# Relative vs Absolute scale: Where does this argument fail?

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.

• What is the basis of premise 2? It seems much easier to control a BB than a cannon ball, from my perspective. I can point it around, instead of shooting indirectly up in the air. I can rifle it more readily. Etc.
– user9166
Commented Jul 27, 2015 at 17:14
• @jobermark If the center of the cannonball is off be 3 inches, the edge will still hit the target. But if the center of the bb is off by that much, it will miss. Thinking more theoretical, not worrying about drop, weight, etc.
– Cain
Commented Jul 27, 2015 at 18:10
• Yes, a cannonball has a larger diameter, and will hit marginally more often than a bb, but I don't think premise 2 really follows from that. If you only count hits that include the midpoint of the bullet the size clearly has no impact. If you want to allow for grazing hits, you should treat them as partial hits, e.g. it's a 10% hit if only 10% of the bullet hits. Then you'll find the partial hits balance out with the misses from the smaller bullet. Commented Jul 28, 2015 at 1:31
• It's ridiculous to make conclusion about hit or miss. The smallness of either may be a function of time (or even the smallness of the other).
– user15473
Commented Jul 31, 2015 at 2:07
• @user15473 Err, that's the point of holding all else constant. The smallness MIGHT be a function of time. Space monkeys MIGHT intercept the bullet. But since these things don't really relate to the actual question, lets assume that they don't happen
– Cain
Commented Jul 31, 2015 at 15:53

Considering just the logic, you argument is fine on a formal level. Worded a little bit differently, you are asserting:

1. A target is harder to hit with a bullet the smaller it is.
2. A bullet has a harder time hitting a target the smaller it is.
3. Therefore, if we scale down both the bullet and the target, the target will be that much harder to hit.

Reworded:

Hittability is a function inverse to the size of the target and size of the bullet.

There's some other kinks going on here. First, your abstractions ask us throw physics out the window, but you also tell us the conclusion seems false. I'm not sure how to square those two claims. If we look at your argument in the abstract, there's nothing wrong with the logic and the conclusion directly follows.

1. f(x) = C1 * ( 1 / size of target ) --> f(x) = C1/s1
2. f(x) = C2 * ( 1 / size of bullet ) --> f(x) = C2/s2 Then it follows that f(x) can be restated more precisely as

Conclusion: C9 / (s1 * s2)

(where C1, C2, C9 are different stand-in constants for factors unknown or kept constant).

With your assumptions, we have no room to question the conclusion. But you tell us this

seems inherently false.

Second, I take this "seems inherently wrong" to be predicated on something where you're going wrong in saying

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.

What's missing is that if you merely shrink the target and gun, you've doubled the relative distance. If you shrink all three, you've fixed it, but then you've invoked physics and you've created a problem for your hittability equation, viz., you need a third premise:

1. f(x) = C3 / (distance between target and gun)

But then what you'll discover (if you do bother moving from a formally unproblematic argument to engaging real-world physics) is that you're missing several parameters related to hittability, and these factors seem to be behind your intuition.

Third, an important unresolved ambiguity is that your original argument does not make clear whether you are specifying that hittability is only a function of these two factors or whether hittability is a function that includes these two factors.

• Ok, this answer certainly makes a lot of sense to me. The distance thing definitely makes sense, and by scaling the distance back you increase hittability and balance out the effects. The other thing I know think is involved, from other's answers, is that we have to assume the spread of the gun decreases proportionally to the size of the bullet as well.
– Cain
Commented Jul 28, 2015 at 15:00

This is an answer based on a sensible mathematical formulation.

We have some `f(t,b)` that describes the probability of hitting the target as a function of the size of the target `t`, and bullet `b`. Premise 1 is that the partial derivative of `f` wrt `t` is `>0` everywhere. Premise 2 is that the partial derivative of `f` wrt `b` is `>0` everywhere. We can express the "scaled down" form for `t,b` as a parametric equation `t=c*u, b=u` so now we have `t/b=c` (a constant) as we vary `u`. If you compute the total derivative of `f` wrt `u` using this, you'll find `df/du>0` everywhere. Therefore, for the straightforward mathematical model, the conclusion follows from the premises.

I believe that your the model/intuition mismatch is due to the our implicit idea of aiming inherent in the use of the term "gun" and "target". For example, imagine that the target is moved around between shots, and the shooter is blindfolded -- a situation that does match the constraints of the premises.

I suppose this is largely contingent upon physics. A smaller target is only harder to hit insofar as the human firing the weapon has (assumingly) developed his ability on the larger target. Increasing the distance (relative size, among several other variables) or actual size of the target affects the human's ability to accurately or consistently hit it. Because there is a bit of perceived randomness when firing a gun, the likelihood that the shot will be within the range of the target drops noticeably. The same is true for the gun; it is no more physically difficult to hit a small or large target, but the intrinsic effects that the shape of the gun, ambient temperature, distance, psychological mindset, wind speeds, etc... have on the bullet would, in fact, change the perceived difficulty.

• I'm not really convinced. Take the human out of the equation. Say the gun has some spread area, and will shoot at a random spot within that area, such that the target is at the center of the spread. The larger the target is, the more likely it will overlap with the shot. No perception necessary, the bullet will hit a larger target more often than a smaller one.
– Cain
Commented Jul 27, 2015 at 16:31
• Let's say, for example, that the wind is blowing at 3 m/s. If we had a bullet the size of a bowling ball, it would be far less affected by this wind than a bullet the size of a DNA molecule. I do see what your concern is. The problem with the argument is not that it is invalid, because it isn't, but that it is unsound. It isn't the inherent size of the gun or target, but rather the proportional variance between them. Commented Jul 27, 2015 at 16:37
• In other words, the smaller target is not harder to hit because it is small, but because it is smaller in proportion than the gun. Imagine a target was shrunk down to 90% its original size and a gun (and a human) was shrunken down to a mere 1% of their original size. If the premises were true, then it should be very hard to hit the target, since both of them shrunk. However, proportionally speaking, you have a much wider area to hit within your field of view. Commented Jul 27, 2015 at 16:39
• So essentially premises 1 and 2 are not exactly false, but not fully complete, and smaller should be replaced with "proportionately smaller"?
– Cain
Commented Jul 27, 2015 at 18:55

This question as asked does not distinguish between accuracy and precision. A projectile might hit the target (and therefore be accurate), but if it hits a bunch of other things, then it is not precise.

If your projectile contains a bomb, then even if the target and projectile get smaller, the likelihood of hitting the target does not seem to change. If you have a more precise projectile, then hitting the target at all might get harder, but it seems more likely that you will just hit the target.

So, the problem is underspecified, and both conclusions you offered above seem possible but not necessarily true.

• I'm assuming 100% accuracy i.e The expected value of the center of the bullet is the center of the target. Obviously not 100% precise or size wouldn't matter at all. But I'm keeping this more abstract than thinking bombs and other things. Abstract it to drawing a circle on a target if that helps.
– Cain
Commented Jul 27, 2015 at 18:13

You ask whether the result of the experiment depends on scaling.

Your scaling parameter is the spatial dimension. But your experimental setting does not seem consequent. Because you scale the target and the gun, but you do not scale the distance between gun and target.

In my opinion, only after scaling all(!) spatial dimension one should reason whether the success probability changes or not.

• I was assuming distance was held constant, but if we hold relative distance constant instead the argument holds
– Cain
Commented Jul 27, 2015 at 21:42

I think your argument may fail because neither of the premises relate directly to the conclusion, since the conclusion assumes both the target size and the bullet size are altered together.

Premise 1 asserts only that the probability decreases as the target size decreases. There is no reference to the bullet size so nothing is being asserted about any change in probability if both the bullet size and target size is changed.

Similarly, premise two deals with the case where only the bullet size is changed.

EDIT

I think, perhaps, that the reasoning you are using in your argument is better suited to a static, mathematical structure than a dynamic, physical system.

In a physical system, the implications of altering one variable may not apply if we alter more than one variable.

There's an implicit additional part of your second premise that you haven't mentioned:

Premise 2: A smaller projectile is less likely to hit a target of the same size.

This is true because the projectile, fired from the same distance and at the same size target, will land fewer grazing hits (where a portion hits, but the midpoint misses) as a percentage of its total hits.