If this can be formulated more rigidly, it might belong on Math instead, but for now I think this is the best place for it.
The context of this is in scaling things. Lets say there exists a target and a gun. The goal of the gun is to hit the target. Where does the following argument fail?
- Premise 1: A smaller target is harder to hit.
- Premise 2: A smaller bullet (gun) is harder to hit the target with.
- Conclusion: If both the target and gun were scaled down equally, the target would become increasingly harder to hit.
This conclusion seems inherently false. If both the target and gun were half the size, it should be like nothing has changed, and the target should certainly not be any harder to hit. Where is this argument wrong?
Abstractions: To really make the issue more clear, we're holding all else constant. No physics, no bystanders etc. Let the gun be 100% accurate, meaning the average of all the shots is always dead on, but not 100% precise, so any given shot can be off from the target by some amount. The conclusion then becomes that the smaller the target, and smaller the projectile, the smaller the percent of hits.
Alternate Another way of thinking about the problem would be what level of precision is needed to always hit the target. As the target and projectile get smaller, a higher level of precision is needed.