So my question is one of a factual nature, I’m not looking for a big debate here.
What established philosophical/mathematical/scientific methodologies exist that can prove cause and effect, that is the proposition “A was caused by B”?
In pure mathematics, there is no real notion of cause and effect, because there’s no notion of time. So mathematical proof is pretty much out.
In science likewise there is no capacity to prove anything. Science has no proven facts, despite what a lot of people believe, there are only theories with varying degrees of evidence.
Broadly speaking scientific methodology can lead to the valid acceptance of something in two ways. The simplified Popperian model is based on falsification, that is science can prove that A does not cause B under circumstance C. Essentially it works by eliminating everything that is impossible and leaving the possible. In most real science the Popperian model is not all that applicable because most things have too many variables to enable falsification of them all. At some stage there is also an element of ‘preponderance of evidence’. Basically what that means is that something is scientifically accepted so long as it can’t be proven to be wrong and has evidence supporting it. Something becomes scientific ‘fact’ when there is so much evidence supporting it that it overwhelms all else, but no scientists would ever talk professionally about something being a fact. Lunar gravity causing tides, viruses causing diseases and chemical reactions causing fires are all just scientific theories. Not one of those things has ever been proven and there’s no conceivable way to prove them.
There are various philosophical proofs that claim to prove that A causes B, but from what I’ve seen of them they are entirely sematic rather than practical. In reality it’s never possible to conclude that A caused B with any absolute certainty AFAIK. That’s because any practical human proof ultimately relies on observations logically interpreted WRT arbitrary axioms. You can prove that every time A is observed B follows and you can prove that B never occurs unless A is observed first, but even that doesn’t constitute proof that A caused B. That’s because either you can never prove whether B would have happened without A from observation, that can only be inferred. It’s also impossible because the arbitrary nature of any axioms means that there is no absolute reference value for true. With no reference value proof becomes a meaningless concept. There’s just no way to prove that A causes B in any practical logical sense.
Now this is a surprise. I’m not a Mathematician like you but not only have I heard of modal logics, I even wrote a paper on the computer application of one such modal logic (PDL). Time and cause/effect can be quite nicely handled in various modal logics. Temporal Logic is one such example.
In my field, real time has no true meaning so events in complex computer systems can best be described via cause and effect only. There’s an implied virtual time (if A caused B then A occurred before B) but most such events are don’t have a cause and effect relation so there is usually no way to give them a temporal order.
Modal logic generally isn’t considered part of math, IME (although I would consider it as such). But I wasn’t familiar with that branch of it, so I’ll have to poke around.
If clock B chimes immediately after clock A, did A cause B? 
If the cock crows, then the sun comes up, did the cock cause the event? :dubious:
If I press “submit” after typing this line, did pressing “submit” cause a “submit”? 
Maybe, No, and Yes.
But I can’t prove it 
[sup]B gets all the blame! and C has a smile on his face.[/sup]
I’m not a mathematician or statistician or philosopher, but the following has always been my understanding:
Correlation is usually a straightforward mathematical concept; two or more variables can (usually) be shown to be correlated, uncorrelated, or somewhere between. Causation is much more difficult to prove, and the researcher must resort to common sense, logic, or just basic assumptions.
This leads to the following axiom: correlation does not prove causation. Many research papers have been published that ignore this basic axiom. By demonstrating correlation, they have made the assumption that correlation proves causation.
Case in point: many sociologists have shown a correlation between poverty and crime. In areas where poverty is high, crime is high. In areas where poverty is low, crime is low. Therefore, poverty & high unemployment cause crime! If we reduced poverty, crime would decrease. The recommendation in these papers is always the same: more welfare and more government intervention.
There’s no doubt a strong correlation exists between poverty and crime. But does poverty really cause crime? How do we know crime doesn’t cause poverty? Or perhaps there’s a third variable (which the researcher refuses to acknowledge) that causes crime and poverty…
My answer is similar to that of Blake with one small addition.
In his book An Introduction to Scientific Research E. Bright Wilson[sup]*[/sup] said of cause and effect [bolding added in all cases]:
“A great deal has been written about the philosophy of cause and effect, much of which is based on redefining the terms in one way or another. Here, however, a rather prosaic viewpoint will be adopted. A practical aim of science is to control nature or, failing that, to predict important events. These aims are closely allied to that of explanation, a complex notion. If a scheme of connections between events has been uncovered such that when one event is made or allowed to occur, another necessarily follows, whereas if the first is not allowed to occur, the second will not happen, then the first event will be said to be the cause of the other. This definition requires that the first event be sufficiently fully specified, a limiting idea in itself since in principle it probably requires an infinite number of specifications. Fortunately, in practice a finite number usually suffices with a high, but not complete, degree of certainty. Some events seem to be capable of arising from several alternative causes. This can be put down either as a defect of the above definition or as a result of insufficiently precise specification of the exact nature of the effect. In practice, again, events often are specified so loosely that several causes are possible. Then the definition given above is conveniently broadened to include such cases.”
So cause isn’t proved but only supported by evidence. Sometimes the supporting evidence is exceedingly strong, sometimes not. When the evidence is strong the connection can be relied upon. As W.K. Clifford, scientist and philosopher of science said, “Truth is not that which may be contemplated without error. Rather it is that which can be relied upon without fear.”
- E. Bright Wilson received the PhD in physics from Caltech in 1933 with Linus Pauling as his advisor. He and Pauling collaborated in scientific work from time to time and coauthored a couple of books on quantum theory.
The basic problem is that nothing proves causation. As has been posted above, there are only varying degrees of certainty. I would say that causation can be established, but not proven.
The trouble arises when (a) someone naively tries to establish causation solely based on correlation, like the “electromagnetic fields cause cancer” people or (b) someone naively tries to disprove causation even when there is plenty of evidence in addition to correlation, by holding it up to an impossible standard, like the “cigarettes don’t cause cancer” people.
I beleive “entailment” is a useful concept in resolving the definition of cause and effect. Entailment (in logic) is the notion that in every case where A is true, B is true. Science hopes to evaluate the statement “Event A happens entails Event B happens.” Notice this says nothing about the statements for all other events C: “Event C happens entails event B happens” and A could describe a confluence of events. One way to prove this statement is to set up a place where only A occurs (excluding as many other events as posible) and see if B also occurs. Science concludes that A entails B by controlled study, attempting to minimize other potential causes. So what you have is correlation, but with minimal other posible causes. Itterate over enough time to increase certainty.
This can be done (as I often do in software tracking the cause of a bug) by introducing exactly one element, and see if the theorized effect occurs. I’ll often disable a feature and a bug doesnt occur, reintroduce it and it occurs. The former case is the “control,” a situation without potential cause A, and the latter an experimental situation, seeing if some B follows.
I hate answering my own question 
Entailment can’t differentiate between A causing B and both A and B being caused by C.
The truth of
C entails A and B vs simply A entails B
can be differentiated by finding situations where A happens and B does not. If these situations don’t exist, then C entails A and B AND A entails B are both true. I suppose, therefore, entailment might not be useful without a notion of time. Intuitively If A and B happen simultaneously after some time from C then maybe we can say C causes A and B, where (A and B) are actually a single event, if C happens then A then B with two gaps of time between C and A and A and B, then perhaps C causes A, causes B.
When a car is in motion (B), the hood vibrates (A). These are both caused by the engine rotating (C). Now you’re claiming that C entails A and B–which is reasonable–and that A entails B. That’s a curious claim, don’t you think?
Ah, but the true cause of (ABC) is a dead brontosaurus (X).