Part I. The Mendacity of Mathematical Certainty

Standard Pennsylvania Jury Instruction 7.01 states, in part:

Although the Commonwealth has the burden of proving that the defendant is guilty, this does not mean that the Commonwealth must prove its case beyond all doubt and to a mathematical certainty, nor must it demonstrate the complete impossibility of innocence. A reasonable doubt is a doubt that would cause a reasonably careful and sensible person to hesitate before acting upon a matter of importance in his or her own affairs. A reasonable doubt must fairly arise out of the evidence that was presented or out of the lack of evidence presented with respect to some element of the crime. A reasonable doubt must be a real doubt; it may not be an imagined one, nor may it be a doubt manufactured to avoid carrying out an unpleasant duty.

(emphasis added). 

I have objected to the use of the phrase “mathematical certainty” in the jury instructions because it is misleading and inaccurate.  Prosecutors often use the phrase “mathematical certainty”  to diminish or lessen the burden of proof that they must meet to “win” a conviction. When, in fact, the phrase “mathematical certainty” means the same thing as proof beyond a reasonable doubt. 

The phrase “mathematical certainty” is ambiguous.   What is a “mathematical certainty” anyway?  Within the context of a criminal jury trial, judges and prosecutors use the phrase “mathematical certainty” as a euphemism for “absolute certainty” or “demonstrative certainty,” and logical or syllogist certainty. These phrases all refer to the type of truth that is inherent in its proof– the type of proof that can be demonstrated, and to which no rational or empirical evidence can refute.  However, this is not what a “mathematical certainty” is.  

In the past, the phrase “mathematical certainty” was juxtaposed to the phrase “moral certainty,” which is derived from “moral evidence.”  The two types of “certainty” were defined by their relationship to each other.  Mathematical Certainty is the standard that applies to evidence that can be measured and quantified.  Moral certainty refers to evidence that cannot be measured. The American legal system gradually came to realize that modern juries no longer understand the concept of “moral evidence” or “moral certainty.”  So, the phrase “moral certainty” was replaced with the phrase “proof beyond a reasonable doubt.”   In 1875, the Supreme Judicial Court of Massachusetts explained, 

Proof ‘beyond a reasonable doubt’ ... is proof ‘to a moral certainty,’ as distinguished from an absolute certainty. As applied to a judicial trial for crime, the two phrases are synonymous and equivalent; each has been used by eminent judges to explain the other; and each signifies such proof as satisfies the judgment and consciences of the jury, as reasonable men, and applying their reason to the evidence before them, that the crime charged has been committed by the defendant, and so satisfies them as to leave no other reasonable conclusion possible.” 

Commonwealth v. Costley, 118 Mass. 1, 24 (1875).  

Criminal trials deal with the affairs of human beings, their interactions, their perceptions, their emotions, etc.  These human facts are not measurable or quantifiable.  In Victor v. Nebraska, the United States Supreme Court noted, 

“everything relating to human affairs, and depending on moral evidence, is open to some possible or imaginary doubt”—in other words, that absolute certainty  is unattainable in matters relating to human affairs. Moral evidence, in this sentence, can only mean empirical evidence offered to prove such matters—the proof  introduced at trial.

511 US 1, 13 (1994).

Outside of philosophy, people who use math in the real world as a tool generally do not reach conclusions to an absolute certainty.  In fact, many of the important facts that can be proven to an absolute certainty already have been so proven.  To the extent that a person might disagree with settled facts, such as the shape or age of the Earth, the disagreement stems from ignorance or ideology.  There is simply no reasonable doubt about these facts, and to the extent that any person harbors doubt, it is unreasonable and imaginary. 

However, when dealing with facts that are not settled, we deal with statistical significance, margins of error, confidence intervals, probability, and industry-wide accepted standards. Laboratories such as the Office of the Medical Examiner do not measure or quantify their conclusions to absolute certainty.  They have accepted margins of error.  Scientists using the most vigorous and controlled experimental designs usually report their conclusions in terms of statistical significance, rather than absolute certainty. A criminal trial is not an experiment under controlled laboratory conditions. No fact which is settled, or can be settled, would ever be the ultimate question to answer at a crimnal trial. So, to instruct a jury that the prosecution need not prove its case to an absolute certainty is a truism.  

Suppose the question at trial is whether the defendant acted maliciously when the car he was driving crashed causing the death of his passenger.   

Mathematical Evidence:

The prosecution could introduce data from the vehicle’s event data recorder.  This data could show the speed of the vehicle at the time of the crash, if and when the brakes were applied, the angle of the steering wheel, the amount of traction available to the wheels, and many other data points.  The accident scene investigators could measure skid marks on the road.  If surveillance video is available, investigators could estimate the vehicle’s speed based on the number of video frames in which the car passed between two defined points, etc.  This type of evidence would have to be proven to a mathematical certainty in order for it to be credible. 

Moral Evidence: 

The prosecution could also  introduce the testimony of witnesses who observed the defendant’s actions, heard the defendant’s statements, and saw or experienced the accident. This type of evidence would have to be proven to a moral certainty for it to be credible.  

A criminal trial might, and probably will,  include both types of evidence.  The factfinder must then consider all the evidence– both the mathematical and moral evidence– and be satisfied that there is no reasonable conclusion other than the defendant’s guilt.  So, the factfinder can take into consideration the vehicle’s speed– a fact proven to a mathematical certainty– and a witness’s testimony that immediately before the accident the defendant attempted to swerve to avoid hitting a deer– a fact proven to a moral certainty.  The factfinder could then conclude that even though the defendant was driving in excess of the speed limit, his conduct was merely reckless, but not malicious.  

Thus, Standard Pennsylvania Jury Instruction 7.01 is simply misleading and inaccurate.  Depending on the type of evidence introduced at trial, the prosecution might very well need to prove guilt to a mathematical certainty in order to meet its burden of proof beyond a reasonable doubt.  Just as the phrase “moral certainty” means the same thing as proof beyond a reasonable doubt, so does the phrase “mathematical certainty.”  The applicable standard depends on the type of evidence upon which the decision must be made.  Thus, I would revise Instruction 7.01 as follows: 

Although the Commonwealth has the burden of proving that the defendant is guilty, this does not mean that the Commonwealth must prove its case beyond all doubt or to an absolute certainty.  However, the Commonwealth must prove the defendant guilty to a moral and mathematical certainty.  In other words, the Commonwealth must prove the defendant’s guilt beyond any and all reasonable doubt.  A reasonable doubt is a doubt that would cause a reasonably careful and sensible person to pause or hesitate before acting upon a matter of importance in his or her own affairs. A reasonable doubt must fairly arise out of the evidence that was presented or out of the lack of evidence presented with respect to some element of the crime. A reasonable doubt must be a real doubt; it may not be an imagined one, nor may it be a doubt manufactured to avoid carrying out an unpleasant duty.  The Commonwealth meets its burden of proving the defendant guilty beyond all reasonable doubt when there is no reasonable conclusion other than the defendant’s guilt.