Fallacies

A fallacy is an error in reasoning having an air of plausibility

Table of Contents

  1. Fallacies
    1. Affirming the Consequent
    2. Appeal to Consequences
    3. Argument from Ignorance
    4. Begging the Question and Circular Reasoning
    5. Biased Sample / Selection Bias
    6. Base Rate Fallacy
    7. Composition
    8. Cum Hoc Ergo Propter Hoc
    9. Denying the Antecedent
    10. Division
    11. Ecological Fallacy
    12. Equivocation
    13. False Dilemma
    14. Gambler’s Fallacy
    15. Hasty Generalization
    16. Intensional Fallacy
    17. Post Hoc Ergo Propter Hoc
    18. Prosecutor’s Fallacy
    19. Simpson’s Paradox
  2. Fallacies Classified as Deductive and Evidential
    1. Deductive Fallacies
    2. Evidential Fallacies
  3. Affirming the Consequent, Denying the Antecedent, Modus Ponens, Modus Tollens
  4. Resources

Fallacies

Affirming the Consequent
  • Affirming the Consequent is an inference of the form:
    • If A, then C
    • C
    • Therefore A
  • The conditional if A then C consists of the antecedent A and the consequent C. The second premise of Affirming the Consequent affirms the consequent C.
  • The argument-form is invalid, per logical analogy:
    • If Matt Damon is over seven feet, he’s over five feet.
    • Damon is over five feet.
    • Therefore, he’s over seven feet.
  • See Affirming the Consequent, Denying the Antecedent, Modus Ponens, Modus Tollens
Appeal to Consequences
  • Appeal to Consequences is arguing that a claim should be accepted or rejected based on the desirable or undesirable consequences of believing it.
  • Example
    • If you don’t accept Jesus Christ as your lord and savior, you’ll be damned for all eternity.
Argument from Ignorance
  • Argument from Ignorance is arguing that something is true because it’s not disproven or that something is false because it’s not proven.
  • Sir Martin Rees: “Absence of evidence is not evidence of absence.”
  • Examples
    • There must be some truth to astrology since for thousands of years no one has succeeded in disproving it.
    • OJ Simpson did not kill Nicole Brown Simpson and Ron Goldman since he was not proven guilty in a court of law.
Begging the Question and Circular Reasoning
  • Begging the Question is assuming that which is to be proved
  • Circular Reasoning is arguing that a statement is true because a second statement is true; but also that the second statement is true because the first is true.
  • Example
    • Believer: I know Jesus is the Son of God because he returned from the dead.
    • Skeptic: Human beings don’t come back to life. 
    • Believer: But Jesus was not an ordinary human being – he was the Son of God.
Biased Sample / Selection Bias
  • Selection bias is procedure for selecting data for calculating a statistic that causes systematic and one-directional errors in the data, resulting in the calculated statistic being wrong.
  • A biased sample is the improperly selected data.
  • The fallacy of biased sample is generalizing from the sample.
  • The classic example:
    • The Literary Digest magazine conducted a poll to determine the likely winner of the 1936 presidential election between Franklin Delano Roosevelt and Alf Landon, Governor of Kansas.  Questionnaires were mailed to 10 million people, the names obtained from telephone books, automobile registrations, and the magazine’s readership.  2.4 million people responded. (The number of respondents in a typical national election poll is in the thousands.)   43% said they planned to vote for Roosevelt.  The magazine predicted a landslide for Landon.
    • Roosevelt won all but two states.
    • The Literary Digest’s poll was biased against those without phones and automobiles, mostly Democrats. 
    • It is also thought that anti-Roosevelt sentiment was strong among Landon supporters, making it more likely they would take the time to mail back a response.
Base Rate Fallacy
  • The Base Rate Fallacy is an inference from a rate or proportion that ignores a background rate or proportion, called the base rate.
  • Example 1
    • Fallacious Argument
      • Joe is given a random drug test that’s 95 percent reliable, meaning 95 percent of drug users test positive and 95 percent of non-drug users test negative. Five percent of the population takes drugs. 
      • He tests positive.
      • Therefore, the probability Joe is a drug user is 95 percent.
    • The Problem
      • The probability is 50 percent, not 95 percent.  This is because of the base rate of 5 percent of the population who takes drugs.  See Random Drug Test for details.
  • Example 2
    • Fallacious Argument
      • After taxes were cut at the beginning of 2017, tax revenues increased from 3.2 trillion in 2017 to 3.3 trillion in 2018
      • By bringing in more tax revenue than the preceding year, therefore, the tax cuts paid for themselves in 2018.
    • The Problem
      • But tax revenues were on track to be higher in 2018 than in 2017 without the tax cuts. Indeed, the CBO had predicted 3.5 trillion for 2018. So the 2018 tax revenues were less than they would have been without the tax cuts, by $200 billion.  The 2017 tax cuts thus did not pay for themselves in 2018.  The argument ignores the base rate of 10% that tax revenues were increasing before the tax cuts.
    • See Right and Wrong Baselines for details.
Composition
  • The first form of Composition is inferring that what’s true individually of a group’s members is thereby true of the group collectively.
    • Example
      • A car emits less toxic waste than a bus.  
      • Therefore cars emit less toxic waste than buses.
  • The second form of Composition is inferring that what’s true of the entities making up a thing is thereby true of the thing itself
    • Example
      • What appears to be a “red” wheelbarrow really has no color because it’s made up of colorless atoms.
Cum Hoc Ergo Propter Hoc
  • Cum Hoc Ergo Propter Hoc (“with this therefore because of this”) is inferring that a causal connection exists between two phenomena merely because of a statistical correlation between them.
  • Example
    • The Pearson correlation coefficient is a number between -1 and +1 calculated from two sets of paired numbers. For example, the correlation coefficient between the series {1, 2, 3, 4, 3, 2, 1} and {2, 5, 7, 9, 8, 6, 4} is 0.94992, a high correlation.
    • Suppose the first series, {1, 2, 3, 4, 3, 2, 1}, is the number of times Joe in New York sneezed each day Monday through Sunday and the second series, {2, 5, 7, 9, 8, 6, 4}, is the number of times Bill in Texas cursed each day of the same week.  The high correlation is no evidence of a causal connection.
    • Inferring a causal connection requires more than mere statistical correlation.
  • The evidence may establish a causal connection without establishing what caused what.
    • Examples:
      • The speed at which a wind turbine rotates is correlated with the speed of the wind: the greater the one, the greater the other.
        • Explanation 1: the wind makes the propeller rotate
        • Explanation 2: the rotating propeller makes the wind blow.
      • If you have the flu you’re more likely to die if you go to the Emergency Room.
        • Explanation 1: emergency rooms, awash in germs, make people sicker.
        • Explanation 2: people with the flu who go to the Emergency Room are sicker than those who don’t. 
Denying the Antecedent
  • Denying the Antecedent is an argument of the form:
    • If A, then C
    • It’s false that A
    • Therefore it’s false that C.
  • The conditional if A then C consists of the antecedent A and the consequent C. The second premise of Denying the Antecedent denies the antecedent A.
  • The argument form is invalid per logical analogy:
    • If Matt Damon is over seven feet, he’s over five feet.
    • Damon is not over seven feet.
    • Therefore, he’s not over five feet.
  • Example
    • Had Saddam Hussein been responsible for the 9/11 attacks, the U.S. invasion of Iraq would have been morally justified.  But Hussein was not responsible for the 9/11 attacks.  Therefore, the U.S. invasion of Iraq was not justified.
  • See Affirming the Consequent, Denying the Antecedent, Modus Ponens, Modus Tollens
Division
  • The first form of Division is inferring that what’s collectively true of a group is thereby true of its members individually.
    • Example
      • People are honest on the whole.  So, Matt is honest on the whole
  • The second form is inferring that what’s true of something is thereby true of the things that make it up.
    • Example
      • Since the brain is conscious so are the neurons it comprises.
  • The third form of Division is inferring that what’s collectively true of a group is collectively true of a subgroup.
    • Example
      • People are honest for the most part.
      • Therefore, criminals are honest for the most part.
Ecological Fallacy
  • The ecological fallacy is drawing a conclusion about individuals from statistics about groups (percentages, proportions, averages, rates).
  • Percentages of the vaccinated and those with Covid for a hypothetical country’s regions:
  • The percentages are correlated: the larger the one, the larger the other. Indeed, the correlation coefficient is +1, a “perfect correlation.”
    • Correlation[{60, 55, 50, 45, 40}, {40, 30, 20, 10, 0}] = 1
  • It’s tempting to infer that getting vaccinated increases your chances of getting Covid.
  • But the inference is logically invalid: the percentages are consistent with vaccinations being 100% effective:
  • The percentages of the vaccinated and those with Covid are the same as above. But no one vaccinated gets Covid.
  • Ecological correlations may suggest connections among individuals, but the evidence is weaker than direct correlations.
  • Links
    • wikipedia.org/wiki/Ecological_fallacy
      • An ecological fallacy is a formal fallacy in the interpretation of statistical data that occurs when inferences about the nature of individuals are deduced from inferences about the group to which those individuals belong.
    • britannica.com/science/ecological-fallacy
      • Ecological fallacy is a failure in reasoning that arises when an inference is made about an individual based on aggregate data for a group.
    • Ecological Inference and the Ecological Fallacy, by David Freedman
      • The ecological fallacy consists in thinking that relationships observed for groups necessarily hold for individuals: if countries with more Protestants tend to have higher suicide rates, then Protestants must be more likely to commit suicide; if countries with more fat in the diet have higher rates of breast cancer, then women who eat fatty foods must be more likely to get breast cancer. These inferences may be correct, but are only weakly supported by the aggregate data.
      • Aggregate data are often easier to obtain than data on individuals, and may offer valuable clues about individual behavior. Ecological inferences will therefore continue to be made. The problems of confounding and aggregation bias, however, are unlikely to be resolved in the proximate future.
Equivocation
  • Equivocation is an inference invalidated by a word or phrase meaning one thing in one part of the argument but something else in another.
  • Example
    • It’s a law of nature that nothing can go faster than light.  
    • Laws can be broken. 
    • A spaceship that goes faster than light is thus possible.
  • The argument equivocates on the word law, which means law of nature in the first premise but statutory law in the second.
False Dilemma
  • False Dilemma is asserting falsely that only two alternatives exist and then arguing (validly) that one is true because the other’s false.
  • Example
    • John Edward is either a fraud or really communicates with the dead.  
    • But he’s not a fraud, as anyone who knows him will tell you.  
    • Therefore, he really communicates with the dead
Gambler’s Fallacy
  • The Gambler’s Fallacy is inferring that:
    • The probability beforehand of a sequence of events H = n.
    • Therefore, all the H events having happened except the last, the probability of the last H = n.
  • In tossing a coin, the probability of heads is ½.  In tossing two coins, the probability of two heads in a row is ½ x ½ = ¼. 
  • The probability of ten heads in a row is 1/1024.  You just flipped nine heads. It makes sense that the probability of heads on toss 10 is 1/1024, since that would make 10 successive heads.
  • But coins have no memory.  The probability of heads on the tenth toss is ½.
  • The ½ probability is proved using probability theory.  Simplifying by tossing the coin only four times:
    • For any toss, P(H) = ½ 
    • The outcome of one coin toss doesn’t affect the probability of the next.  Therefore
      • P(H1&H2&H3&H4) = P(H1) x P(H2) x P(H3) x P(H4) = ½ x ½ x  ½ x ½ = 1/16
      • P(H1&H2&H3) = P(H1) x P(H2) x P(H3) = ½ x ½ x  ½ = 1/8
    • The conditional probability P(H4|(H1&H2&H3)) = P(H1&H2&H3&H4) / P(H1&H2&H3) = (1/16)/(1/8) = 1/2
Hasty Generalization
  • Generalization is inferring that what is true of a sample is likely true of the population at large. 
  • Hasty Generalization is generalizing from a sample that’s too small.
Intensional Fallacy
  • Compare:
    • Anna Argument
      • Anna selected the ace of hearts.
      • The ace of hearts is an ace.
      • Therefore Anna selected an ace
    • Probability Argument
      • The probability of selecting the ace of hearts is 1/52
      • The ace of hearts is an ace
      • Therefore, the probability of selecting an ace is 1/52.
  • The first is valid.  The second is invalid, the conclusion being false.
  • The difference is that the phrase selected the ace of spades in the Probability Argument occurs in what’s called an intensional context, within the scope of the word probability. Intensional contexts don’t permit deductive inference. 
  • The Intensional Fallacy is making a deductive inference from within an intensional context.
  • Intensional contexts include believing that, thinking that, asserting that, certain that, looking for, and quotation marks.  For example, suppose Matt, happily married, is looking for Sandra, a colleague at work married to someone else. Then:
    • Matt is looking for Sandra.
    • Sandra is a wife.
    • Therefore Matt is looking for a wife.
Post Hoc Ergo Propter Hoc
  • Post Hoc Ergo Propter Hoc (“after this therefore because of this”) is inferring a causal connection merely because one thing happened after another.
  • Examples
    • I’ve worn an anti-elephant bracelet my entire life and never been attacked.
    • In sight of tourists at Yellowstone college students turned a fake valve wheel just before an eruption of Old Faithful.
    • Hours after getting a flu shot a person has a stroke.
  • Establishing a causal connection requires more than temporal succession.
Prosecutor’s Fallacy

The Fallacy

  • An example of the Prosecutor’s Fallacy:
    • The suspect’s DNA profile matches that of the blood on the murder weapon.
    • The probability of a match, given that it’s not suspect’s blood on the murder weapon, is 1 in a quadrillion.
    • Therefore the probability it’s the suspect’s blood on the murder weapon is 0.999999999999999, a virtual certainty.
  • The Prosecutor’s Fallacy is an argument of the form:
    • M is the case.
    • The probability of M’s being the case given that G is false is a tiny fraction n.
    • Therefore the probability that G is true is a large fraction 1–n.
  • In the notation of probability theory:
  • An alternative form of the fallacy:
    • The probability of M’s being the case given that G is false is a tiny fraction n.
    • Therefore the probability of G’s being true given that M is the case is a a large fraction 1–n.
  • In symbols:
    • P(M | ~G) =  n
    • So, P(G | M) = 1 – n
  • An analogous piece of reasoning shows there’s something wrong with the Prosecutor’s argument:
    • Anna won the Lottery.
    • The probability of Anna’s winning the Lottery given that she didn’t cheat is 1 in 300 million.
    • Therefore the probability Anna cheated is 299, 999,999 / 300,000,000, a virtual certainty
  • The problem is that the argument is one-sided, a form of cherry picking.  The tiny probability of winning the lottery without cheating must be weighed against the tiny probability of a person who plays the lottery cheating. Bayes Theorem (below) incorporates both probabilities.
  • The Prosecutor’s Fallacy was described and named by William Thompson and Edward Schumann in their 1987 paper Interpretation of Statistical Evidence in Criminal Trials)

The Prosecutor’s Fallacy is half a Bayesian Inference

  • The argument that the suspect’s DNA is on the murder weapon can be formulated using conditional probabilities:
    • P(M | ~B) = n,
    • Therefore, P(B | M) = 1 – n
    • where
      • M = The suspects’s DNA profile matches that of the blood on the murder weapon.
      • B = The suspect’s blood is on the murder weapon
      • n = 1 / 1,000,000,000,000,000
  • The reformulated argument is deductively invalid.
    • Let
      • P(B&M) = 0.1
      • P(B&~M) = 0.1
      • P(M&~B) = 0.2
      • P(~M&~B) = 0.6
    • Then
      • P(M | ~B) = 0.2 / (0.2 + 0.6) = ¼
      • P(B | M) = 0.1 / (0.1 + 0.2) = ⅓
  • By adding two premises, however, the reformulated argument can be made into a valid Bayesian inference.
    • P(M | B) = 1
      • The probability that there’s a match given the suspect’s blood on the murder weapon is 1.0
    • P(B) = about 1/2, specifically (n – 1) / (n – 2)
      • The probability that it’s the suspects’s blood on the murder weapon, apart from the matching profiles, is about 1/2.
  • The resulting Bayesian inference is valid
    • P(B) = (n –1) / (n – 2)
    • P(M | B) = 1
    • P(M | ~B) = n
    • Therefore, P(B | M) =  1 – n
  • The conclusion follows by Bayes Theorem.
    • P(B | M) = P(B) P(M|B)  /  ( P(B) P(M|B) + P(~B) P(M|~B) )
    • Suppose n = 1/100, for example;
    • Then
      • P(B | M) = P(B) P(M|B)  /  ( P(B) P(M|B)  +  P(~B) P(M|~B) )
      • P(B | M) = (99/199 * 1) / (  (99/199 * 1) + (1 – 99/199) * 1/100  )
      • P(B | M) = 99/100
      • P(B | M) = 1 – (1/100)
      • P(B | M) = 1 – n
  • Thus the original argument that the suspect’s blood is on the murder weapon, given a match, presupposes that the probability that the suspect’s blood is on the murder weapon, apart from the DNA match, is approximately ½.  But the assumption is made without evidence. If the suspect is in jail at the time, P(B) = 0.
Simpson’s Paradox

View Simpson’s Paradox

Fallacies Classified as Deductive and Evidential

Deductive Fallacies
  • A deductive fallacy is an invalid piece of reasoning that seems deductively valid.
Evidential Fallacies
  • An evidential fallacy is an inference based on insufficient or defective evidence.

Affirming the Consequent, Denying the Antecedent,
Modus Ponens, Modus Tollens

  • Denying the Antecedent and Affirming the Consequent are invalid forms of argument:
    • Affirming the Consequent
      • If A, then C
      • C
      • Therefore A
    • Denying the Antecedent
      • If A, then C
      • It’s false that A
      • Therefore it’s false that C.
  • In contrast, Modus Ponens and Modus Tollens are valid forms of argument:
    • Modus Ponens
      • If A, then C
      • A
      • Therefore C
    • Modus Tollens
      • If A then C
      • It’s false that C
      • Therefore it’s false that A.

Resources