The Logical Before

Since Paul C. is having difficulty understanding why causality is linked to a logical order, not a temporal order, and since others might be interested in seeing why this is the case, I decided to write another post spelling it out clearly. Before I get into the main point, we already know that temporal order is not sufficient to infer causality because that is the post hoc ergo propter hoc (after this, therefore because of this) fallacy. A simple example will suffice: the Oklahoma City bombing happened before 9/11 happened, therefore 9/11 was caused by Timothy McVeigh. This is an obvious example of the post hoc fallacy. Others are not so obvious, and we see this many times in movies about crime. For example: The victim is killed moments before the defendant leaves the premise.

So we know that temporal order is not sufficient to infer causality. In this post, I am going to take it one step further. To do so, I must talk a bit about Einstein’s theory of Relativity. In order to follow what will occur, the most important aspect to grasp is the fact that light moves at a constant velocity regardless of the framework of the observer. This is counter-intuitive, and a simple example should show why.

Suppose you are travelling in a car that is moving at 60 miles per hour. If you throw a baseball at 60 miles an hour in the same direction that you are travelling, the ball will look (from your perspective) like it is travelling 60 miles per hour. From someone on the ground, however, the ball will look like it is travelling at 120 miles per hour. That is because the observer outside your car sees the ball’s velocity as the sum of your throw (60 miles per hour) plus your vehicles velocity (60 miles per hour).

Suppose that you saw the observer on the side of the road and wanted to throw the ball back at him after you’ve already passed. Your car is still travelling at 60 miles an hour, but you give a little extra effort and throw the ball at the observer at 70 miles per hour. The observer on the side of the road will have the ball come toward him at 10 miles per hour. (The car is moving 60 miles per hour in one direction, and you throw the ball in the opposite direction (indicated by a – sign), so the result is 60 – 70 = -10 miles per hour; or 10 miles per hour in the opposite direction that the car is moving.)

This makes sense to us because we’ve seen it in action. Suppose, however, that instead of a baseball, the person in the car turns on a flashlight. Relativity states that light will appear at approximately 3 x 10⁸ m/sec for both the observer in the vehicle and the observer outside the vehicle. That is, there is no adding on the velocity of the observer to light. It moves at a constant speed through all frames of reference.

So with this in mind, let me give a slightly different version of Einstein’s train. Suppose there are two people on opposite ends of a train and these people are named Adam and Bill. At the midpoint of the train is a bomb. Adam and Bill both have buttons they can press. This will send an electrical signal that travels at the speed of light to the bomb in the middle of the train. Adam wants to blow up the train; Bill wants to keep Adam from blowing up the train. As a result, Adam’s signal will cause the bomb to detonate while Bill’s signal will keep the bomb from detonating. Furthermore, let us stipulate that Bill is at the front of the train (i.e., toward the engine) while Adam is at the back of the train (i.e., the caboose).

For ease of math, let us stipulate that light moves at exactly 3 x 10⁸ m/s. Let us also stipulate that the distance between Bill and the bomb is exactly 1,000 meters. However, due to an error when the experiment was set up, Adam is slightly closer to the bomb: he’s only 900 meters away. Let us stipulate that from the train’s framework, Adam and Bill press their buttons at the exact same time.

Now it is obvious without even doing math that because Adam is closer to the bomb and because light travels at a constant velocity that the bomb will detonate if both press their buttons at the same time. Nevertheless, let us do the math on it.

If light travels at 300,000,000 m/s, how long does it take light to go 1,000 meters? This is a simple physics problem: t = d/v. In this case, t = 1,000 / 300,000,000 or 3.3 x 10^-6 seconds.

How long does it take light to travel 900 meters? In this case, t = 900 / 300,000,000 or 3.0 x 10^-6 seconds. This means that Adam’s signal will reach the bomb 3 x 10^-7 (0.0000003) seconds before Bill’s signal will reach the bomb.

Now suppose that there is an outside observer named Charlie. The train is moving. But because light has a constant velocity irrespective of the observer, he will see both signals travel at 3 x 10⁸ m/s just like those inside the train. Suppose that at the exact instant (from Charlie’s perspective) the bomb is in front of him, both Adam and Bill press their buttons, what does Charlie see? He sees two signals travelling at 3 x 10⁸ m/s. But he also sees the bomb travelling toward Bill's position (Bill is at the engine) and away from Adam's (Adam is in the caboose).

This means from Charlie’s perspective, if Adam and Bill were exactly the same distance apart and pressed their buttons at exactly the same instant, the signal from Bill’s button would reach the bomb before the signal from Adam’s button would reach the bomb. But because we know that Adam is 100 meters closer to the bomb than Bill, we ask a question: how fast must the train be moving so that from Charlie’s perspective both signals will reach the bomb at the same time?

As we calculated above, Bill’s signal will reach the bomb 3 x 10^-7 seconds after Adam’s. And we know that the difference in distance is 100 meters. So we need the train to cover 100 meters in 3 x 10^-7 seconds. However, this distance is split between Adam and Bill. That is, because the signal is moving toward Bill and away from Adam, the train needs to actually only cover 50 meters in 3 x 10^-7 seconds. This gives us 50m /0.0000003s = 1.67 x 10⁸ m/s, or just over 50% the speed of light.

So let us suppose that the train is moving at 2 x 10⁸ m/s, or 2/3s the speed of light. What will Charlie see?

He sees Adam press his button. The signal moves out at 3 x 10⁸ m/s and covers 900 meters. However, when it hits the 900 meter mark (from Charlie’s perspective) 3.0 x 10^-6 seconds later, the bomb has moved. The bomb is moving at 2 x 10⁸ m/s, and it does so for the same 3.0 x 10^-6 seconds. That means the bomb has moved 600 meters further down the track after that 3.0 x 10 ^-6 seconds. Ultimately, this means it takes Adam’s signal 9.0 x 10^-6 seconds to actually read the bomb.

At the same instant, Charlie sees Bill press the button. Bill’s signal travels out at 3 x 10⁸ m/s and the bomb has moved toward him at 2 x 10⁸ m/s too. This means that it takes only 2.0 x 10^-6 for Bill’s signal to reach the bomb. From Charlie’s perspective, Bill’s signal reaches the bomb 7.0 x 10^-6 seconds before Adam’s does.

What will the train do? Answer: it will explode. Even from Charlie’s perspective, it will still explode. Why is that? Because on the train, which is where the bomb is located, Adam’s signal reaches the bomb 3 x 10^-7 seconds before Bill’s signal does. Charlie observes Bill’s signal arriving 7.0 x 10^-6 seconds before Adam’s does, however. From Charlie’s perspective, the signal that causes the bomb to explode arrives after the signal to keep the bomb from exploding should have neutralized it.

So what caused the train to explode? Adam’s signal did. But from Charlie’s perspective, it shouldn’t have. But Charlie is still left with an exploding train, one that does not fit in a temporal causative sense. It does, however, fit logically. He knows that logically Adam’s signal must have caused the train to explode, and that Bill’s counter-signal did not neutralize the bomb.

Naturally, the train had to be going extremely fast: 2/3 the speed of light. Since we never reach those speeds on Earth, cause and effect usually follow the temporal scheme. However, it is a fallacy for us to believe that causes are temporal causes for the reasons illustrated above. The only thing that matters is whether logically they are causes. If we know that A and only A logically causes B, then even if we observe B occurring before A we know that A is the cause of B. This must be the case.

This is also why we can have logical precedence (that is, a logical before) without having a temporal before. This is commonly seen in theology when, for instance, we talk about the decrees of God. The difference between Infralapsarians and Supralapsarians boils down to the logical order of the decrees of God, not the temporal order (since all agree that temporally each decree occurred before the foundation of the world, in eternity past; that is, outside of time). There is no temporal before in causality; there is only a logical before.