>If P is true, its negation is false.

Not exactly sure what you're referring to here. Still let me give you some detail on how I negated that statement:

I'm essentially negating phrases within the sentence using the distributive property from algebra:

~(<Correlation> <does not imply> <causation>) = (<No Correlation> <does imply> <no causation>)

If the positive statement is valid the negated statement is also valid. Negating +2 into -2 doesn't make the -2 any less true then the +2, in fact it's not even applicable.

You should note that the negated statement (No Correlation does imply no causation) is a fundamental pillar in science experiments. It's part of the reason why researchers always convert their hypothesis to null hypothesis and then try to disprove it.

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>If this were the case, the word "true" would have no meaning. We prove things in science all the time. We proved that you can cross a river without getting wet, for instance. We proved that humans can travel to the moon. The examples are endless, really.

I want to emphasize that what I'm talking about isn't some theory I made up. This is established scientific philosophy within academia. It's actually one of the harder things to understand because it flies in the face of our natural intuition.

None of your examples have been "proven." What is happening is that you have formed a very reliable conclusion based on intuition, you haven't gathered proof. Lets take a look at your river example...

One way in attempt to gather proof that crossing a river makes you wet is to observe 500 people crossing the river. If you observe that 500 people cross a river and all 500 got wet, what you have established is not a proof but a correlation that crossing the river is correlated to getting wet. Intuitively we assume that this is a proof, but from a rigorous and more philosophical standpoint not only have we not proven anything, it is actually fundamentally impossible to establish a "proof." Try to wrap your head around that.

Let me give you an example that's easier to understand and more inline with our intuitive understanding of the world. The following link is a research study on the correlation between birthrate and the population of storks: http://web.stanford.edu/class/hrp259/2007/regression/storke....

They arrive at the conclusion that baby's are delivered by storks because of the positive correlation. Our intuition tells us that this conclusion is improbable despite the correlation. Note that correlation, intuition and proof are completely separate things.

It is however possible to disprove things. If I establish that there is no correlation between cancer rates and heights then it is by logic impossible for height to cause higher cancer rates. What I have done is disproven height as a causative factor for cancer.

It is fundamentally impossible to prove anything. It is only possible to establish correlations. Correlations can only be used to falsify things or demonstrate the possibility of causation. This is not some weird thing I made up, this is established knowledge. I don't blame people for voting my OP down; Many researchers don't consider or even learn about the implications that the scientific method has on reality as we know it.