Irrationally Yours—An Introduction to Logical Fallacies

By Carlos Fuentes

“Contrariwise, continued Tweedledee, if it were so, it might be, and if it were so, it would be; but as it isn’t, it ain’t. That’s logic!”

—Lewis Carroll, Through the Looking Glass

I believe logical flaws and cognitive biases plague our thinking in part because we are genetically programmed to justify beliefs with emotions. Emotions are evolutionary adaptive^[1] tools like fear, happiness, hunger, sweating, and walking on two legs. Emotions are the catalysts of actions that allowed our species to survive when acting on impulse was the difference between life and death. No doubt in the early history of our species^[2] there were countless cases where impulse led to erroneous actions, but false negatives were a fair price to pay for staying alive.

Technological revolution and social evolution in the last few thousand years have created environments with little resemblance to the conditions in which our ancestors lived during pre-historic times. Human traits that served the fundamental purpose of survival are still part of our genetic makeup, but some of them have become social burdens.

Neurological research confirms the evolutionary perspective. It shows the correspondence between sections of the brain and functions. In particular, the most primitive section, the brain stem,^[3] controls many of our emotions while the most evolved section, the neocortex,^[4] is responsible for higher functions like sensory perception, spatial reasoning, conscious thought, and language.

The fact that we are hardwired to respond with emotions explains our tendency to fallacious thinking. It also explains our innate predisposition to believe advertising,^[5] propaganda,^[6] and conspiracy theories.^[7] Neurologists and psychologists have explained how the human brain functions^[8] and why social behavior that appears baffling, including blind belief,^[9] is consistent with our genetic makeup.

With these remarks we embark in a brief discussion of logical, probabilistic, and statistical fallacies.

Logical Fallacies

“If you cannot prove a man wrong, don’t panic.
You can always call him names”

—Oscar Wilde

Before discussing logical fallacies, it is useful to remind the readers about the fundamental elements of logic:

1. Logic is the study of rules of inference and demonstration. For instance, it can be proved from first principles that the sum of the internal angles of any triangle is 180°.
2. A proposition is a statement that can be true or false such as “there are no prime numbers greater than 125,358,425,102.”
3. An argument is a collection of propositions that support a conclusion. For examples, see the birthday problem below.
4. Premises are statements assumed to be true on which an argument is based or from which a conclusion is drawn. One premise in the 180°-triangle proof is that the triangle is in a flat (Euclidian) surface
5. A warrant is the logical connection between the evidence or reason and the claim. For instance:
6. Claim: Elizabeth has accomplished much this year
7. Evidence: Elizabeth graduated with a 4.0 GPA
8. Warrant: Graduating with a 4.0 GPA is an important achievement
9. A tautology is an assertion that is always true such as “x = y or x ≠ y.”

Logical fallacies are, at best, flawed logic, but more commonly are illegitimate arguments intended to mislead. Here is a description of the most common types:

1. Rationalizing is the process of starting with the conclusion and then finding arguments to defend it. For instance: “I did not get the job, but I didn’t want it in the first place.” Rationalization is common in politics, public policy, and economics.
2. Ad Hominem (Latin for “against the man”) involves focusing on the failings of the adversary rather than on the merits of the case. For instance: “scientific consensus should be rejected because scientists are arrogant people who believe they are better than the common person.”The ubiquitous presence of ad hominem arguments attests their effectiveness in certain areas such as politics and economics.^[10]
3. Poisoning the Well is a type of ad hominem argument under which one of the parties attempts to tie the opponent’s argument to something that is unpopular. Here is an example: when Medicare was introduced in 1965, the American Medical Association (AMA) warned that Medicare would lead to absolutism and the destruction of American democracy.^[11]
4. Tu Quoque (Latin for “you too”) consists of attacking someone for doing what he is arguing for or against. For instance: “how can you tell me not to experiment with drugs when you did it as a teenager?” or “the Ancient Greeks were outstanding thinkers. They had slaves, so we should have slaves too.”
5. Ipse Dixit (Latin for “he said it”), or Argumentum Ad Verecundiam (Latin for “argument to modesty”), or more commonly known as the Appeal to False or Irrelevant Authority seeks to persuade by appealing to some authority, real or assumed. For example, “COVID-19 can be cured with hydroxychloroquine. This is true because MDs have told us so.”
6. In the No-True-Scotsman Argument one attempts to protect a generalization from counter-examples by changing the definition, as necessary. For instance: “All actuaries are good in probability. Alfred is an FSA but he cannot solve the probability. This means that Alfred is not a real actuary.”
7. Under Argumentum Ad Ignorantiam (Latin for “argument from ignorance”) a particular belief is assumed to be true when its falsehood has not been determined. For example, “Peter had more votes only because the election was rigged. Can you prove otherwise?” This fallacy is commonly used by believers in supernatural phenomena. They argue along the following lines: “if you cannot identify a light in the sky, the light must have been created by an unidentified flying object. Therefore, it must be an alien spacecraft.”
8. Conspiracy Theories typically rely on the argument from ignorance. Their “logic” focuses on certain aspects of a situation that appear anomalous, or on explanations that although reasonable are not complete—a process called anomaly hunting. For instance, if you live in Rhode Island and by chance you see your neighbor in California when she had informed you that she would be vacationing in Florida, you may conclude that she is living the double life of a secret agent.
9. The Argumentum Ad Populum, or Wagon Fallacy, or Appeal to Common Belief or Appeal to the Masses relies on the assumption that the opinion of the majority is correct. For example: “books in the best seller list are well-written. If they were not, people would not buy them”
10. The Red Herring Fallacy is a common technique among politicians to derail the discussion from the relevant issue to a different one. For instance: “While you may be concerned about my votes on the economic proposal, I can assure you that I am open-minded. We should really discuss my record on votes that safeguard freedom.”
11. In a Straw Man Fallacy, an opponent’s argument is overstated or misrepresented. The technique often takes quotes out of context or incorrectly paraphrases an opponent’s position. For example: “The theory of evolution holds that humans evolved from chimpanzees. If that is true, why are chimpanzees still around? Shouldn’t they have evolved into humans by now? This means that the theory of evolution is incorrect.”
12. The Fallacy of Relevance consists of supporting or discrediting an argument based on information irrelevant to the argument. The fallacy can take different forms such as ad hominem attacks, appeals to authority, and appeals to ignorance, but also poor logic. For example:

• Statement: “Historians cast doubt about the popular belief that Nero set Rome on fire.”
• Refutation: “Nero did not care about loss of lives. Therefore, he started the fire.”

13. Moving the Goalposts consists of changing the criteria for acceptance or rejection of a claim. For instance, if a fortune teller agrees to submit his predictive abilities to a scientific test and he fails it, he could argue that the test was unfair because by not believing in his abilities people negatively influence his psychic prowess.

14. The Unexplained = The Unexplainable Fallacy. The argument is that if something has not been explained completely, it cannot be explained. The theory of evolution is frequently “refuted” using this logic: “According to the theory of evolution, changes in organisms are minute and occur over exceedingly long periods of time. But given the fact that the fossil record does not always show the gradual changes in morphology and function, the theory of evolution cannot be correct.”

15. False Dichotomy or False Choice consists of incorrectly reducing the number of possible outcomes to two and then choosing between them, usually as follows:

• Either X is true, or Y is true;
• X is not true;
• Therefore, Y must be true.

   Here is an example: “Either you are for no taxes, or you are a tyrannical communist. You support taxes. Therefore, you oppose freedom.”

16. The False Continuum fallacy is committed when a claim is rejected because a concept within the claim is, to some degree, vague, that is, not quantified, such as belief/disbelief, justice/injustice. Here is an example: “The fortune teller stated that Sherry’s lack of effort made her an average student. Sherry informed the fortune teller that she had the second highest GPA in her grade. The fortune teller responded that not having the highest GPA proved his point.”

17. Under the Slippery Slope Fallacy, if a position is accepted then the most extreme version of that position must also be accepted, leading to an unintended consequence. Politicians and interest groups frequently make use of this fallacy. Ronald Reagan famously stated that “all of us can see what happens once you establish the precedent that the government can determine a man’s working place and his working methods, determine his employment. From here it is a short step to all the rest of socialism, to determining his pay…. He will wait for the government to tell him where he will go to work and what he will do.”

18. Inconsistent Criteria or Kettle Logic is a series of arguments that are valid on their own but conflict with each other. Sigmund Freud coined the term in a story about a man accused of damaging a kettle. The man gave three defenses:

• He returned the kettle undamaged;
• The kettle was already damaged when he borrowed it;
• He did not borrow the kettle.

   Each statement is a potentially valid defense. However, taken together, they contradict each other and invalidate the defense.

   Kettle logic may be difficult to confront when people do not understand the problem and the inconsistent arguments seem to supplement each other.

19. In a False Tautology the argument is the conclusion although the premise and the argument may be stated in different terms. For instance, “Either the ancient Egyptians moved huge stones to build the pyramids or they did not. They could not because the required technology did not exist then. Therefore, aliens built the pyramids.”

20. Petitio Principii (Latin for “an assumption at the outset,” commonly translated as “begging the question”) is a logical fallacy in which a premise is assumed to be true without a warrant, or in which what is to be proved is taken for granted. For example: “BMWs are the most reliable cars because nobody builds more reliable cars than BMW.”

21. Circulus in Probando (Latin for “circle in proving”—circular reasoning), a particular case of Petitio Principii, is often described as a convoluted argument: What is to be demonstrated has already been assumed to be truth. Excel circular references come to mind. The structure is as follows:

• p is true and implies q
• q is true and implies r
• r is true and implies p

For example:

• “I need a car to transport myself to the shop where I work;”
• “I need a part-time job to pay for the car;”
• “I need to pay for the car, because I must own a car.”

22. Hasty or Faulty Generalization refers to conclusions reached based on a few examples that support a statement. Here is an argument used extensively in the U.S.: “The Canadian health care system is bad because I know seven people who didn’t receive proper treatment.”^[12]

23. In the Fallacy Fallacy, the assumption is that if an argument is incorrect the conclusion must also be incorrect. For example: “Matt asserts that the rule to divide two numbers with common digits is to remove the common digits. For example, 31/23 should equal 1/2 (the common digit 3 is removed). This example shows that the rule is false. Therefore 64/16 cannot equal 4/1.”

24. The Causal Fallacy occurs when a causal relationship between two variables is incorrectly assumed. There are several variations:

• Post Hoc (Latin for “after the fact”) Fallacy: inferring that X causes Y because X is followed by Y. For instance: “The day after John started practicing yoga, he was diagnosed with cancer. This shows that yoga caused his cancer.”
• Mistaking Correlation with Causation. This is one of the favorite actuarial fallacies.^[13]
• Reversing Causal Direction: Assuming that X causes Y without considering the possibility that Y is the cause of X. For example: “if you own a Rolex you are financially successful.”
• The Genetic^[14] Fallacy or Fallacy of Origins is an illogical argument for or against an idea based on its origin. It usually takes one of two forms:
  • Person A claims X. Person A is an unreliable source or has a bad reputation. Therefore, X is false
  • Person A claims X. Person A is a reliable source or has a good reputation. Therefore, X is true

For example: “Nike shoes are of high quality because ‘Magic’ Johnson recommends them on TV.”

• The Fallacy of the Single Cause wrongly presupposes that an event has a single cause when many are involved. For instance: “More women have joined the workforce because the average family size has decreased.”

25. Reductio Ad Absurdum (Latin for “reduction to absurdity”) is a sound logical method used to demonstrate that a hypothesis is not true because if it were, its logical consequences would be absurd. Reductio Ad Absurdum is utilized in mathematical proofs such as to show that is an irrational number.^[15] The proof is as follows:

Assume that √2 is rational. Then, there should exist positive integers m and n such that √­­2 = ^m⁄n. Furthermore, m and n can be chosen so that they do not have common factors. This means that at least one of m and n must be odd (if both were even then they would have 2 as a common factor). By squaring both sides and solving for m it follows that m² = 2n². This means that m is even^[16] and can be represented as m = 2k, so that √2 = ^2k⁄n. Solving for n it follows that n² = 2k². Thus, n must be even. So, m and n are even. This fact contradicts the assumption that at least one of m and n is odd. Consequently, the original hypothesis (√2 is rational) cannot be true, and 2 must be an irrational number.Unfortunately, reductio ad absurdum logic can be misapplied by forcing a conclusion that does not follow from the premises. For instance: “You state that you do not believe in aliens because you have never seen one. But since you have never been to Paris you have not seen the Eiffel Tower. This must mean that you believe the Eiffel Tower does not exist.”

An example will illustrate the point: There is a barber in town that shaves men who do not shave themselves, and only those men. Then:

• If the barber shaves himself, then he shaves one of those men that do not shave themselves. This is a contradiction.
• If the barber does not shave himself, then the barber must shave himself, because the barber shaves men who do not shave themselves. This is a contradiction.

Therefore, a barber that shaves men who do not shave themselves and only those men cannot exist.

The following two videos vividly illustrate the use of logical fallacies. The first also shows how what appears to be an educated audience reacts to emotional appeals:

• A Firing Line Debate: Resolved: That It’s Time to Abolish the Welfare State (https://www.youtube.com/watch?v=RhbDbAr7JsQ)–d
• Fallacious Debate (https://www.youtube.com/watch?v=F87T-1Po3D8)

Concluding Remarks

“Circular logic is self-validating.
Therefore, it is correct.”

— Anonymous

The wide variety of logical fallacies described in this article, whether the result of ignorance or malice, would amuse the trained professional were it not for the fact that they are ubiquitous and influential. Is economic policy based on careful thinking or on political and personal calculation? Are the arguments for and against a single health care payer the result of serious analysis or misunderstandings of economic concepts, slogans, and psychological games? Do arguments to expand or reduce Social Security rely on a careful analysis or on propaganda?

More generally: to what extent actions are led by the invisible hand of irrationality, emotions, and personal interests, and to what cost to society? What can be done to improve our collective thinking abilities? The answer to the first question is that when it matters, emotions and manipulation generally trump rational thinking. The answer to the second question is that we can become better thinkers by pondering each situation carefully, but it must be acknowledged that we tend to gravitate toward the irrational. Unfortunately, these assertions are supported by experience, human history, our evolutionary past, and neurological research.  

Carlos Fuentes, MAAA, MBA, MS, FSA, FCA, is president of Axiom Actuarial Consulting. He can be reached at carlos-fuentes@axiom-actuarial.com.

The opinions expressed in this article are the sole responsibility of the author. They do not express the official views of the American Academy of Actuaries, nor do they necessarily reflect the opinions of the Academy’s officers, members, or staff.

References

[1]   Evolutionary adaptation refers to the alteration or adjustment in structure or habits by which a species or individual improves its ability to survive and pass on its genes.

[2]  See “Timeline: Human Evolution,” New Scientists, September 4, 2006.

[3]  The brain stem is an area at the base of the brain that lies between the deep structures of the cerebral hemispheres and the cervical spinal cord. The brain stem acts as an automatic control center for important involuntary actions of the body, including heartbeat, breathing, blood pressure, and swallowing.

[4]   The neocortex is a set of layers of the mammalian cerebral cortex responsible for higher-order brain functions. In humans it accounts for 76% of the brain’s volume.

[5]   Do rubber bands with holograms receive the energy of cosmic rays or other celestial forces and by virtue of such connection improve performance? For a fee, endorsers, including professional athletes, maintain that performance is enhanced; scientific experiments prove otherwise. Judging by the volume of sales, many people are skeptical about science.

[6]  Propaganda is the systematic effort to manipulate people’s beliefs, attitudes and actions. Here is advice from a practitioner: “The great masses of people are not made up […] of people capable of making a judgement based on reason and logic […] The masses cannot tell where the enemy’s wrongs end and their own begin.” (“Mein Kampf,” Adolf Hitler, Ford Translation (2009-2019), p. 171).

[7]   A conspiracy theory is an attempt to explain with little or no evidence harmful or tragic events as the result of the action of a small, powerful group. Conspiracy theories are emotionally loaded and tend to become prevalent in periods of widespread anxiety, uncertainty, or hardship.

[8]   See “The Believing Brain: From Ghosts and Gods to Politics and Conspiracies—How We Construct Beliefs and Reinforce Them as Truths,” Michael Shermer.

[9]   See “88 Reasons Why the Rapture Will Be in 1988” by NASA Engineer Edgar Whisenant, and its sequel “The final Shout: Rapture Report 1989.” The sequel explains why certain data mining errors blemished his otherwise fine research work. Judging by the 4.5 million copies sold and the actions of some people (e.g., selling possessions to prepare for the end of the world), the books were influential in some circles.

[10] See “On Economics,” Contingencies, Jul/Aug 2020, p. 22.

[11] Interestingly, the AMA opposed “Medicare for All” in 2019 (https://www.cnn.com/2019/06/08/politics/medical-students-rally-medicare-for-all/index.html) but it eventually dropped from the industry coalition opposed to Medicare expansion (https://www.politico.com/story/2019/08/15/ama-drops-out-of-industry-coalition-opposed-to-medicare-expansion-1664604).

[12] See in YouTube “The Truth About Canadian Health Care,” Nov 16, 2009.

[13] See “Making Sense of the Unexpected,” The Actuary, Dec 2017/Jan 2018.

[14] Genetic in the sense of source, not related to genes.

[15] A rational number is the quotient ^p⁄q of two integer numbers, p and q, with q ≠ 0.

[16] If m² is even, must be even. The reader can verify this statement.

[17] Bertrand Arthur William Russell (1872-1970) was a British philosopher, logician, essayist, and social critic, best known for his works in mathematical logic and analytic philosophy. He received the Order of Merit in 1949 and the Nobel Prize in Literature in 1950.