Probability Arguments

probability argument is an argument from evidence
to a probable hypothesis

Probability Arguments
• probability argument is an argument from evidence to a probable hypothesis.
Example
• You’re not sure whether black swan is a figure of speech or a real bird.  So you look up swan in the Britannica, where you see a photo of a black swan and read that “The Southern Hemisphere has the black swan….”
• You thus infer the existence of black swans.
1. The Britannica says there are black swans.
2. The Britannica is a reliable source of information.
3. So, there are black swans.
Evidence
• Evidence is an established fact supporting or casting doubt on a claim.
• Evidence makes a hypothesis more (or less) likely than it would have been otherwise.
• Federal Rule of Evidence (FRE-401)
• Relevant evidence means evidence having any tendency to make the existence of any fact that is of consequence to the determination of the action more probable or less probable than it would be without the evidence.
Degrees of Probability
• A variety of terms are used to express degrees of probability.
• For example:
• 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.
Kinds of Probability Arguments
Reliable-Process Arguments
• A reliable-process argument is an argument whose conclusion is made probable by a reliable process.
Abductive Arguments
• An abductive argument is an argument whose conclusion is made probable by explaining and/or predicting the evidence.
Analogical Probability Argument
• An analogical probability argument is an argument whose conclusion is made probable by similarities among things.
Reliable-Process Arguments

A reliable-process argument is an argument whose conclusion is made probable by a reliable process, mechanism, procedure, practice, device, or naturally occurring regularity.

Basic Form
1. E, the evidence, is true
2. E-ness is reliable evidence of H-ness
• Because of a reliable process
3. So H is true
Examples
• Testimony
• The Britannica says black swans exist. So black swans exist
1. The Britannica says black swans exist.
2. The Britannica is a reliable source of information.
3. Therefore, black swans exist
• Perception
• The napkin appeared green to Matty. Therefore Matty saw a green napkin.
1. The napkin appeared green to Matty
2. Visual perception is reliable.
3. Therefore Matty saw a green napkin.
• Memory
• Allen talked to Evan at the party because he remembers talking to him.
1. Allen remembers talking to Evan at the party
2. Remembering P is reliable evidence that P is true.
3. So, Allen talked to Evan at the party
• Natural Phenomena
• Luke is exhibiting pain behavior. So he’s in pain
1. Luke is exhibiting pain behavior.
2. Pain behavior is reliable evidence of being in pain.
3. So he’s in pain
• Mechanisms
• There’s somebody at the door — I hear the doorbell is ringing.
1. The doorbell is ringing
2. A doorbell’s ringing is reliable evidence that someone’s at the door.
3. Therefore, someone’s at the door
• Computer Forecasts
• The Congressional Budget Office forecasts that real GDP growth will be under 2% next year. So, real GDP growth will be under 2% next year.
1. The CBO forecasts that real GDP growth will be under 2% next year.
2. The CBO’s forecasts of GDP are reliable.
3. Therefore, real GDP growth will be under 2% next year.
• Forensic Methods
• The crime lab matched the left index fingerprint on the murder weapon to the suspect’s. So the suspect’s left index finger came in contact with the murder weapon.
1. The crime lab matched the left index fingerprint on the murder weapon to the suspect’s.
2. A fingerprint match is reliable evidence that the fingerprints are of the same person.
3. Therefore the suspect’s left index finger came in contact with the murder weapon.
Evaluating Reliable-Process Arguments
• Factors in evaluating a reliable-process argument:
• The process’s track record, buttressed by an explanation why the process is reliable
• The probability or improbability of the conclusion apart from the support by the reliable process.
• Whether additional facts nullify the argument’s reliability
• Track record, buttressed by an explanation why the process is reliable.
• For example:
• DNA and fingerprinting have impressive track records. Other methods have been proven unreliable, e.g. Bitemark Analysis.
• For DNA, scientists understand why and how it’s reliable.
• Probability or improbability of the conclusion apart from the support by the reliable process.
• Suppose the police charge your friend Evan with burglary. But Evan and his wife were at your home playing bridge when the crime occurred. The reliability of the police is irrelevant because you know Evan could not have committed the crime. The probability that Evan is guilty, independent of the police accusation, is zero.
• Bottom line:
• In evaluating a reliable-process argument, the reliability of the process must be weighed against the independent improbability of the conclusion.
• Argument’s reliability nullified.
• Additional facts can nullify the reliability of a reliable-process argument.
• Consider:
• In 1998 CNN reported that the U.S. had used sarin nerve gas in Laos in 1970 as part of Operation Tailwind during the Vietnam War.
• CNN is a reliable source of news.
• Therefore, the US used sarin gas in Laos in 1970.
• The argument’s reliability was later undercut by CNN’s 54-page retraction.
Hume’s Argument on Miracles
• In his essay on miracles, David Hume argues that the improbability of a miracle always outweighs the reliability of a report of the miracle’s occurrence.
Abductive Arguments

An abductive argument is an argument whose conclusion is made probable by explaining and/or predicting the evidence.

• Abduction is also called inference to the best explanation.
• The underlying principle is that a hypothesis that explains and/or predicts the evidence better than competing hypotheses is more likely, other things being equal.
• The mathematical formulation of the principle is Bayes Theorem.
Basic Form
1. E, the evidence, is true
2. Hypothesis H explains and/or predicts E better than competing hypotheses.
3. Therefore H is more likely than the competing hypotheses
Subway Brain Teaser
• Ernie, a Manhattanite, has two girlfriends, one in Brooklyn, the other in the Bronx.  He visits one or the other every Saturday, taking the subway. Liking them equally, he lets chance (or fate) decide whom he visits, by showing up at the subway station at a random time on Saturday. The trains on one side of the platform arrive every ten minutes going to the Bronx; those on the other side arrive every ten minutes going to Brooklyn.  He takes the first train to arrive, no matter its direction. Curiously, Ernie visits his Brooklyn girlfriend nine times out of ten. How can this be?
• View Ernie’s Hypothesis
Harry Markopolos’ Argument to the SEC
1. Bernie Madoff’s fund Fairfield Sentry had only 7 losing months out of 174; its largest monthly loss was only 0.55%; and its longest losing streak was one month every few years.
• Let H be the hypothesis that Bernie Madoff runs a Ponzi scheme.
2. Hypothesis H explains and predicts Fairfield Sentry’s performance better than the hypothesis Madoff is an honest investor.
3. Therefore it’s more likely Madoff runs a Ponzi scheme than he’s an honest investor.
Statistical Estimation
1. In a random poll of 1,000 Americans, 600 said they liked the ACA.
• Let H be the hypothesis that about 60 percent of Americans in general like the ACA.
2. Hypothesis H explains and predicts the 60 percent sample figure better than any competing hypothesis, e.g. that 50 percent of Americans like the ACA.
3. Therefore, H is more likely than competing hypotheses.
Coke Taste Test
• Ava claims she can taste the difference between regular coke (in the red can) and caffeine-free coke (in the gold can).  Skeptical, you arrange a taste test with 12 unmarked glasses of regular and caffeine-free coke. Ava identifies 10 of the 12 cokes.
• The argument that Ava can taste the difference:
1. The result of the test is that Ava identified 10 of the 12 cokes.
2. The hypothesis that Ava can taste the difference explains and predicts the result of the test better than the competing hypothesis that she is randomly guessing.
• If Ava is randomly guessing, the probability of her getting 10, 11, or 12 right is 1/50.
• If Ava can taste the difference, getting 10, 11, or 12 right is expected.
3. It’s therefore more likely that Ava can taste the difference between cokes than she is randomly guessing.
Last Twelve Verses of Mark
• Biblical scholars have put forth the hypothesis that the last twelve verses of the Gospel of Mark were not in the original autograph, but added later. The verses relate post-mortem appearances of Jesus of Nazareth and his ascension into Heaven.  It’s important to recognize that scholars agree that none of the manuscripts of the Bible, Old Testament and New, are autographs, i.e. handwritten by the author him or herself; rather, the manuscripts are copies of copies of copies of copies. So the hypothesis is that the writer of the original manuscript of the Gospel of Mark did not write the last twelve verses included in most versions of the gospel; rather, the verses were added later by copyists.
• Bart Ehrman lays out the evidence:
• “The verses are absent from our two oldest and best manuscripts of Mark’s Gospel [Codex Sinaiticus and Codex Vatanicus], along with other important witnesses; the writing style varies from what we find elsewhere in Mark; the transition between this passage and the one preceding is hard to understand (e.g. Mary Magdalene is introduced in verse 9 as if she hadn’t been mentioned yet, even though she is discussed in the preceding verses; there is another problem with the Greek that makes the transition even more awkward); and there are a large number of words and phases in the passage that are not found elsewhere in Mark.” (Bart D Ehrman, Misquoting Jesus, page 67)
• Ehrman’s argument, reconstructed:
1. The evidence is that:
• The verses do not appear in the Codex Vatinicus and Codex Sinaiticus;
• The writing style of the last twelve verses doesn’t match the style of the rest of Mark;
• There’s an awkward transition from the previous passage.
2. The hypothesis that the verses were added later explains the evidence better than the hypothesis they were in the original autograph.
• That the last 12 verses were not in the original autograph explains why
• The verses do not appear in the Codex Vatinicus and Codex Sinaiticus;
• The writing style of the last twelve verses doesn’t match the style of the rest of Mark;
• There’s an awkward transition from the previous passage.
• By contrast, if the verses were part of the original text, the opposite would be expected:
• The verses would appear in the Codex Vatinicus and Codex Sinaiticus;
• The writing style of the last twelve verses would match the style of the rest of Mark;
• There would be a smooth transition from the previous passage.
3. Therefore it’s more likely that the last twelve verses of Mark were added afterwards than they were part of the original autograph.
Evaluating Abductive Arguments
• Factors in evaluating abductive arguments.
• How the hypotheses compare in explaining and/or predicting evidence E
• How likely the hypothesis are, apart from explaining and/or predicting evidence E
• How the hypotheses compare in explaining and/or predicting evidence E
• A Numerical Example:
• In the Subway Brain Teaser example:
• The evidence E is that Ernie visits his Brooklyn girlfriend nine out of ten times
• Ernie’s hypothesis is that the Bronx train arrives one minute after the Brooklyn train.
• The competing hypothesis is that there’s no correlation between the arrival times of the trains.
• The likelihood of the evidence E given Ernie’s hypothesis = 1.0
• The likelihood of the evidence E given the competing hypothesis = 0.5
• So Ernie’s hypothesis predicts evidence E better than the competing hypothesis.
• A Qualitative Example:
• In the Last Twelve Verses of Mark example, the evidence E was that:
• The verses do not appear in the Codex Vatinicus and Codex Sinaiticus;
• The writing style of the last twelve verses doesn’t match the style of the rest of Mark;
• There’s an awkward transition from the previous passage.
• The hypothesis that the verses were added later explains why each piece of evidence is true.
• The hypothesis that the verses were part of the original text not only fails to account for the evidence E but also predicts the opposite of the evidence:
• that the verses would appear in the Codex Vatinicus and Codex Sinaiticus;
• that the writing style of the last twelve verses would match the style of the rest of Mark;
• that there would be a smooth transition from the previous passage.
• Thus the added-later hypothesis explains and predicts the evidence better than the part-of-autograph hypothesis.
• How likely the hypothesis are, apart from explaining and/or predicting evidence E
• An abductive argument focuses on certain evidence E. But additional evidence may cast doubt on or support the inferred and competing hypotheses.
• Consider the conspiracy theory that the moon-landing was a hoax.
• The evidence consists of news reports, TV transmissions, photos, interviews with astronauts, moon rocks, takeoffs and landings
• The straightforward hypothesis is that the astronauts walked on the Moon
• The hoax hypothesis is that NASA made fake photos, telemetry records, radio and TV transmissions, moon rocks, etc.
• Both hypotheses may explain the evidence equally well. The difference is that, due to its complexity, the hoax hypothesis is inherently far more improbable than the straightforward hypothesis. Per Ockham’s Razor, the simpler of two competing hypotheses is more likely, other things equal.
Single-Hypothesis Abduction
• Single-Hypothesis Abduction
• We’ve considered only comparative abductive arguments, where a hypothesis is compared to competing hypotheses.
• Abductive arguments can also be non-comparative:
• E, the evidence, is true
• Hypothesis H explains and/or predicts E.
• Therefore H is probable.
• The problem is that single-hypothesis arguments provide only one side of the story.
• Consider this explanation for high inflation:
• Biden’s American Rescue Plan poured \$1.9 trillion into the economy. “Flush with government money, millions of Americans decided they could afford to stay on the sidelines and not to return to work — producing a historic labor shortage. Demand for goods and services soared as people emerged from pandemic lockdowns and started spending again, but supply could not keep up — in large part because businesses could not find workers. The result is inflation the likes of which we have not seen since the 1970s.” (Marc A. Thiessen, WaPo)
• Argument Reconstructed:
• Inflation is high
• Let H = the hypothesis that Biden’s American Rescue Plan added \$1.9 trillion into the economy, which resulted in a labor shortage while demand for goods and services soured, which caused prices to increase dramatically.
• H explains why inflation is high.
• Therefore H is likely.
• The obvious question: how does this theory stack up against the competitors.
• However, a version of single-hypothesis abduction can provide strong evidence for a theory.
• The form:
• E, the evidence, is true
• Hypothesis H explains and/or predicts E.
• Except for H, E is completely unexpected.
• Therefore H is probable.
• For example, Einstein’s theory of gravitation, General Relativity, predicted phenomena no one had even conceived: Gravitational Redshift, Gravitational Time Dilation, Gravity Waves.
Analogical Probability Arguments

An analogical probability argument is an argument whose conclusion is probable based on things being alike.

• Form
• A and B are alike in respects X, Y, Z
• A is a Y
• Therefore B is likely a Y
• Example, from Ben Franklin
1. Lightning and sparks are alike in the following respects:
• 1. Giving light, 2. Color of the light, 3. Crooked direction, 4. Swift motion, 5. Being conducted by metals, 6. Crack or noise in exploding, 7. Subsisting in water or ice, 8. Rending bodies it passes through,  9. Destroying animals, 10. Melting metals, 11. Firing inflammable substances, 12. Sulphureous smell
2. A spark is electrical in nature
3. Therefore, it’s likely that lightning is electrical in nature.
Probability Synonyms
• The term probable is synonymous with likely and chances are.
• It is probable that A
• It is likely that A
• Chances are that A
• It’s more probable that A than B
• It’s more likely that A than B
• Chances are greater that A than B
• It is improbable that A
• It’s unlikely that A
• Chances are that A is false
Conditional Probability
• The deductive form of probability arguments can be expressed:
1. E
2. [epistemic operator] C given E, other things being equal.
3. Other things are equal.
4. Therefore, [epistemic operator] C.
• The second premise expresses the idea of conditional probability: P(C|E) = n, that is:
• the probability of C given E, other things being equal, = n.
• An example using a Venn Diagram for Aces and Hearts of a standard deck of cards:
• The diagram establishes the following probabilities for a randomly selected single card:
• P(Ace) = 4/52
• P(Heart) = 13/52
• P(Ace & Heart) = 1/52
• P(Ace & ~Heart) = 3/52
• P(~Ace & Heart) = 12/52
• P(~Ace & ~Heart) = 36/52
• The following are conditional probabilities:
• P(Ace | Heart) = 1/13
• The probability of drawing an ace given that the card is a heart = 1/13
• P(Heart | Ace) = 1/4
• The probability of drawing a heart given that the card is an ace = 1/4

View Aces and Hearts

Ernie’s Hypothesis
• Ernie’s hypothesis is that the Bronx train arrives one minute after the Brooklyn train.
• The competing hypothesis is that there’s no correlation between the arrival times of the trains.
• The observed fact is that Ernie visits his Brooklyn girlfriend nine times out of ten.
• The argument
1. Ernie visits his Brooklyn girlfriend nine times out of ten.
2. Ernie’s hypothesis explains and predicts the observed fact far better than the competing hypothesis.
• Ernie’s hypothesis predicts the observed fact:
• Since Ernie arrives at the station at random times, the Bronx train arrives first 1/10 of the time and the Brooklyn train 9 times of 10.
• The diagram shows the ten-minute period between arrivals of the Brooklyn train.
• The probability Ernie arrives at the station in the gray time zone is 9/10, for which the Brooklyn train is next to arrive.
• The competing hypothesis predicts that Ernie takes the Brooklyn train half the time.
3. Thus Ernie’s hypothesis is far more likely than the competing hypothesis.
Bayesian Calculation of the Subway Brain Teaser
• In the Subway Brain Teaser:
• The likelihood of the evidence E given Ernie’s hypothesis = 1.0
• The likelihood of the evidence E given the competing hypothesis = 0.5
• where E is the fact that Ernie visits his Brooklyn girlfriend nine out of ten times.
• So, Ernie’s hypothesis is more probable, other things being equal.
• To determine the probabilities of the two hypotheses all things considered we have to factor in their probabilities apart from their predictions of E.
• Bayes Theorem can then be used to calculate what we want to know: the probabilities of the two hypotheses based on
• their predictions of evidence E, and
• the probabilities of the hypotheses apart from their prediction of E.
• To do the calculation we need figures for:
• The likelihood of the evidence E given Ernie’s hypothesis
• The likelihood of the evidence E given the competing hypothesis
• The probability of Ernie’s hypothesis apart from predicting evidence E
• The probability of the competing hypothesis apart from predicting evidence E
• Let’s suppose that the hypotheses are equally likely apart from their prediction of E.
• Then:
• The likelihood of the evidence E given Ernie’s hypothesis = 1.0
• The likelihood of the evidence E given the competing hypothesis = 0.5
• The probability of Ernie’s hypothesis apart from predicting evidence E = 0.5
• The probability of the competing hypothesis apart from predicting evidence E = 0.5
• The result of the calculation is:
• The probability of Ernie’s hypothesis all things considered = 2/3.
• The probability of the competing hypothesis all things considered = 1/3.
• View Bayes Theorem Calculator
• View Bayes Theorem
The Term ‘Induction’ is Obsolete
• In philosophy the term “induction” is obsolete.
• britannica.com/topic/induction-reason
• Induction is the method of reasoning from a part to a whole, from particulars to generals, or from the individual to the universal. As it applies to logic in systems of the 20th century, the term is obsolete.
• In the 19th century John Stuart Mill characterized deduction as reasoning from the general to the particular and induction as reasoning from the particular to the general. Today ‘deduction’ means reasoning from premises to a logically entailed conclusion. The word ‘induction’ has been replaced by terms such as: probabilistic reasoning, ampliative inference, and abduction.
Evaluating Abduction Arguments
• Procedure:
1. Formulate the argument so it’s premises, conclusion, and logic are clear
• The basic form of an abductive argument:
1. H explains and/or predicts the evidence better than competing hypotheses.
2. Therefore H is more likely than the competing hypotheses, other things being equal.
• View Argument Reconstruction
2. Determine whether the premises are beyond a reasonable doubt.
• The key premise is that hypothesis H explains and/or predicts the evidence better than the competing hypothesis.
• A Numerical Example:
• In the Subway Brain Teaser example:
• The evidence is that Ernie visits his Brooklyn girlfriend nine out of ten times
• Ernie’s hypothesis is that the Bronx train arrives one minute after the Brooklyn train.
• The competing hypothesis is that there’s no correlation between the arrival times of the trains.
• The likelihood of the evidence given Ernie’s hypothesis = 1.0
• The likelihood of the evidence given the competing hypothesis = 0.5
• So Ernie’s hypothesis predicts the evidence better than the competing hypothesis.
• A Quantitative Example:
• In the Last Twelve Verses of Mark example, the evidence is that:
• The verses do not appear in the Codex Vatinicus and Codex Sinaiticus;
• The writing style of the last twelve verses doesn’t match the style of the rest of Mark;
• There’s an awkward transition from the previous passage.
• The hypothesis that the verses were added later explains the evidence
• The hypothesis that the verses were part of the original text not only fails to account for the evidence but also predicts the opposite of the evidence:
• that the verses would appear in the Codex Vatinicus and Codex Sinaiticus;
• that the writing style of the last twelve verses would match the style of the rest of Mark;
• that there would be a smooth transition from the previous passage.
• Thus the added-later hypothesis explains and predicts the evidence better than the part-of-autograph hypothesis.
3. Determine whether the premises, if true, adequately support the conclusion
• As with reliable-process arguments, additional facts can affect abductive support (AFAS).
• Suppose a doctor is trying to diagnose the cause of a strange set of symptoms. Disease D explains the symptoms better than disease E. So, D is likelier than E, other things being equal. However, D is so rare that only five cases are diagnosed annually, whereas E is common. Its rarity thus offsets D’s explanatory success.
• To determine whether the premises of an abductive argument, if true, support its conclusion, we need to find out whether additional facts affect the abductive support.
• A Numerical Example:
• In the Subway Brain Teaser case we determined that
• The likelihood of the evidence E given Ernie’s hypothesis = 1.0
• The likelihood of the evidence E given the competing hypothesis = 0.5
• where E is the fact that Ernie visits his Brooklyn girlfriend nine out of ten times.
• So, Ernie’s hypothesis is more probable, other things being equal.
• The question is: are there other facts that affect the probabilities of the two hypotheses. Let’s assume not. Bayes Theorem can then be used to calculate the probabilities of Ernie’s hypothesis and the competing hypothesis given these facts:
• The likelihood of the evidence E given Ernie’s hypothesis = 1.0
• The likelihood of the evidence E given the competing hypothesis = 0.5
• The probability of Ernie’s hypothesis apart from evidence E = 0.5
• The probability of the competing hypothesis apart from evidence E = 0.5
• The result:
• The probability of Ernie’s hypothesis = 2/3.
• The probability of the competing hypothesis = 1/3.
• If the probabilities apart from evidence E had been different, for example:
• The probability of Ernie’s hypothesis apart from evidence E = 0.1
• The probability of the competing hypothesis apart from evidence E = 0.9
• The result would have been:
• The probability of Ernie’s hypothesis = 1/5
• The probability of the competing hypothesis = 4/5
• View Bayes Theorem Calculator
• View Bayes Theorem
• A Quantitative Example
• In the Last Twelve Verses of Mark example, the evidence consisted of three facts:
• The verses do not appear in the Codex Vatinicus and Codex Sinaiticus;
• The writing style of the last twelve verses doesn’t match the style of the rest of Mark;
• There’s an awkward transition from the previous passage.
• We concluded that the hypothesis that the verses were added later explains and predicts the evidence better than the hypothesis that the verses were part of the original text. Thus, the added-later hypothesis is more probable than the part-of-autograph hypothesis, other things being equal.
• But are other things equal? Are there additional facts affecting the probabilities of the two hypotheses? The burden of proof seems to be on anyone who thinks the argument wrong.

View Bayes Theorem