Our friend Christopher Monckton of Brenchley wrote a piece over at Anthony Watt’s place in which he said (Reader KA Rodgers asked me to have a look):
CO2 concentration continues to climb. Global temperature doesn’t. Absence of correlation necessarily implies absence of causation. Game over, logically speaking.
The key proposition—Absence of correlation necessarily implies absence of causation—pleased some and set others off into a standard internet tizzy. Some said, “Right!”, others “Wrong!” Since it’s Advent, a joyous time, I’m happy to report both sides are right and can cease fighting. Although, those who said “Right!” were righter than those who said “Wrong!”
Correlation has a technical definition in statistical theory and roughly means that two “variables” when plotted one against the other show something resembling a straight line. Imagine a cosine wave from 0 to 2π: one variable time, the other amplitude. The statistical, technical correlation of this series is zero. But any civilian who looks at this plot would say that something is going on; i.e. that these two variables are “correlated.”
To distinguish these two meanings of the word, let st.correlation represent statistical, technical correlation, and let cv.correlation be the civilian’s “something is going on.” Keep in mind that cv.correlation encompasses st.correlation: if you have st.correlation you automatically have cv.correlation, but (as we have just seen) the converse is not true.
Obviously, one can have causation (if we insist time causes the amplitude to take the values it does) and no st.correlation. Lord Monckton’s modified phrase “Absence of st.correlation necessarily implies absence of causation” is therefore false. Why? Because absence of st.correlation does not imply absence of causation.
But given causation, we see cv.correlation, and vice versa. However, as is well known, we can have cv.correlation without causation, i.e. cv.correlation does not imply causation.
Causation in this case says the probability the amplitude takes the value y given that time equals t is either 1 or 0, and that this condition holds for all values of y and t. In other words, if we know the value of t we know with certainty the value of y.
But don’t forget we’re insisting that time makes amplitude take its values. In most real-life examples, it wouldn’t be time causing something, but the state of some other thing at time t which is causing the thing of interest (here amplitude) to take the value it does. Time is merely a proxy for this other variable. Thus knowledge of causation is something greater than knowledge of probability.
Lord Monckton’s example is thus just like ours, using time and temperature, with time being a proxy for carbon dioxide. I think it is also clear he had in mind cv.correlation and not st.correlation because nobody expects carbon dioxide to everywhere and always cause linear effects on temperature.
Now some logic. “If causation implies correlation” is true, then (via elementary rules) the contrapositive “If no correlation then no causation” is also true.
We have already seen that “causation implies st.correlation” is not necessarily true, which is why “no st.correlation then no causation” is not necessarily true. But if—as the full logical statement goes—“causation implies st.correlation” is true, then “If no st.correlation then no causation” must be true. So Lord Monckton was on solid logical (but perhaps not solid physics) ground if he had meant st.correlation—which, again, I don’t think he did.
Now “causation implies cv.correlation” is true. Causation, by definition, has some effect which must be evinced in cv.correlation. Thus it must be so that “if no cv.correlation then no causation”. Monckton scores again.
Could anything else booger up the works? Well, there could be some third (or third plus forth, plus fifth…) thing which somehow (we needn’t know how) blocks carbon dioxide from causally changing temperature. But since this third thing is blocking carbon dioxide, carbon dioxide is not causing changing temperatures.
Or it could be that carbon dioxide does cause temperature to change but that some third (etc.) thing immediately (or at least before we can measure temperature) changes temperature back to where it was before carbon dioxide changed it. Kind of like an evil anti-carbon dioxide. But how could we prove this? This brings us back to the epistemology (probability)—ontology (actual cause) distinction made above. If all we have are the measurements, it would look like there was no causality. That is, our knowledge of causality would be missing, even though it’s there.
The third (fourth, etc.) thing could also cause temperatures to change but to all sorts of other values and not just to the exact place to where it was before being affected by carbon dioxide. This is the normal case, as seen in nature. The cv.correlation will not be exact, and again we musn’t confuse epistemology with ontology. In any case, the man from Brenchley is still right.