movement space is πM2. The area occupied by A is thus πM2 − (1/4)πM2 = (3/4)πM2. Thus, A occupies three times the area occupied by not-A. Hence, not-A is a positive action and A is a negative action.

This seems quite wrong.