Logic and Reality

A implies B, B implies C, C implies D, D implies E.
A is true therefore E is true.
Impeccable logic.

Reality may be a little different.
 If A is true, then B is probably true.
 If B is true, then C is probably true.
 If C is true, then D is probably true.
 If D is true, then E is probably true.

A is probably true. Does this imply that E is probably true?

That depends on what values you assign to the probabilities.

If you reckon each probability is 95%, then the combined probability of E being true is 0.955, which is 77%. E is probably true.
If you reckon each probability is 90%, then the combined probability is 0.95, which is 59%. E is still probably true, but you’re a lot less confident of it.
But take your estimates of the probabilities down to 85%, and E is probably NOT true – just 44%.

If there are more probabilities to multiply together, your estimates of the probabilities have to be even higher to make the probability of the combination over 50%.

“Estimates” – an important word, that. Probabilities in the real world are usually estimates. Not many things are like fair dice or fair roulette wheels, where you can be reasonably confident about the values of probabilities.

There’s often a bit of wishful thinking in the estimation of probabilities, and it doesn’t take much over-estimation of combined probabilities to make an unlikely scenario appear likely. (Or, conversely, it doesn’t take much underestimation of combined probabilities to make a very possible scenario look highly unlikely.)

This is a problem that afflicts cosmology particularly – with no serious consequences down here on Earth. It affects psychology badly, even neuropsychology, with potentially serious consequences. Other areas of science can be affected too, particularly where there’s a lot of deductions from indirect evidence, or where observations are to any degree subjective.

It’s also a problem that afflicts risk evaluation in engineering projects. With potentially extremely serious consequences.