Since I don't understand this topic very well, I blog about it. Nescio, ergo blogo! Caveat lector! The following notes are a blend of what I have gleaned from Nicholas Rescher and Douglas Walton and my own reflections.
1. Burden of Proof and Defeasible Presumption are correlative notions. If there is a defeasible presumption in favor of not-p, then the burden of proof rests on the one who asserts p. And if p is such that the burden of proof rests on the one who asserts it, then there is a defeasible presumption in favor of not-p. BOP and DP are two sides of the same coin.
For example, in Anglo-American courts of law there is a defeasible presumption in favor of the innocence of the accused. One is presumed innocent until proven guilty. This throws the onus probandi upon the state in criminal cases and upon the plaintiff in civil cases. The presumption of non-guilt induces the burden of proving guilt.
For a second example consider the practice of safety-conscious gun handlers in non-combat situations. Their presumption is that every gun is loaded; this puts the BOP on the one who claims the opposite. In a combat situation, or just prior to one, however, it is the other way around: the wise soldier does not presume that his weapon is ready to fire; he checks and makes sure. There is a defeasible presumption that his weapon is unloaded, and the burden is on him to prove that it is loaded. Either way we have the correlativity of BOP and DP.
This suggests the context-relativity of judgments as to where the BOP lies.
2. Presumption that p is true is not to be confused with (high) probability that p is true. If a gun dealer has just received a shipment of tactical shotguns from Remington the manufacturer, then the probability is very high that none of these guns is loaded. And yet his safety-conscious presumption will be that they are loaded. Similarly in a court of law. The accused is presumed innocent even when the probability of his being innocent is low or even zero. (E.g., Jack Ruby's shooting of Lee Harvey Oswald.)
3. Proof is a logical concept, but burden of proof is not. Perhaps we could say that BOP lays down a rule of proper conduct in dialectical situations. The rule pertains to the 'ethics of argumentation.' The rule is that he who advances a thesis, by so doing, incurs the obligation to substantiate his thesis by adducing reasons or considerations in in its favor, and by answering objections.
4. Accordingly, there is both a burden of proof and a burden of reply. The proponent of a thesis has the initial burden of defending his thesis. This remains constant throughout the dialectical proceedings. But if his opponent lodges a good objection, then the proponent has the additional burden of replying to the objection. A further complication is that the opponent in the course of objecting to the proponent's contention may make a claim that itself needs defense, in which case the burden of proof shifts onto the opponent in respect of that claim.
Bearing this in mind, we see the need to nuance the claim advanced in #1 above according to which the onus probandi in Anglo-American law rests on the state or on the plaintiff. That is true with respect to the initial allegation, but the defense may assume burdens of proof depending on how it builds its case.
5. Presumptions make up the doxastic status quo. And so it appears that a certain conservatism is inherent in laying the burden proof on those who would defeat presumptions. This needs to be explored.
6. Wherein resides the rationality of a presumption? Rescher claims in his book on presumptions that the rationality of a presumption consists in its conformity to a well-established practive, and that it is not a matter of evidence. This too needs to be explored.
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