Legal Standards Of Proof And Probability

Different courts and legal systems use diverse phrases about evidentiary standards of proof1. Few to none are precise in the sense that they offer a fixed probability number for this proof. For example, there is no certain rule that proof “beyond a reasonable doubt” means that proof must have probability greater than, say, 0.95. As to this trend towards vagueness, I say amen.

Onus probandiThis fact is consonant with objective, or logical probability, where precise numerical values are not always known. For example, given the evidence, “Most male statisticians are remarkably virile and WMB is a male statistician” then the probability that WMB is remarkably virile is greater than 0.5. We cannot be more exact given this evidence: the probability is not, for example, 0.732; the best we can do is the interval greater than 0.5 to, but not reaching, 1.

In British and American courts and those similar, the highest standard of proof is “beyond a reasonable doubt.” Importantly, this is not “beyond any doubt.” There is some evidence that some jurors commit this misinterpretation.

The reason the substitution cannot be made is simple: questions of guilt are contingent and the probability of any contingent event, given any non-tautological evidence, is less than 1. It can be as close to 1 as you like, but it can never reach it. This is a philosophical truth.

Therefore, there will always be, for any event, evidence, which if accepted, will imply a probability of guilt less than 1 (this is what “contingent” means). Since this is so, the standard “beyond any doubt” can never—as in never—be reached.

“Beyond reasonable doubt” acknowledges both the contingency of the event and the fallibility of knowledge. We can assume, given the history of the courts and the behavior of juries, that the probability implied by this is quite high. But just how high is not, cannot, and (I say) should not be specified.

The next most stringent category is “clear and convincing evidence,” which is used in some criminal trials and often in civil cases. The probability implied by this standard is meant to be less than beyond reasonable doubt,” but how much less is never (and should never be) stated.

Concentrate instead on the next lower rung, which is “preponderance of the evidence,” which some interpret as “as likely as not.” The later, by the English alone, allows us to fix a number for this probability, which is 0.5. If, given the evidence, it is adjudged the probability of the event is “as likely as not”, this means the probability is greater than 0.5. Substitute the words “more probable than not” and the same conclusion is reached.

This is a very low standard of evidence. It is on par, quite literally, with a coin flip. This is so low a standard that you would not credit any lower exists. But one does.

It is “some credible evidence”, which unlike the previous standards, inverts (the temptation is to say “perverts”) the standard. This standard is used and is advocated by, inter alia, the National Association of Social Workers.

They say, “If an investigation determines that some credible evidence of abuse or maltreatment exists, the report is indicated (i.e., substantiated) and the family is offered appropriate services.” You have to enjoy the use of the euphemism offered.

Let’s see why this is backwards. John Doe is on trial for murder. Evidence for his guilt and innocence is presented. Considering the whole of this evidence, each juror forms an opinion of the probability of Doe’s guilt. The order is important and natural: given the evidence, what is the probability of guilt?

This same order is found for “clear and convincing evidence” and “preponderance of the evidence.” It can also be used for the “some credible evidence” standard, but here it can be inverted too easily.

Suppose (as Rumpole would say) a member of the caring profession arrives at the home of John and Jane Doe and their daughter Jill. The social worker notices on a wall a dent in the plaster, which if examined in just the right light might be head-shaped. The social worker says to herself, “Given that John is an abuser, what is the chance of finding a head-shaped dent in the wall, of the kind formed by smashing a child’s head into it?”

If John Doe is guilty, the probability of finding this evidence is at least greater than 0. It is therefore credible evidence. Mr Doe is then “offered” services which he must accept.

The order is backwards: guilt is assumed, then the probability the evidence is calculated. It is far too easy to find an argument for the credibility of nearly any piece of evidence, which is why the “some credible evidence” standard is so minor a burden.

The example is fanciful and somewhat absurd. But not too absurd. Anybody recall the Satanic Panic of late last century (a.k.a. Satanic Ritual Abuse)? The “some credible evidence” standard was wielded as a blunt instrument to “prove” that Johns and Janes Doe were eating babies, made as offerings to the Evil One. A sad tale of broken families, innocents convicted, and ruined reputations and fortunes.


1This post is sketch, to be filled in later. By no means is it complete or even mostly complete.


  1. DAV

    I’ve always been under the impression that “credible evidence” meant maybe “sufficient to go to trial” but more often “okay to that search warrant”. Do you know of any state that convicts solely on “credible evidence”? People have been convicted on “incredible evidence” in child abuse cases.

    In traffic court there is generally “no evidence” excepting the accusation of the cop handing out the ticket.

  2. BobN

    Minor correction – From my grand jury duty days, we were instructed that “preponderance of evidence” was to be interpreted as “more likely than not”, i.e., > 50%. It was the standard used to determine if there was sufficient evidence for an indictment.

  3. Ken

    A concise, thoughtful article about logic & truth. And mention of spouse abuse (as an example).

    Curiously, there’s a women’s support group (MSV) that has deemed use of “the truth” and “using logic” as methods of discussion as tactics that constitute “abuse.”

    Apparently, under the criteria applied this only applies to a man’s use of “the truth” and/or a man “using logic” toward a woman…apparently either a woman can apply these levers of control & domination against a man with impunity, or, it never occured to the MSV group that a woman could do such a horrid thing.

    In other words, Briggs, if you’re directing the above essay–its use of truth & logic–at one of the women associated with the Men Stopping Violence (MSV) organization, you may well have condemned yourself as an “abuser.”

    I wish I was making that up; its discussed at:

    The issue discussed is the use of “emotional reasoning” — a symptom of a psychological defense mechanism that is quite intrinsic to the “liberal mind” in addition to the theme at the above website.

  4. Doug M

    Credible evidence may not be sufficient to convict someone of a crime, but it may be sufficient for an agent of the state to take action. The state may separate a man from his children without trial based on credible evidence. With further investigation and legal wrangling, he will get “his day in court” and a higher standard of evidence will be applied.

    I am reminded of the OJ Simpson Trial. The State was unable to convince a jury that OJ had killed his wife, using the standard of “beyond a reasonable doubt.” Yet, the father of the victim managed to put fort a case “proved” that OJ was liable for the wrongful death of his son, using the standard of “preponderance of the evidence.”

  5. DAV

    Much as I think OJ was guilty, I think his being subject to a tort case after the trial is just plain wrong. Allowing these things amounts to saying “If we can’t get you in criminal court we will find a way”. Having to endure a criminal trial is bad enough without the potential of being subjected to an endless stream of lawsuits.

  6. Eric Anderson

    “The example is fanciful and somewhat absurd.”

    Perhaps, but it happens all the time, even in “science,” which is a somewhat different field than what you’re discussing right now. I’ve had some interesting discussions with people about whether there was evidence “supporting” a particular theory. Those who are convinced of a theory’s correctness often believe they have mounds of supporting evidence, but I’ve tried to explain that it is important to look at the whole of the evidence (at least getting into the “preponderance” standard). Simply finding evidence that is credible, doesn’t do it. Another term that might be used is “consistent with.” Is there evidence that is consistent with theory x? This is an extremely low standard, so we should not be too impressed with evidence that is merely consistent with a theory.

    The example I like to use is this: There is all kinds of evidence that is consistent with the idea that the earth is flat. Indeed, in almost all of our daily interactions, nearly everything we see and do is consistent with this idea. It is only in those unusual situations when we are extremely high up, or get into space, or carefully follow astronomical phenomena, that we see evidence that the earth is a spheroid. So evidence simply being consistent with a theory is not enough. We need to look at all the evidence available and at least come to a “preponderance” level.

  7. DAV, I believe your feelings are misplaced. The reason more convicted criminals are not sued for damages by their victims [or victims’ families] in civil law is because they have little or no money. When criminal defendants are well-to-do tort actions are quite common. Just reviewed my local Superior Court calendar and out of 22 trials scheduled this week, two were tort actions against convicted criminal defendants. A conviction in a criminal trial virtually never truly covers [repays] the unsolicited expenses of a crime’s human victim. Fines levied there go to the county’s or state’s general fund.

    In the mentioned case [and many others] there were at least two separate acts thus two causes of action. The alleged “crime” was a violation of criminal law, where charges could only be brought on behalf of “the people” – or the community, while as you noted “torts” are a violation of civil law, where charges can be brought by the actual persons victimized. How else could they personally receive justice and recover their lost wages, etc.?

    The two types of “legal wrongs” usually have separate and common elements, so they tend to confuse citizens into thinking they may cover the same exact thing. They don’t. Justice demands they be treated as separate acts.

  8. Noblesse Oblige

    Nice piece.

    The vernacular also spills into the world of verbs. For example, “The body of evidence SUGGESTS that….” or “The body of evidence INDICATES that….” In scientific publications, it seems that “indicates” is stronger than “suggests.” Or is it the other way around. And what does it say about the probablity that (the following phrase) is true? And then what does it mean to say “The body of evidence POINTS TO….” And what kind of body is a body of evidence, anyway.

  9. Eric Anderson


    Thanks for the careful explanation. You are certainly correct that between the criminal and civil cases covering the same events, there are separate parties bringing the actions and, potentially, separate elements to the claims. Further, germane to this thread, there is often a different evidentialy level that must be attained, so it will typically be easier for the civil plaintiff to win something than for The People to prevail in the criminal case.

    I think DAV is expressing a general unease with the idea that you could be (let’s say, wrongly) accused of a crime, found not guilty, and then still have to endure a civil trial about the same event. In addition to having to go through the trauma of two trials, the defendent (if not indigent) is left paying for their defense as well.

    It seems like it is the middle-class person who is in the worst position. They might have enough that they are worth going after in a civil case, but not enough to be able to pay huge sums in attorneys’ fees defending two actions. The whole idea is pretty scary to a lot of folks.

    Tort reform, anyone?

  10. Eric Anderson
    Thanks. Agreed it can be scary and is mostly redundant. It is a bad system, but better than any other currently in existence. Don’t believe many bar association members would buy into a “tort reform” campaign, but put me down for supporting the idea.

  11. Rich

    Concentrate instead on the next lower rung, which is “preponderance of the evidence,” which some interpret as “as likely as not.” The later, by the English alone, allows us to fix a number for this probability, which is 0.5. If, given the evidence, it is adjudged the probability of the event is “as likely as not”, this means the probability is greater than 0.5. Substitute the words “more probable than not” and the same conclusion is reached.

    It seems to me that this paragraph contradicts itself. Either “as likely as not” means p = 0.5 or it means p > 0.5. It cannot mean both.

    It is also clearly a mistake (which you avoid) to assume that “lower standard of evidence” means “lower probability”. The lowest standard of evidence must carry a probability of 0.5 since this is the point of maximum ignorance. Lower probabilities than this imply higher probabilities for the converse proposition.

  12. Briggs


    I meant the cutoff.

  13. Rob

    @Eric Anderson:

    Not true. Mankind has known for a very long time that the Earth is round. As detailed in the linked article, early Greeks calculated its dimensions. One simply needs to watch a ship sail over the horizon to infer that its not flat. One might then think “the Earth appears to be round but with a circumference very large in comparison to my day to day activities.” A flat Earth works very well for laying out the footings for a house and similar. This is analogous to Newtonian gravity working well for many astronomical calculations. The simpler theories work well in their domains and approximate to more sophisticated theories within those domains. ALL theories are subject to this type of modification as more sophisticated observational and experimental techniques and theoretical constructs are developed.

  14. Eric Anderson

    Rob, my point has nothing to do with how long people have known the earth was round. My point is that there is lots and lots of evidence that is “consistent with” the idea of a flat earth (“credible evidence” in Briggs’ terminology above). There is, of course, evidence that points another direction. So it is not adequate to note that there is some evidence that is “consistent with” your theory, or even that there is “credible evidence” supporting your theory. The only way to make a reasoned conclusion is to look at all the evidence, as a whole.

  15. Rich

    Rob says, “One simply needs to watch a ship sail over the horizon to infer that its not flat.” Patrick Moore in Can You Speak Venusian has a nice take on this. He actually talked to the chairman of the Flat Earth Society. The response to this remark was, “Have you ever actually seen this happen?” He (Moore) reckoned 99% of people would have to say, “No”. For the other 1% the answer was, “Refraction”.

    Just sayin.

  16. Graeme

    be careful – in criminal law the criterion is “beyond a reasonable doubt”. However, in civil law, it becomes “on the balance of probabilities” – a very different standard of evidence.

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