Epistemic Probability

The epistemic probability of a proposition is how reasonable it is to believe

Contents

Epistemic Probability

  • The epistemic probability of a proposition is how reasonable it is to believe, relative to a body of knowledge.

A Matter of Degree

  • “In our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of [probable] evidence. A wise man, therefore, proportions his belief to the evidence.” (David Hume, 1758)

Examples

  • It’s certain that 2+2=4.
  • It’s beyond a reasonable doubt that Russia interfered with the 2016 election in support of Trump.
  • In all likelihood Oswald acted alone in assassinating President Kennedy.
  • Echinacea probably does not prevent colds.
  • It’s reasonable to believe that Jefferson fathered at least some of  Sally Hemings’ children.
  • The Tunguska Event of 1908 was most likely caused by a meteor.
  • The idea the CIA assassinated Kennedy is farfetched.
  • It’s unlikely the tumor is malignant.
  • His statistics are questionable.
  • Chances are he died instantly.
  • It’s doubtful the legislation will pass.
  • It’s impossible that Frederick Douglass founded the NAACP.  

Epistemic Operators

  • An epistemic operator is a sentential operator expressing a degree or range of epistemic probability.
  • It’s certain that
  • The probability is 1.0 that
  • It’s virtually certain that
  • It is beyond a reasonable doubt that
  • It’s reasonable to believe that
  • It’s very likely that
  • There’s clear and convincing evidence that
  • There is a preponderance of evidence that
  • It’s a toss-up whether
  • It’s an open question whether
  • It’s uncertain whether
  • It’s unknown whether
  • There’s a chance that
  • It’s possible that
  • It’s plausible that
  • It’s credible that 
  • It’s doubtful that
  • It’s unlikely that
  • There’s a remote possibility that
  • It’s far-fetched that
  • It’s impossible that
  • The probability is zero that

Relative to a Body of Knowledge

  • Epistemic probability is relative to a body of knowledge.
  • A standard deck of 52 cards is shuffled and placed face-down.  The probability the top card is the ace of spades is 1/52, relative to what you know. But for someone who has peeked, the probability is either one or zero.
  • The peeker and you don’t have the same body of knowledge.

Certain, Impossible, and Toss-up

  • Certainty and impossibility are the upper and lower bounds of epistemic operators, corresponding to probabilities one and zero. 
  • A proposition is certain if its falsehood is impossible, if there’s no chance of it being false.  
  • A proposition is impossible if it’s certainly false, if there’s no chance of it being true.  As you read this sentence, for example, it’s certain you have a functioning brain and impossible you don’t.
  • Midway between the extremes is toss-up, where the probability of being true equals the probability of being false.  Flip a coin and it’s a toss-up whether it lands heads.
  • Certainty logically entails truth, as GE Moore noted in “Certainty.” 
  • Impossibility entails falsehood.  
  • The epistemic operators between entail neither.
  • It is certain that is logically equivalent to:
    • it is known that
    • it is known for certain that
    • it is an established fact that
  • Two Senses of Impossible
    • Incapability Sense
      • It’s impossible to trisect an angle using only a straightedge and compass
      • It’s impossible for a human being to run 100 miles an hour.
      • It’s impossible that an object with mass be accelerated beyond the speed of light.
      • It’s impossible for Smith to rob a bank and be in jail at the time.
    • Epistemic Sense
      • It’s impossible that Amy is in Alaska, since she’s right next to me here in Texas.
      • It’s Impossible that Smith robbed the bank, since he was in jail at the time.

Preponderance of Evidence and Very Likely

  • Certainty, impossibility and toss-up are points on the probability continuum. By contrast, preponderance of evidence and very likely are ranges of probabilities, line-segments rather than points. 
  • A preponderance of evidence exists for a proposition if it’s more probable than not. There’s a preponderance of evidence, for example, that New Year’s resolutions to lose weight don’t survive past February. Preponderance of evidence is the standard of evidence in civil actions, where a plaintiff sues a defendant.
  • A proposition is very likely if its probability is significantly greater than 0.5, if it’s in all likelihood true.  Flip a coin three times and it’s very likely it lands heads at least once.

Beyond a Reasonable Doubt

  • Beyond a reasonable doubt is the standard of evidence for criminal trials in the United States and Great Britain. It’s also the standard used in everyday affairs.
  • Washington Post Fact Checker
    • We will adopt a “reasonable person” standard for reaching conclusions. We do not demand 100 percent proof.
  • Beyond a reasonable doubt belongs to a family of terms expressing epistemic probability
    • View A Matter of Degree
    • Beyond a reasonable doubt is weaker than certain: everything certain is beyond a reasonable doubt, but not the reverse.  As I leave work my belief my car is where I parked it is beyond a reasonable doubt, given that locked cars generally stay put.  But it’s not certain, since even locked cars are stolen or towed.
    • Beyond a reasonable doubt is stronger than very likely: every belief beyond a reasonable doubt is very likely, but not vice versa. It’s very likely you won’t roll two sixes, but not beyond a reasonable doubt.
  • In the law reasonable doubt is described as the kind of doubt that would make a reasonable person hesitate to act in the most important of his affairs.
  • Beyond a reasonable doubt seems synonymous with certain for all intents and purposes.
  • Reasonable doubt exists if there’s credible evidence the statement is false, if there’s a plausible, competing hypothesis.
  • A statement is either:
    • true beyond a reasonable doubt, or
    • false beyond a reasonable doubt, or
    • an open question.
  • A person who states something as fact assumes the burden of proof, the burden of proving the statement beyond a reasonable doubt.

True and False

  • It is true that and it is false that are not epistemic probabilities
    • The epistemic probability of a proposition depends on the evidence.
    • Truth and falsehood depend on reality.  
  • Consider the hypothesis that Aunt Flo’s tumor is malignant. Its truth depends on whether the tumor is in fact malignant. Its epistemic probability, by contrast, is a function of the evidence: symptoms, family history, blood tests, scans, biopsies.  
  • Truth and falsehood depend on what’s real; epistemic operators on what’s known.

Epistemic Operators in the Law

  • Beyond a reasonable doubt:  This is the standard of evidence in a criminal trial.  The prosecution has the burden of proof, i.e. to prove the defendant guilty beyond a reasonable doubt; otherwise the defendant is found innocent.  The law doesn’t define the phrase.
  • Clear and convincing evidence:  Weaker than beyond a reasonable doubt, this is the standard for removing a child from a home or terminating parental rights.  The operator is sometimes said to mean substantially more probable than not.
  • Preponderance of evidence:  This is the standard used in civil cases, where a plaintiff sues a defendant.  The operator means more probable than not
  • Probable cause:  This is the standard used by police to arrest or search a person without a warrant.
  • Reasonable suspicion:  A police officer is justified in stopping and briefly detaining a person if there’s a reasonable suspicion of involvement in a crime

Epistemic Operators in Intelligence Assessment
AKA Words of Estimative Probability (Wikipedia)

  • National Intelligence Estimates are summaries of the intelligence assessments of sixteen agencies of the United States Government, including the CIA, the Department of Defense, and the Department of Justice. 
  • Intelligence assessments have the form:
    • We judge (assess) with high (moderate, low) confidence that X
    • We judge (assess) with high (moderate, low) confidence that it is likely (unlikely, very likely, very unlikely, almost certainly, remotely possible, an even chance) that X
  • For example:
    • We assess with low confidence that Iran probably has imported at least some weapons-usable fissile material, but still judge with moderate-to-high confidence it has not obtained enough for a nuclear weapon.
    • We judge with moderate confidence that the earliest possible date Iran would be technically capable of producing enough HEU for a weapon is late 2009, but that this is very unlikely.
  • From a National Intelligence Estimate:

Epistemic Operators in Philosophy
Theory of Knowledge, 3rd Edition, Roderick Chisholm

  • Basic Locution:
    • Doxastic attitude A is at least as justified for S as doxastic attitude B, where doxastic attitudes are:
      • believing P
      • disbelieving P (believing ~P)
      • withholding P (neither believing nor disbelieving P)
  • Doxastic attitude A is more justified for S than doxastic B =df it is false that B is at least as justified for S as is A
  • P is certain for S =df for every Q, believing P is more justified for S than withholding Q, and believing P is at least as justified for S as is believing Q
  • P is obvious for S =df for every Q, S is more justified in believing P than in withholding Q
  • P is evident for S =df for every proposition Q, believing P is at least as justified for S as withholding Q
  • P is beyond reasonable doubt for S =df S is more justified in believing P than in withholding P
  • P is epistemically in the clear for S =df S is not more justified in withholding P than in believing P
  • P is probable for D =df S is more justified in believing P than in believing ~P
  • P is counterbalanced for S =df S is no more justified in believing P than believing ~P and S is no more justified in believing ~P than believing P

Addenda

Knowledge

  • Two millennia ago Plato argued that knowledge is not mere true belief.
    • “Suppose a jury has been justly persuaded of some matter which only an eyewitness could know, and which cannot otherwise be known; suppose they come to their decision based on hearsay, forming a true belief: then they have decided the case without knowledge.”
      • Plato’s Theaetetus
  • Plato thus set forth a counterexample to the idea that knowledge is the same thing as true belief: a logically coherent scenario where people, the jurors, have a true belief that isn’t knowledge.
  • Hence, knowledge is more than mere true belief.  What more?
  • Philosophers long believed a person knows that such-and-such if
    • it’s true that such-and-such, and
    • the person believes that such-and-such, and
    • the person is justified in believing that such-and-such.
      • Or, for the person, it is beyond a reasonable doubt that such-and-such
  • Two problems with the analysis:
    • It’s subject to so-called Gettier counterexamples:
      • Seeing what appears to be your friend Mark jogging, you naturally  believe he’s is jogging. The person you see, though, is Mark’s twin brother.  It happens that Mark is also jogging, though you haven’t seen him. Hence, you don’t know Mark is jogging, yet the three conditions above are met.
    • Justified belief is too weak
      • As I leave work, I am justified in believing my car is where I parked earlier.  Though beyond a reasonable doubt; I don’t know my car is still in its spot. My evidence — that parked cars usually stay put — doesn’t exclude its having been stolen or towed.  Though unlikely, I can be mistaken. If I can be mistaken, I don’t know. Knowledge thus requires certainty. 
  • Thus it seems that a person knows that such-and-such if
    • it’s true that such-and-such, and
    • the person believes that such-and-such, and
    • for the person, it is certain that such-and-such
      • This is different from: the person is certain that such-and-such

Prove, Show, Establish, Disprove, Refute

Principle of Indifference and Indeterminate Probabilities

Easy Problem
  • An urn has 100 balls, 50 red, 50 white. You randomly select a ball.  It’s clear that the probability the ball is red is 0.5, other things being equal.  Or, in other words:
    • The odds are even the ball is red.
    • There’s a 50-50 chance the ball is red.
    • It’s a tossup whether the ball is red.
Hard Problem
  • An urn has 100 balls, each red or white.  But it’s unknown how many are red and how many are white. You randomly select a ball. What’s the probability the ball is red?
  • One Answer
    • The probability the ball is red is 0.5.
      • The argument uses the Principle of Indifference:
        • Competing hypotheses are equally likely if there’s no evidence that one is more likely than another.
      • Since there’s no evidence that makes it more likely that the ball is one color rather than the other, red and white are equally likely.
    • But the argument fails:
      • If it’s reasonable to believe there’s a 50-50 chance the ball is red, it’s reasonable to believe that at least one ball is red.  The latter is false since it’s unknown how many are red and how many are white. It’s therefore not reasonable to believe the probability of the ball’s being red is 1/2.
  • A Better Answer
    • The probability the ball is red is unknown and indeterminate.
      • The probability is unknown since the probability depends on the number of red and white balls in the urn, which is unknown.
      • The probability is indeterminate since there’s no way of determining the probability based on the evidence we have, which does not include evidence regarding the number of red and white balls.

Sentential Operators

A sentential operator is a word or phrase that generates an independent clause when combined other independent clauses.