I don't have much experience of the use of Baysian reasoning, but I would say that it is a mathematically formalised and quantified method for applying the principle of inductive reasoning. Inductive reasoning gives you the general qualitative idea that the reliability of a pattern of past observations is strengthened by future observations that fit the pattern and weakened or destroyed by those that don't fit the pattern. Baysian reasoning seems to allow you to actually quantify - to attach probabilities to - the strength of the pattern. But, as I say, my experience of Baysian reasoning is limited. so I may be wrong!

James:

Bayesian reasoning is merely saying, "it is more probable than not, therefore it is true."

This is not my understanding at all. If by "true" you mean "certain" then the incorrectness of your statement is obvious. Your statement would then amount to this: "If probability > 0.5 then probability = 1". Clearly the conclusion does not necessarily follow from the premise.

If you mean something else, you'll have to explain it.

In your example of people being "murdered", do you mean people being executed on flimsy evidence? If so, how does this relate to Baysian reasoning? In a legal setting, there is the concept of "proof beyond reasonable doubt" because the probability of guilt can never be shown to be 1.