Fallacies Flashcards
(93 cards)
Argument from fallacy
Assumes that if an argument for some conclusion is fallacious then the conclusion itself is false.
Argument from fallacy is the formal fallacy of analyzing an argument and inferring that, since it contains a fallacy, its conclusion must be false.K. S. Pope (2003) “Logical Fallacies in Psychology: 21 Types” Fallacies & Pitfalls in Psychology It is also called argument to logic (argumentum ad logicam), fallacy fallacy, or fallacist’s fallacy.
Fallacious arguments can arrive at true conclusions, so this is an informal fallacy of relevance.
Form
It has the general argument form:
If P, then Q. P is a fallacious argument. Therefore, Q is false. c since A A is fallacious ¬c Thus, it is a special case of denying the antecedent where the antecedent, rather than being a proposition that is false, is an entire argument that is fallacious. A fallacious argument, just as with a false antecedent, can still have a consequent that happens to be true. The fallacy is in concluding the consequent of a fallacious argument has to be false.
That the argument is fallacious only means that the argument cannot succeed in proving its consequent.John Woods, The death of argument: fallacies in agent based reasoning, Springer 2004, pp. XXIII–XXV But showing how one argument in a complex thesis is fallaciously reasoned does not necessarily invalidate the proof; the complete proof could still logically imply its conclusion if that conclusion is not dependent on the fallacy:
Examples:
Tom: All cats are animals. Ginger is an animal. This means Ginger is a cat.
Bill: Ah, you just committed the affirming the consequent logical fallacy. Sorry, you are wrong, which means that Ginger is not a cat.
Tom: OK – I’ll prove I’m English – I speak English so that proves it.
Bill: But Americans and Canadians, among others, speak English too. You have committed the package-deal fallacy, assuming that speaking English and being English always go together. That means you are not English.
Both of Bill’s rebuttals are arguments from fallacy, because Ginger may or may not be a cat, and Tom may or may not be English. Of course, the mere fact that one can invoke the argument from fallacy against a position does not automatically “prove” one’s own position either, as this would itself be yet another argument from fallacy. An example of this false reasoning follows:
Joe: Bill’s assumption that Ginger is not a cat uses the argument from fallacy. Therefore, Ginger absolutely must be a cat.
An argument using fallacious reasoning is capable of being consequentially correct.
Base rate fallacy
Making a probability judgement based on conditional probabilities, without taking into account the effect of prior probabilities.
The base rate fallacy, also called base rate neglect or base rate bias, is an error that occurs when the conditional probability of some hypothesis H given some evidence E is assessed without taking into account the prior probability (“base rate”) of H and the total probability of evidence E. The conditional probability can be expressed as P(H|E), the probability of H given E, and the base rate error happens when the values of sensitivity and specificity, which depend only on the test itself, are used in place of positive predictive value and negative predictive value, which depend on both the test and the baseline prevalence of event.
Example
In a city of 1 million inhabitants there are 100 terrorists and 999,900 non-terrorists. To simplify the example, it is assumed that the only people in the city are inhabitants. Thus, the base rate probability of a randomly selected inhabitant of the city being a terrorist is 0.0001, and the base rate probability of that same inhabitant being a non-terrorist is 0.9999. In an attempt to catch the terrorists, the city installs an alarm system with a surveillance camera and automatic facial recognition software. The software has two failure rates of 1%:
The false negative rate: If the camera scans a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time.
The false positive rate: If the camera scans a non-terrorist, a bell will not ring 99% of the time, but it will ring 1% of the time.
Suppose now that an inhabitant triggers the alarm. What is the chance that the person is a terrorist? In other words, what is P(P|B), the probability that a terrorist has been detected given the ringing of the bell? Someone making the ‘base rate fallacy’ would infer that there is a 99% chance that the detected person is a terrorist. Although the inference seems to make sense, it is actually bad reasoning, and a calculation below will show that the chances they are a terrorist are actually near 1%, not near 99%.
The fallacy arises from confusing the natures of two different failure rates. The ‘number of non-bells per 100 terrorists’ and the ‘number of non-terrorists per 100 bells’ are unrelated quantities. One does not necessarily equal the other, and they don’t even have to be almost equal. To show this, consider what happens if an identical alarm system were set up in a second city with no terrorists at all. As in the first city, the alarm sounds for 1 out of every 100 non-terrorist inhabitants detected, but unlike in the first city, the alarm never sounds for a terrorist. Therefore 100% of all occasions of the alarm sounding are for non-terrorists, but a false negative rate cannot even be calculated. The ‘number of non-terrorists per 100 bells’ in that city is 100, yet P(T|B) = 0%. There is zero chance that a terrorist has been detected given the ringing of the bell.
Imagine that the city’s entire population of one million people pass in front of the camera. About 99 of the 100 terrorists will trigger the alarm—-and so will about 9,999 of the 999,900 non-terrorists. Therefore, about 10,098 people will trigger the alarm, among which about 99 will be terrorists. So the probability that a person triggering the alarm is actually a terrorist is only about 99 in 10,098, which is less than 1%, and very very far below our initial guess of 99%.
The base rate fallacy is so misleading in this example because there are many more non-terrorists than terrorists. If, instead, the city had about as many terrorists as non-terrorists, and the false-positive rate and the false-negative rate were nearly equal, then the probability of misidentification would be about the same as the false-positive rate of the device. These special conditions hold sometimes: as for instance, about half the women undergoing a pregnancy test are actually pregnant, and some pregnancy tests give about the same rates of false positives and of false negatives. In this case, the rate of false positives per positive test will be nearly equal to the rate of false positives per nonpregnant woman. This is why it is very easy to fall into this fallacy: by coincidence it gives the correct answer in many common situations.
In many real-world situations, though, particularly problems like detecting criminals in a largely law-abiding population, the small proportion of targets in the large population makes the base rate fallacy very applicable. Even a very low false-positive rate will result in so many false alarms as to make such a system useless in practice.
Findings in psychology
In experiments, people have been found to prefer individuating information over general information when the former is available.
In some experiments, students were asked to estimate the grade point averages (GPAs) of hypothetical students. When given relevant statistics about GPA distribution, students tended to ignore them if given descriptive information about the particular student, even if the new descriptive information was obviously of little or no relevance to school performance. This finding has been used to argue that interviews are an unnecessary part of the college admissions process because interviewers are unable to pick successful candidates better than basic statistics.
Psychologists Daniel Kahneman and Amos Tversky attempted to explain this finding in terms of a simple rule or “heuristic” called representativeness. They argued that many judgements relating to likelihood, or to cause and effect, are based on how representative one thing is of another, or of a category. Richard Nisbett has argued that some attributional biases like the fundamental attribution error are instances of the base rate fallacy: people underutilize “consensus information” (the “base rate”) about how others behaved in similar situations and instead prefer simpler dispositional attributions.
Kahneman considers base rate neglect to be a specific form of extension neglect.
Conjunction fallacy
Assumption that an outcome simultaneously satisfying multiple conditions is more probable than an outcome satisfying a single one of them.
The conjunction fallacy is a formal fallacy that occurs when it is assumed that specific conditions are more probable than a single general one.
The most often-cited example of this fallacy originated with Amos Tversky and Daniel Kahneman:Tversky & Kahneman (1982, 1983)
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Which is more probable?
Linda is a bank teller.
Linda is a bank teller and is active in the feminist movement.
90% of those asked chose option 2. However the probability of two events occurring together (in “conjunction”) is always less than or equal to the probability of either one occurring alone—formally, for two events A and B this inequality could be written as \Pr(A \and B) \leq \Pr(A), and \Pr(A \and B) \leq \Pr(B).
For example, even choosing a very low probability of Linda being a bank teller, say Pr(Linda is a bank teller)
- 05 and a high probability that she would be a feminist, say Pr(Linda is a feminist)
- 95, then, assuming independence, Pr(Linda is a bank teller and Linda is a feminist) = 0.05 × 0.95 or 0.0475, lower than Pr(Linda is a bank teller).
Tversky and Kahneman argue that most people get this problem wrong because they use a heuristic called representativeness to make this kind of judgment: Option 2 seems more “representative” of Linda based on the description of her, even though it is clearly mathematically less likely.Tversky & Kahneman (1983)
In other demonstrations they argued that specific scenario seemed more likely because of representativeness, but each added detail would actually make the scenario less and less likely. In this way it could be similar to the misleading vividness or slippery slope fallacies. More recently Kahneman has argued that the conjunction fallacy is a type of extension neglectKahneman (2003)
Joint versus separate evaluation
In some experimental demonstrations the conjoint option is evaluated separately from its basic option. In other words, one group of participants is asked to rank order the likelihood that Linda is a bank teller, a high school teacher, and several other options, and another group is asked to rank order whether Linda is a bank teller and active in the feminist movement versus the same set of options (without Linda is a bankteller as an option). In this type of demonstration different groups of subjects rank order Linda as a bank teller and active in the feminist movement more highly than Linda as a bank teller.
Separate evaluation experiments preceded the earliest joint evaluation experiments, and Kahneman and Tversky were surprised when the effect was still observed under joint evaluation.Kahneman (2011) chapter 15
In separate evaluation the term conjunction effect may be preferred.
Criticism of the Linda problem
Critics such as Gerd Gigerenzer and Ralph Hertwig criticized the Linda problem on grounds such as the wording and framing. The question of the Linda problem may violate conversational maxims in that people assume that the question obeys the maxim of relevance. Gigerenzer argues that some of the terminology used have polysemous meanings, the alternatives of which he claimed were more “natural”. He argues that the meaning of probable “what happens frequently”, corresponds to the mathematical probability people are supposed to be tested on, but the meanings of probable “what is plausible”, and “whether there is evidence” do not.Gigerenzer (1996), Hertwig & Gigerenzer (1999) The term “and” has even been argued to have relevant polysemous meanings.Mellers, Hertwig & Kahneman (2001) Many techniques have been developed to control for this possible misinterpretation but none of them has dissipated the effect.Moro, 2009; Tentori & Crupi, 2012
Many variations in wording of the Linda problem were studied by Tversky and Kahneman. If the first option is changed to obey conversational relevance, i.e., “Linda is a bank teller whether or not she is active in the feminist movement” the effect is decreased, but the majority (57%) of the respondents still commit the conjunction error. If the probability is changed to frequency format (see debiasing section below) the effect is reduced or eliminated. However, studies exist in which indistinguishable conjunction fallacy rates have been observed with stimuli framed in terms of probabilities versus frequencies.see, for example, Tentori, Bonini, & Osherson, 2004 or Weddell & Moro, 2008
The wording criticisms may be less applicable to the conjunction effect in separate evaluation.Gigerenzer (1996) The “Linda problem” has been studied and criticized more than other types of demonstration of the effect (some described below).Kahneman (2011) ch. 15, Kahneman & Tversky (1996), Mellers, Hertwig & Kahneman (2001)
Other demonstrations
Policy experts were asked to rate the probability that the Soviet Union would invade Poland, and the United States would break off diplomatic relations, all in the following year. They rated it on average as having a 4% probability of occurring. Another group of experts was asked to rate the probability simply that the United States would break off relations with the Soviet Union in the following year. They gave it an average probability of only 1%.
In an experiment conducted in 1980, respondents were asked the following:
Suppose Bjorn Borg reaches the Wimbledon finals in 1981. Please rank order the following outcomes from most to least likely.
Borg will win the match
Borg will lose the first set
Borg will lose the first set but win the match
Borg will win the first set but lose the match
On average, participants rated “Borg will lose the first set but win the match” more highly than “Borg will lose the first set”.
In another experiment, participants were asked:
Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequence of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you choose appears on successive rolls of the die.
RGRRR
GRGRRR
GRRRRR
65% of participants chose the second sequence, though option 1 is contained within it and is shorter than the other options. In a version where the $25 bet was only hypothetical the results did not significantly differ. Tversky and Kahneman argued that sequence 2 appears “representative” of a chance sequence (compare to the clustering illusion).
Debiasing
Drawing attention to set relationships, using frequencies instead of probabilities and/or thinking diagrammatically sharply reduce the error in some forms of the conjunction fallacy.Tversky & Kahneman (1983), Gigerenzer (1991), Hertwig & Gigerenzer (1999), Mellers, Hertwig & Kahneman (2001)
In one experiment the question of the Linda problem was reformulated as follows:
There are 100 persons who fit the description above (that is, Linda’s). How many of them are:
Bank tellers? __ of 100
Bank tellers and active in the feminist movement? __ of 100
Whereas previously 85% of participants gave the wrong answer (bank teller and active in the feminist movement) in experiments done with this questioning none of the participants gave a wrong answer.Gigerenzer (1991)
Masked man fallacy
The substitution of identical designators in a true statement can lead to a false one.
The masked man fallacy is a fallacy of formal logic in which substitution of identical designators in a true statement can lead to a false one.
One form of the fallacy may be summarized as follows:
Premise 1: I know who X is.
Premise 2: I do not know who Y is.
Conclusion: Therefore, X is not Y.
The problem arises because Premise 1 and Premise 2 can be simultaneously true even when X and Y refer to the same person. Consider the argument, “I know who my father is. I do not know who the thief is. Therefore, my father is not the thief.” The premises may be true and the conclusion false if the father is the thief but the speaker does not know this about his father. Thus the argument is a fallacious one.
The name of the fallacy comes from the example, “I do not know who the masked man is”, which can be true even though the masked man is Jones, and I know who Jones is.
If someone were to say, “I do not know the masked man,” it implies, “If I do know the masked man, I do not know that he is the masked man.” The masked man fallacy omits the implication.
Note that the following similar argument is valid:
X is Z
Y is not Z
Therefore, X is not Y
But this is because being something is different from knowing (or believing, etc.) something.
Affirming a disjunct
Concluded that one disjunct of a logical disjunction must be false because the other disjunct is true. A or B; A; therefore not B.
The formal fallacy of affirming a disjunct also known as the fallacy of the alternative disjunct or a false exclusionary disjunct occurs when a deductive argument takes the following logical form:
A or B
A
Therefore, it is not the case that B
Explanation
The fallacy lies in concluding that one disjunct must be false because the other disjunct is true; in fact they may both be true because “or” is defined inclusively rather than exclusively. It is a fallacy of equivocation between the operations OR and XOR.
Affirming the disjunct should not be confused with the valid argument known as the disjunctive syllogism.
Example
The following argument indicates the invalidity of affirming a disjunct:
Max is a cat or Max is a mammal.
Max is a cat.
Therefore, Max is not a mammal.
This inference is invalid. If Max is a cat then Max is also a mammal. (Remember “or” is defined in an inclusive sense not an exclusive sense.)
The car is red or the car is large.
The car is red.
Therefore, the car is not large.
The above example of an argument also demonstrates the fallacy.
Affirming the consequent
The antecedent in an indicative conditional is claimed to be true because the consequent is true.
If A, then B; B, therefore A.
Affirming the consequent, sometimes called converse error or fallacy of the converse, is a formal fallacy of inferring the converse from the original statement. The corresponding argument has the general form:
If P, then Q.
Q.
Therefore, P.
An argument of this form is invalid, i.e., the conclusion can be false even when statements 1 and 2 are true. Since P was never asserted as the only sufficient condition for Q, other factors could account for Q (while P was false).
To put it differently, if P implies Q, the only inference that can be made is non-Q implies non-P. (Non-P and non-Q designate the opposite propositions to P and Q.) Symbolically:
(P ⇒ Q) ⇔ (non-Q ⇒ non-P)
The name affirming the consequent derives from the premise Q, which affirms the “then” clause of the conditional premise.
Examples
One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:
If Bill Gates owns Fort Knox, then he is rich.
Bill Gates is rich.
Therefore, Bill Gates owns Fort Knox.
Owning Fort Knox is not the only way to be rich. Any number of other ways exist to be rich.
However, one can affirm with certainty that “if Bill Gates is not rich” (non-Q) then “Bill Gates does not own Fort Knox” (non-P)
Arguments of the same form can sometimes seem superficially convincing, as in the following example:
If I have the flu, then I have a sore throat.
I have a sore throat.
Therefore, I have the flu.
But having the flu is not the only cause of a sore throat since many illnesses cause sore throat, such as the common cold or strep throat.
Denying the antecedent
The consequent in an indicative conditional is claimed to be false because the antecedent is false.
If A, then B; not A, therefore not B.
Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form:
If P, then Q.
Not P.
Therefore, not Q.
Arguments of this form are invalid. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.
The name denying the antecedent derives from the premise “not P”, which denies the “if” clause of the conditional premise.
One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:
If Queen Elizabeth is an American citizen, then she is a human being.
Queen Elizabeth is not an American citizen.
Therefore, Queen Elizabeth is not a human being.
That argument is obviously bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example offered, with apologies for its lack of logical rigour, by Alan Turing in the article “Computing Machinery and Intelligence”:
However, men could still be machines that do not follow a definite set of rules. Thus this argument (as Turing intends) is invalid.
It is possible that an argument that denies the antecedent could be valid, if the argument instantiates some other valid form. For example, if the claims P and Q express the same proposition, then the argument would be trivially valid, as it would beg the question. In everyday discourse, however, such cases are rare, typically only occurring when the “if-then” premise is actually an “if and only if” claim (i.e., a biconditional/equality). For example:
If I am President of the United States, then I can veto Congress.
I am not President.
Therefore, I cannot veto Congress.
The above argument is not valid, but would be if the first premise ended thus: “…and if I can veto Congress, then I am the U.S. President” (as is in fact true). More to the point, the validity of the new argument stems not from denying the antecedent, but modus tollens (denying the consequent).
Existential fallacy
An argument has a universal premise and a particular conclusion.
The existential fallacy, or existential instantiation, is a formal fallacy. In the existential fallacy, we presuppose that a class has members when we are not supposed to do so; that is, when we should not assume existential import.
An existential fallacy is committed in a medieval categorical syllogism because it has two universal premises and a particular conclusion with no assumption that at least one member of the class exists, which is not established by the premises.
In modern logic, the presupposition that a class has members is seen as unacceptable. In 1905, Bertrand Russell wrote an essay entitled “The Existential Import of Proposition”, in which he called this Boolean approach “Peano’s interpretation”.
The fallacy does not occur in enthymemes, where hidden premises required to make the syllogism valid assume the existence of at least one member of the class.
One central concern of the Aristotelian tradition in logic is the theory of the categorical syllogism. This is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid. The theory of the syllogism partly constrains the interpretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the valid patterns (Darapti) is:
Every C is B’’
Every C is A’’
So, some A is B
This is invalid if the A form lacks existential import, and valid if it has existential import. It is held to be valid, and so we know how the A form is to be interpreted. One then naturally asks about the O form; what do the syllogisms tell us about it? The answer is that they tell us nothing. This is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the corresponding universal. For example, he does not mention the form:
No C is B
Every A is C
So, some A is not B
If people had thoughtfully taken sides for or against the validity of this form, that would clearly be relevant to the understanding of the O form. But the weakened forms were typically ignored.
Affirmative conclusion from a negative premise
When a categorical syllogism has a positive conclusion, but at least one negative premise.
Affirmative conclusion from a negative premise (illicit negative) is a formal fallacy that is committed when a categorical syllogism has a positive conclusion, but one or two negative premises.
For example:
No fish are dogs, and no dogs can fly, therefore all fish can fly.
The only thing that can be properly inferred from these premises is that some things that are not fish cannot fly, provided that dogs exist.
Or:
We don’t read that trash. People who read that trash don’t appreciate real literature. Therefore, we appreciate real literature.
It is a fallacy because any valid forms of categorical syllogism that assert a negative premise must have a negative conclusion.
Appeal to probability
Takes something for granted because it would probably be the case.
An appeal to probability (or appeal to possibility) is the logical fallacy of taking something for granted because it would probably be the case, (or might possibly be the case). Inductive arguments lack deductive validity and must therefore be asserted or denied in the premises.
Example
A fallacious appeal to possibility:
Something can go wrong .
Therefore, something will go wrong .
A deductively valid argument would be explicitly premised on Murphy’s law, (see also, modal logic).
Anything that can go wrong, will go wrong .
Something can go wrong .
Therefore, something will go wrong .
Fallacy of exclusive premises
A categorical syllogism that is invalid because both of its premises are negative.
The fallacy of exclusive premises is a syllogistic fallacy committed in a categorical syllogism that is invalid because both of its premises are negative.
Example of an EOO-4 invalid proposition:
E Proposition: No mammals are fish.
O Proposition: Some fish are not whales.
O Proposition: Therefore, some whales are not mammals.
Fallacy of four terms
A categorical syllogism that has four terms.
The fallacy of four terms is the formal fallacy that occurs when a syllogism has four (or more) terms rather than the requisite three. This form of argument is thus invalid.
Explanation
Categorical syllogisms always have three terms:
Major premise: All fish have fins.
Minor premise: All goldfish are fish.
Conclusion: All goldfish have fins.
Here, the three terms are: “goldfish”, “fish”, and “fins”.
Using four terms invalidates the syllogism:
Major premise: All fish have fins.
Minor premise: All goldfish are fish.
Conclusion: All humans have fins.
The premises do not connect “humans” with “fins”, so the reasoning is invalid. Notice that there are four terms: “fish”, “fins”, “goldfish” and “humans”. Two premises are not enough to connect four different terms, since in order to establish connection, there must be one term common to both premises.
In everyday reasoning, the fallacy of four terms occurs most frequently by equivocation: using the same word or phrase but with a different meaning each time, creating a fourth term even though only three distinct words are used:
Major premise: Nothing is better than eternal happiness.
Minor premise: A ham sandwich is better than nothing.
Conclusion: A ham sandwich is better than eternal happiness.
The word “nothing” in the example above has two meanings, as presented: “nothing is better” means the thing being named has the highest value possible; “better than nothing” only means that the thing being described has some value. Therefore, “nothing” acts as two different words in this example, thus creating the fallacy of four terms.
Another example of equivocation, a more tricky one:
Major premise: The pen touches the paper.
Minor premise: The hand touches the pen.
Conclusion: The hand touches the paper.
This is more clear if one uses “is touching” instead of “touches”. It then becomes clear that “touching the pen” is not the same as “the pen”, thus creating four terms: “the hand”, “touching the pen”, “the pen”, “touching the paper”. A correct form of this statement would be:
Major premise: All that touches the pen, touches the paper.
Minor premise: The hand touches the pen.
Conclusion: The hand touches the paper.
Now the term “the pen” has been eliminated, leaving three terms. this argument is now valid but nonsensical because the major premise is untrue
The fallacy of four terms also applies to syllogisms that contain five or six terms.
Reducing terms
Sometimes a syllogism that is apparently fallacious because it is stated with more than three terms can be translated into an equivalent, valid three term syllogism. For example:
Major premise: No humans are immortal.
Minor premise: All Greeks are people.
Conclusion: All Greeks are mortal.
This EAE-1 syllogism apparently has five terms: “humans”, “people”, “immortal”, “mortal”, and “Greeks”. However it can be rewritten as a standard form AAA-1 syllogism by first substituting the synonymous term “humans” for “people” and then by reducing the complementary term “immortal” in the first premise using the immediate inference known as obversion (that is, “No humans are immortal.” is equivalent to “All humans are mortal.”).
Classification
The fallacy of four terms is a syllogistic fallacy. Types of syllogism to which it applies include statistical syllogism, hypothetical syllogism, and categorical syllogism, all of which must have exactly three terms. Because it applies to the argument’s form, as opposed to the argument’s content, it is classified as a formal fallacy.
Equivocation of the middle term is a frequently cited source of a fourth term being added to a syllogism; both of the equivocation examples above affect the middle term of the syllogism. Consequently this common error itself has been given its own name: the fallacy of the ambiguous middle. An argument that commits the ambiguous middle fallacy blurs the line between formal and informal fallacies, however it is usually considered an informal fallacy because the argument’s form appears valid.
Illicit major
A categorical syllogism that is invalid because it’s major term is not distributed in the major premise but distributed in the conclusion.
Illicit minor
A categorical syllogism that is invalid because it’s minor term is not distributed in the minor premise but distributed in the conclusion.
Illicit minor is a formal fallacy committed in a categorical syllogism that is invalid because its minor term is undistributed in the minor premise but distributed in the conclusion.
This fallacy has the following argument form:
All A are B.
All A are C.
Therefore, all C are B.
Example:
All cats are felines.
All cats are mammals.
Therefore, all mammals are felines.
The minor term here is mammal, which is not distributed in the minor premise “All cats are mammals,” because this premise is only defining a property of possibly some mammals (i.e., that they’re cats.) However, in the conclusion “All mammals are felines,” mammal is distributed (it is talking about all mammals being felines). It is shown to be false by any mammal that is not a feline; for example, a dog.
Example:
Pie is good.
Pie is unhealthy.
Thus, all good things are unhealthy.
Negative conclusion from affirmative premises.
When a categorical syllogism has a negative conclusion but affirmative premises.
Negative conclusion from affirmative premises is a syllogistic fallacy committed when a categorical syllogism has a negative conclusion yet both premises are affirmative. The inability of affirmative premises to reach a negative conclusion is usually cited as one of the basic rules of constructing a valid categorical syllogism.
Statements in syllogisms can be identified as the following forms:
a: All A is B. (affirmative)
e: No A is B. (negative)
i: Some A is B. (affirmative)
o: Some A is not B. (negative)
The rule states that a syllogism in which both premises are of form a or i (affirmative) cannot reach a conclusion of form e or o (negative). Exactly one of the premises must be negative to construct a valid syllogism with a negative conclusion. (A syllogism with two negative premises commits the related fallacy of exclusive premises.)
Example (invalid aae form):
Premise: All colonels are officers.
Premise: All officers are soldiers.
Conclusion: Therefore, no colonels are soldiers.
The aao-4 form is perhaps more subtle as it follows many of the rules governing valid syllogisms, except it reaches a negative conclusion from affirmative premises.
Invalid aao-4 form:
All A is B.
All B is C.
Therefore, some C is not A.
This is valid only if A is a proper subset of B and/or B is a proper subset of C. However, this argument reaches a faulty conclusion if A, B, and C are equivalent. In the case that A B C, the conclusion of the following simple aaa-1 syllogism would contradict the aao-4 argument above:
All B is A.
All C is B.
Therefore, all C is A.
Fallacy of the undistributed middle
The middle term in a categorical syllogism is not distributed.
The fallacy of the undistributed middle is a formal fallacy, that is committed when the middle term in a categorical syllogism is not distributed in either the minor premise or the major premise. It is thus a syllogistic fallacy.
Classical formulation
In classical syllogisms, all statements consist of two terms and are in the form of “A” (all), “E” (none), “I” (some), or “O” (some not). The first term is distributed in A statements; the second is distributed in O statements; both are distributed in E statements; and none are distributed in I statements.
The fallacy of the undistributed middle occurs when the term that links the two premises is never distributed.
In this example, distribution is marked in boldface:
All Z is B (All) y is B Therefore (All) y is Z B is the common term between the two premises (the middle term) but is never distributed, so this syllogism is invalid.
Also, a related rule of logic is that anything distributed in the conclusion must be distributed in at least one premise.
All Z is B Some Y is Z Therefore All Y is B The middle term - Z - is distributed, but Y is distributed in the conclusion and not in any premise, so this syllogism is invalid.
Pattern
The fallacy of the undistributed middle takes the following form:
All Z is B
Y is B
Therefore, Y is Z
This can be graphically represented as follows:
Undistributed middle argument map.jpg
where the premises are in the green box and the conclusion is indicated above them.
Here, B is the middle term, and it is not distributed in the major premise, “all Z is B”.
It may or may not be the case that “all Z is B,” but this is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that “all B is Z,” which is ignored in the argument. The fallacy is similar to affirming the consequent and denying the antecedent. However, the fallacy may be resolved if the terms are exchanged in either the conclusion or in the first co-premise. Indeed, from the perspective of first-order logic, all cases of the fallacy of the undistributed middle are, in fact, examples of affirming the consequent or denying the antecedent, depending on the structure of the fallacious argument.
Examples
For example:
All students carry backpacks. My grandfather carries a backpack. Therefore, my grandfather is a student. All students carry backpacks. My grandfather carries a backpack. Everyone who carries a backpack is a student. Therefore, my grandfather is a student. Grandpa backpack undistributed middle.jpg
The middle term is the one that appears in both premises — in this case, it is the class of backpack carriers. It is undistributed because neither of its uses applies to all backpack carriers. Therefore it can’t be used to connect students and my grandfather — both of them could be separate and unconnected divisions of the class of backpack carriers. Note below how “carries a backpack” is truly undistributed:
grandfather is someone who carries a backpack; student is someone who carries a backpack
Specifically, the structure of this example results in affirming the consequent.
However, if the latter two statements were switched, the syllogism would be valid:
All students carry backpacks.
My grandfather is a student.
Therefore, my grandfather carries a backpack.
In this case, the middle term is the class of students, and the first use clearly refers to ‘all students’. It is therefore distributed across the whole of its class, and so can be used to connect the other two terms (backpack carriers, and my grandfather). Again, note below that “student” is distributed:
grandfather is a student and thus carries a backpack
Argument from ignorance
Assuming that a claim is true because it has not been proven false or cannot be proven false.
Argument from repetition
Signifies that it has been discussed extensively until nobody cares to discuss it anymore.
Ad nauseam is a Latin term for something unpleasurable that has continued “to point of nausea”. “ad nauseam” definitions from Dictionary.com For example, the sentence, “This topic has been discussed ad nauseam”, signifies that the topic in question has been discussed extensively, and that those involved in the discussion have grown tired of it.
Etymology
This term is defined by the American Heritage Dictionary as:
Argumentum ad nauseam or argument from repetition or argumentum ad infinitum is an argument made repeatedly (possibly by different people) until nobody cares to discuss it any more. This may sometimes, but not always, be a form of proof by assertion
Argument from silence
Where the conclusion is based on the absence of evidence, rather than the existence of evidence.
An argument from silence (also called argumentum a silentio in Latin) is generally a conclusion drawn based on the absence of statements in historical documents.”argumentum e silentio noun phrase” The Oxford Essential Dictionary of Foreign Terms in English. Ed. Jennifer Speake. Berkley Books, 1999.John Lange, The Argument from Silence, History and Theory, Vol. 5, No. 3 (1966), pp. 288-301 In the field of classical studies, it often refers to the deduction from the lack of references to a subject in the available writings of an author to the conclusion that he was ignorant of it.”silence, the argument from”. The Concise Oxford Dictionary of the Christian Church. Ed. E. A. Livingstone. Oxford University Press, 2006.
Thus in historical analysis with an argument from silence, the absence of a reference to an event or a document is used to cast doubt on the event not mentioned. While most historical approaches rely on what an author’s works contain, an argument from silence relies on what the book or document does not contain. This approach thus uses what an author “should have said” rather than what is available in the author’s extant writings.Historical evidence and argument by David P. Henige 2005 ISBN 978-0-299-21410-4 page 176.Seven Pillories of Wisdom by David R. Hall 1991 ISBN 0-86554-369-0 pages 55-56.
Historical analysis
An argument from silence can be convincing when mentioning a fact can be seen as so natural that its omission is a good reason to assume ignorance. For example, while the editors of Yerushalmi and Bavli mention the other community, most scholars believe these documents were written independently. Louis Jacobs writes, “If the editors of either had had access to an actual text of the other, it is inconceivable that they would not have mentioned this. Here the argument from silence is very convincing.”“Talmud”. A Concise Companion to the Jewish Religion. Louis Jacobs. Oxford University Press, 1999.
Errietta Bissa, professor of Classics at University of Wales flatly state that arguments from silence are not valid.Governmental intervention in foreign trade in archaïc and classical Greece by Errietta M. A. Bissa ISBN 90-04-17504-0 page 21: “This is a fundamental methodological issue on the validity of arguments from silence, where I wish to make my position clear: arguments from silence are not valid.” David Henige states that, although risky, such arguments can at times shed light on historical events. Yifa has pointed out the perils of arguments from silence, in that although no references appear to the “Rules of purity” codes of monastic conduct of 1103 in the Transmission of the Lamp, or any of the Pure Land documents, a copy of the code in which the author identifies himself exists.The origins of Buddhist monastic codes in China by Yifa, Zongze 2002 ISBN 0-8248-2494-6 page 32: “an argumentum ex silencio is hardly conclusive”
Yifa points out that arguments from silence are often less than conclusive, e.g. the lack of references to a compilation of a set of monastic codes by contemporaries or even by disciples does not mean that it never existed. This is well as illustrated by the case of Changlu Zongze’s “Rules of purity” which he wrote for the Chan monastery in 1103.
One of his contemporaries wrote a preface to a collection of his writings neglected to mention his code. And none of his biographies nor the documents of the Transmission of the Lamp, nor the Pure Land documents (which exalt him) refer to Zongze’s collection of a monastic code. However a copy of the code in which the author identifies himself
exists.The origins of Buddhist monastic codes in China by Yifa, Zongze 2002 ISBN 0-8248-2494-6 page 32.
Frances Wood based her controversial book Did Marco Polo go to China? on arguments from silence. Woods argued that Marco Polo never went to China and fabricated his accounts because he failed to mention elements from the visual landscape such as tea, did not record the Great Wall and neglected to record practices such as foot-binding. She argued that no outsider could spend 15 years in China and not observe and record these elements. Most historians disagree with Wood’s reasoning.Historical evidence and argument by David P. Henige 2005 ISBN 978-0-299-21410-4 page 176.
Legal aspects
Jed Rubenfeld, professor of Law at Yale Law School, has shown an example of the difficulty in applying arguments from silence in constitutional law, stating that although arguments from silence can be used to draw conclusions about the intent of the Framers of the US Constitution, their application can lead to two different conclusions and hence they can not be used to settle the issues.Jed Rubenfeld Rights of Passage: Majority Rule in Congress Duke Law Journal, 1996 Section B: Arguments from silence, “From this silence one can draw clear plausible inferences about the Framers’ intent. The only difficulty is that one can draw two
different inferences…. The truth is that the argument from silence is not dispositive”.
In the context of Morocco’s Truth Commission of 1999 regarding torture and secret detentions, Wu and Livescu state that the fact that someone remained silent is no proof of their ignorance about a specific piece of information. They point out that the absence of records about the torture of prisoners under the secret detention program is no proof that such detentions did not involve torture, or that some detentions did not take place.Human Rights, Suffering, and Aesthetics in Political Prison Literature by Yenna Wu, Simona Livescu 2011 ISBN 0-7391-6741-3 pages 86-90.
Begging the question
The failure to provide what is essentially the conclusion of an argument as a premise, if so required.
Begging the question (Latin petitio principii, “assuming the initial point”) is a type of informal fallacy in which an implicit premise would directly entail the conclusion. Begging the question is one of the classic informal fallacies in Aristotle’s Prior Analytics. Some modern authors consider begging the question to be a species of circulus in probando (Latin, “circle in proving”) or circular reasoning. Were it not begging the question, the missing premise would render the argument viciously circular, and while never persuasive, arguments of the form “A therefore A” are logically valid petitio principii or Begging the question is studied in Prior Analytics II, 64b, 34 – 65a, 9 and it is considered a material fallacy. Circulus in probando, or circular reasoning, is explained in Prior Analytics II, 57b, 18 – 59b, 1. Some authors consider begging the question to be a form of circular reasoning, for example: Bradley Dowden, “Fallacies” in Internet Encyclopedia of Philosophy. because asserting the premise while denying the self-same conclusion is a direct contradiction. In general, validity only guarantees the conclusion must follow given the truth of the premises. Absent that, a valid argument proves nothing: the conclusion may or may not follow from faulty premises—although in this particular example, it’s self-evident that the conclusion is false if and only if the premise is false (see logical equivalence and logical equality).The reason petitio principii is considered to be a fallacy is not that the inference is invalid (because any statement is indeed equivalent to itself), but that the argument can be deceptive. A statement cannot prove itself. A premise must have a different source of reason, ground or evidence for its truth from that of the conclusion: Lander University, “Petitio Principii”.
In modern days, English speakers are prone to use “beg the question” as a way of saying “raises the question”. However, the former denotes a failure to explicitly raise an essential premise, so that it may be taken as given, whereas the latter simply functions as a segue for whatever comes to mind.
Definition
The fallacy of petitio principii, or “begging the question”, is committed “when a proposition which requires proof is assumed without proof”; in order to charitably entertain the argument, it must be taken as given “in some form of the very proposition to be proved, as a premise from which to deduce it”.Welton (1905), 279. One must take it upon oneself that the goal, taken as given, is essentially the means to that end.
When the fallacy of begging the question is committed in a single step, it is sometimes called a hysteron proteron,Davies (1915), 572.Welton (1905), 280-282. as in the statement “Opium induces sleep because it has a soporific quality”.Welton (1905), 281. Such fallacies may not be immediately obvious due to the use of synonyms or synonymous phrases; one way to beg the question is to make a statement first in concrete terms, then in abstract ones, or vice-versa. Another is to “bring forth a proposition expressed in words of Saxon origin, and give as a reason for it the very same proposition stated in words of Norman origin”,Gibson (1908), 291. as in this example: “To allow every man an unbounded freedom of speech must always be, on the whole, advantageous to the State, for it is highly conducive to the interests of the community that each individual should enjoy a liberty perfectly unlimited of expressing his sentiments”.Richard Whately, Elements of Logic (1826) quoted in Gibson (1908), 291.
When the fallacy of begging the question is committed in more than one step, some authors consider it circulus in probando or reasoning in a circle however, there is no fallacy if the missing premise is acknowledged, and if not, there is no circle.
“Begging the question” can also refer to an argument in which the unstated premise is essential to, but not identical with the conclusion, or is “controversial or questionable for the same reasons that typically might lead someone to question the conclusion”.Kahane and Cavender (2005), 60.
History
The term was translated into English from Latin in the 16th century. The Latin version, petitio principii, can be interpreted in different ways. Petitio (from peto), in the post-classical context in which the phrase arose, means “assuming” or “postulating”, but in the older classical sense means “petition”, “request” or “beseeching”. Principii, genitive of principium, means “beginning”, “basis” or “premise” (of an argument). Literally petitio principii means “assuming the premise” or “assuming the original point”, or, alternatively, “a request for the beginning or premise” that is, the premise depends on the truth of the very matter in question.
The Latin phrase comes from the Greek en archei aiteisthai in Aristotle’s Prior Analytics II xvi:
Begging or assuming the point at issue consists (to take the expression in its widest sense) failing to demonstrate the required proposition. But there are several other ways in which this may happen; for example, if the argument has not taken syllogistic form at all, he may argue from premises which are less known or equally unknown, or he may establish the antecedent by means of its consequents; for demonstration proceeds from what is more certain and is prior. Now begging the question is none of these. If, however, the relation of B to C is such that they are identical, or that they are clearly convertible, or that one applies to the other, then he is begging the point at issue…. begging the question is proving what is not self-evident by means of itself…either because predicates which are identical belong to the same subject, or because the same predicate belongs to subjects which are identical. Thomas Fowler believed that Petitio Principii would be more properly called Petitio Quæsiti, which is literally “begging the question”.Fowler, Thomas (1887). The Elements of Deductive Logic, Ninth Edition (p. 145). Oxford, England: Clarendon Press.
Related fallacies
Circular reasoning is a fallacy in which “the reasoner begins with what he or she is trying to end up with”. The individual components of a circular argument can be logically valid because if the premises are true, the conclusion must be true, and will not lack relevance. However, circular reasoning is not persuasive because, if the conclusion is doubted, the premise which leads to it will also be doubted.
Begging the question is similar to the complex question or fallacy of many questions: questioning that presupposes something that would not be acceptable to everyone involved. For example, “Is Mary wearing a blue or a red dress?” is fallacious because it artificially restricts the possible responses to a blue or red dress. If the person being questioned wouldn’t necessarily consent to those constraints, the question is thus fallacious.
Modern usage
Many English speakers use “begs the question” to mean “raises the question”, “impels the question”, or even “invites the question”, and follow that phrase with the question raised,see definitions at Wiktionary and at The Free Dictionary (accessed 30th May 2011); each source gives both definitions. for example, “this year’s deficit is half a trillion dollars, which begs the question: how are we ever going to balance the budget?” Philosophers and many grammarians deem such usage incorrect.Follett (1966), 228; Kilpatrick (1997); Martin (2002), 71; Safire (1998).Brians, Common Errors in English Usage: Online Edition (full text of book: 2nd Edition, November, 2008, William, James & Company) (accessed 1 July 2011) Academic linguist Mark Liberman recommends avoiding the phrase entirely, noting that because of shifts in usage in both Latin and English over the centuries, the relationship of the literal expression to its intended meaning is unintelligible and therefore it is now “such a confusing way to say it that only a few pedants understand the phrase.”
Circular reasoning
When the reasoner begins with what he or she is trying to end up with.
Circular reasoning (also known as paradoxical thinking or circular logic), is a logical fallacy in which “the reasoner begins with what he or she is trying to end up with”. The individual components of a circular argument will sometimes be logically valid because if the premises are true, the conclusion must be true, and will not lack relevance. Circular logic cannot prove a conclusion because, if the conclusion is doubted, the premise which leads to it will also be doubted. Begging the question is a form of circular reasoning.
Circular reasoning is often of the form: “a is true because b is true; b is true because a is true.” Circularity can be difficult to detect if it involves a longer chain of propositions.
Academic Douglas Walton used the following example of a fallacious circular argument:
Wellington is in New Zealand.
Therefore, Wellington is in New Zealand.
He notes that, although the argument is deductively valid, it cannot prove that Wellington is in New Zealand because it contains no evidence that is distinct from the conclusion. The context – that of an argument – means that the proposition does not meet the requirement of proving the statement, thus it is a fallacy. He proposes that the context of a dialogue determines whether a circular argument is fallacious: if it forms part of an argument, then it is. Citing Cederblom and Paulsen 1986:109) Hugh G. Gauch observes that non-logical facts can be difficult to capture formally:
“Whatever is less dense than water will float, because whatever is less dense than water will float” sounds stupid, but “Whatever is less dense than water will float, because such objects won’t sink in water” might pass.
Circular reasoning and the problem of induction
Joel Feinberg and Russ Shafer-Landau note that “using the scientific method to judge the scientific method is circular reasoning”. Scientists attempt to discover the laws of nature and to predict what will happen in the future, based on those laws. However, per David Hume’s problem of induction, science cannot be proven inductively by empirical evidence, and thus science cannot be proven scientifically. An appeal to a principle of the uniformity of nature would be required to deductively necessitate the continued accuracy of predictions based on laws that have only succeeded in generalizing past observations. But as Bertrand Russell observed, “The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil”.
Circular cause and consequence
Where the consequence of the phenomenon is claimed to be its root cause.
Correlation does not imply causation (cum hoc propter hoc, Latin for “with this, because of this”) is a phrase used in science and statistics to emphasize that a correlation between two variables does not necessarily imply that one causes the other. Many statistical tests calculate correlation between variables. A few go further and calculate the likelihood of a true causal relationship; examples are the Granger causality test and convergent cross mapping.
The opposite assumption, that correlation proves causation, is one of several questionable cause logical fallacies by which two events that occur together are taken to have a cause-and-effect relationship. This fallacy is also known as cum hoc ergo propter hoc, Latin for “with this, therefore because of this”, and “false cause”. A similar fallacy, that an event that follows another was necessarily a consequence of the first event, is sometimes described as post hoc ergo propter hoc (Latin for “after this, therefore because of this”).
In a widely studied example, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD. Re-analysis of the data from the epidemiological studies showed that women undertaking HRT were more likely to be from higher socio-economic groups (ABC1), with better-than-average diet and exercise regimens. The use of HRT and decreased incidence of coronary heart disease were coincident effects of a common cause (i.e. the benefits associated with a higher socioeconomic status), rather than cause and effect, as had been supposed.
As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not imply that the resulting conclusion is false. In the instance above, if the trials had found that hormone replacement therapy caused a decrease in coronary heart disease, but not to the degree suggested by the epidemiological studies, the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed.
Usage
In logic, the technical use of the word “implies” means “to be a sufficient circumstance.” This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of logical implication: if p then q symbolized as p → q. That is “if circumstance p is true, then q necessarily follows.” In this sense, it is always correct to say “Correlation does not imply causation.”
However, in casual use, the word “imply” loosely means suggests rather than requires. The idea that correlation and causation are connected is certainly true; where there is causation, there is likely to be correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are made after correlations are confirmed to be real and all causational relationship are systematically explored using large enough data sets.
Edward Tufte, in a criticism of the brevity of “correlation does not imply causation,” deprecates the use of “is” to relate correlation and causation (as in “Correlation is not causation”), citing its inaccuracy as incomplete. While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following:
“Empirically observed covariation is a necessary but not sufficient condition for causality.”
“Correlation is not causation but it sure is a hint.”
General pattern
For any two correlated events A and B, the following relationships are possible:
A causes B;
B causes A;
A and B are consequences of a common cause, but do not cause each other;
There is no connection between A and B, the correlation is coincidental.
Less clear-cut correlations are also possible. For example, causality is not necessarily one-way; in a predator-prey relationship, predator numbers affect prey, but prey numbers, i.e. food supply, also affect predators.
The cum hoc ergo propter hoc logical fallacy can be expressed as follows:
A occurs in correlation with B.
Therefore, A causes B.
In this type of logical fallacy, one makes a premature conclusion about causality after observing only a correlation between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B, even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:
A may be the cause of B.
B may be the cause of A.
some unknown third factor C may actually be the cause of both A and B.
there may be a combination of the above three relationships. For example, B may be the cause of A at the same time as A is the cause of B (contradicting that the only relationship between A and B is that A causes B). This describes a self-reinforcing system.
the “relationship” is a coincidence or so complex or indirect that it is more effectively called a coincidence (i.e. two events occurring at the same time that have no direct relationship to each other besides the fact that they are occurring at the same time). A larger sample size helps to reduce the chance of a coincidence, unless there is a systematic error in the experiment.
In other words, there can be no conclusion made regarding the existence or the direction of a cause and effect relationship only from the fact that A and B are correlated. Determining whether there is an actual cause and effect relationship requires further investigation, even when the relationship between A and B is statistically significant, a large effect size is observed, or a large part of the variance is explained.
Examples of illogically inferring causation from correlation
B causes A (reverse causation)
The more firemen fighting a fire, the bigger the fire is observed to be.
Therefore firemen cause an increase in the size of a fire.
In this example, the correlation between the number of firemen at a scene and the size of the fire does not imply that the firemen cause the fire. Firemen are sent according to the severity of the fire and if there is a large fire, a greater number of firemen are sent; therefore, it is rather that fire causes firemen to arrive at the scene. So the above conclusion is false.
A causes B and B causes A (bidirectional causation)
Increased pressure is associated with increased temperature.
Therefore pressure causes temperature.
The ideal gas law, PV=nRT, describes the direct relationship between pressure and temperature (along with other factors) to show that there is a direct correlation between the two properties. For a fixed volume and mass of gas, an increase in temperature will cause an increase in pressure; likewise, increased pressure will cause an increase in temperature. This demonstrates bidirectional causation. The conclusion that pressure causes temperature is true but is not logically guaranteed by the premise.
Third factor C (the common-causal variable) causes both A and B
All these examples deal with a lurking variable, which is simply a hidden third variable that affects both causes of the correlation; for example, the fact that it is summer in Example 3. A difficulty often also arises where the third factor, though fundamentally different from A and B, is so closely related to A and/or B as to be confused with them or very difficult to scientifically disentangle from them (see Example 4).
;Example 1
Sleeping with one’s shoes on is strongly correlated with waking up with a headache.
Therefore, sleeping with one’s shoes on causes headache.
The above example commits the correlation-implies-causation fallacy, as it prematurely concludes that sleeping with one’s shoes on causes headache. A more plausible explanation is that both are caused by a third factor, in this case going to bed drunk, which thereby gives rise to a correlation. So the conclusion is false.
;Example 2
Young children who sleep with the light on are much more likely to develop myopia in later life.
Therefore, sleeping with the light on causes myopia.
This is a scientific example that resulted from a study at the University of Pennsylvania Medical Center. Published in the May 13, 1999 issue of Nature, the study received much coverage at the time in the popular press.CNN, May 13, 1999. Night-light may lead to nearsightedness However, a later study at Ohio State University did not find that infants sleeping with the light on caused the development of myopia. It did find a strong link between parental myopia and the development of child myopia, also noting that myopic parents were more likely to leave a light on in their children’s bedroom.Ohio State University Research News, March 9, 2000. Night lights don’t lead to nearsightedness, study suggests In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.
;Example 3
As ice cream sales increase, the rate of drowning deaths increases sharply.
Therefore, ice cream consumption causes drowning.
The aforementioned example fails to recognize the importance of time and temperature in relationship to ice cream sales. Ice cream is sold during the hot summer months at a much greater rate than during colder times, and it is during these hot summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water-based activities, not ice cream. The stated conclusion is false.
;Example 4
A hypothetical study shows a relationship between test anxiety scores and shyness scores, with a statistical r value (strength of correlation) of +.59.The Psychology of Personality: Viewpoints, Research, and Applications. Carducci, Bernard J. 2nd Edition. Wiley-Blackwell: UK, 2009.
Therefore, it may be simply concluded that shyness, in some part, causally influences test anxiety.
However, as encountered in many psychological studies, another variable, a “self-consciousness score,” is discovered which has a sharper correlation (+.73) with shyness. This suggests a possible “third variable” problem, however, when three such closely related measures are found, it further suggests that each may have bidirectional tendencies (see “bidirectional variable,” above), being a cluster of correlated values each influencing one another to some extent. Therefore, the simple conclusion above may be false.
;Example 5
Since the 1950s, both the atmospheric CO2 level and obesity levels have increased sharply.
Hence, atmospheric CO2 causes obesity.
Richer populations tend to eat more food and consume more energy
;Example 6
HDL (“good”) cholesterol is negatively correlated with incidence of heart attack.
Therefore, taking medication to raise HDL will decrease the chance of having a heart attack.
Further researchOrnish, Dean. “Cholesterol: The good, the bad, and the truth” ]1] (retrieved 3 June 2011) has called this conclusion into question. Instead, it may be that other underlying factors, like genes, diet and exercise, affect both HDL levels and the likelihood of having a heart attack; it is possible that medicines may affect the directly measurable factor, HDL levels, without affecting the chance of heart attack.
Coincidence
With a decrease in the wearing of hats, there has been an increase in global warming over the same period.
Therefore, global warming is caused by people abandoning the practice of wearing hats.
A similar example is used by the parody religion Pastafarianism to illustrate the logical fallacy of assuming that correlation equals causation.
Relation to the Ecological fallacy
There is a relation between this subject-matter and the ecological fallacy, described in a 1950 paper by William S. Robinson. Robinson shows that ecological correlations, where the statistical object is a group of persons (i.e. an ethnic group), does not show the same behaviour as individual correlations, where the objects of inquiry are individuals: “The relation between ecological and individual correlations which is discussed in this paper provides a definite answer as to whether ecological correlations can validly be used as substitutes for individual correlations. They cannot.” (…) “(a)n ecological correlation is almost certainly not equal to its corresponding individual correlation.”
Determining causation
David Hume argued that causality is based on experience, and experience similarly based on the assumption that the future models the past, which in turn can only be based on experience – leading to circular logic. In conclusion, he asserted that causality is not based on actual reasoning: only correlation can actually be perceived. David Hume (Stanford Encyclopedia of Philosophy)
In order for a correlation to be established as causal, the cause and the effect must be connected through an impact mechanism in accordance with known laws of nature.
Intuitively, causation seems to require not just a correlation, but a counterfactual dependence. Suppose that a student performed poorly on a test and guesses that the cause was his not studying. To prove this, one thinks of the counterfactual – the same student writing the same test under the same circumstances but having studied the night before. If one could rewind history, and change only one small thing (making the student study for the exam), then causation could be observed (by comparing version 1 to version 2). Because one cannot rewind history and replay events after making small controlled changes, causation can only be inferred, never exactly known. This is referred to as the Fundamental Problem of Causal Inference – it is impossible to directly observe causal effects.Paul W. Holland. 1986. “Statistics and Causal Inference” Journal of the American Statistical Association, Vol. 81, No. 396. (Dec., 1986), pp. 945-960.
A major goal of scientific experiments and statistical methods is to approximate as best as possible the counterfactual state of the world.Judea Pearl. 2000. Causality: Models, Reasoning, and Inference, Cambridge University Press. For example, one could run an experiment on identical twins who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation.
Well-designed experimental studies replace equality of individuals as in the previous example by equality of groups. This is achieved by randomization of the subjects to two or more groups. Although not a perfect system, the likeliness of being equal in all aspects rises with the number of subjects placed randomly in the treatment/placebo groups. From the significance of the difference of the effect of the treatment vs. the placebo, one can conclude the likeliness of the treatment having a causal effect on the disease. This likeliness can be quantified in statistical terms by the P-value .
When experimental studies are impossible and only pre-existing data are available, as is usually the case for example in economics, regression analysis can be used. Factors other than the potential causative variable of interest are controlled for by including them as regressors in addition to the regressor representing the variable of interest. False inferences of causation due to reverse causation (or wrong estimates of the magnitude of causation due the presence of bidirectional causation) can be avoided by using explanators (regressors) that are necessarily exogenous, such as physical explanators like rainfall amount (as a determinant of, say, futures prices), lagged variables whose values were determined before the dependent variable’s value was determined, instrumental variables for the explanators (chosen based on their known exogeneity), etc. See Causality#Statistics and Economics. Spurious correlation due to mutual influence from a third, common, causative variable, is harder to avoid: the model must be specified such that there is a theoretical reason to believe that no such underlying causative variable has been omitted from the model; in particular, underlying time trends of both the dependent variable and the independent (potentially causative) variable must be controlled for by including time as another independent variable.
Use of correlation as scientific evidence
Much of scientific evidence is based upon a correlation of variables – they tend to occur together. Scientists are careful to point out that correlation does not necessarily mean causation. The assumption that A causes B simply because A correlates with B is a logical fallacy – it is not a legitimate form of argument. However, sometimes people commit the opposite fallacy – dismissing correlation entirely, as if it does not imply causation. This would dismiss a large swath of important scientific evidence.
In conclusion, correlation is an extremely valuable type of scientific evidence in fields such as medicine, psychology, and sociology. But first correlations must be confirmed as real, and then every possible causational relationship must be systematically explored. In the end correlation can be used as powerful evidence for a cause and effect relationship between a treatment and benefit, a risk factor and a disease, or a social or economic factor and various outcomes. But it is also one of the most abused types of evidence, because it is easy and even tempting to come to premature conclusions based upon the preliminary appearance of a correlation.
Continuum fallacy
Improperly rejecting a claim for being imprecise.
The continuum fallacy (also called the fallacy of the beard,David Roberts: Reasoning: Other Fallacies line drawing fallacy, bald man fallacy, fallacy of the heap, the fallacy of grey, the sorites fallacy) is an informal fallacy closely related to the sorites paradox, or paradox of the heap. The fallacy causes one to erroneously reject a vague claim simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity.
The fallacy appears to demonstrate that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states. According to the fallacy, differences in quality cannot result from differences in quantity.
There are clearly reasonable and clearly unreasonable cases in which objects either belong or do not belong to a particular group of objects based on their properties. We are able to take them case by case and designate them as such even in the case of properties which may be vaguely defined. The existence of hard or controversial cases does not preclude our ability to designate members of particular kinds of groups.
Relation with sorites paradox
Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomism, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.
For the purpose of the continuum fallacy, one assumes that there is in fact a continuum, though this is generally a minor distinction: in general, any argument against the sorites paradox can also be used against the continuum fallacy. One argument against the fallacy is based on the simple counterexample: there do exist bald people and people who aren’t bald. Another argument is that for each degree of change in states, the degree of the condition changes slightly, and these “slightly”s build up to shift the state from one category to another. For example, perhaps the addition of a grain of rice causes the total group of rice to be “slightly more” of a heap, and enough “slightly”s will certify the group’s heap status – see fuzzy logic.
Examples
Fred can never be called bald
Fred can never be called bald. Fred isn’t bald now. However, if he loses one hair, that won’t make him go from not bald to bald either. If he loses one more hair after that, this loss of a second hair also does not make him go from not bald to bald. Therefore, no matter how much hair he loses, he can never be called bald.
The heap
The fallacy can be described in the form of a conversation:
Q: Does one grain of wheat form a heap?
A: No.
Q: If we add one, do two grains of wheat form a heap?
A: No.
Q: If we add one, do three grains of wheat form a heap?
A: No.
…
Q: If we add one, do one hundred grains of wheat form a heap?
A: No.
Q: Therefore, no matter how many grains of wheat we add, we will never have a heap. Therefore, heaps don’t exist!
Correlation proves causation
A faulty assumption that correlation between two variables implies that one causes the other.
Correlation does not imply causation (cum hoc propter hoc, Latin for “with this, because of this”) is a phrase used in science and statistics to emphasize that a correlation between two variables does not necessarily imply that one causes the other. Many statistical tests calculate correlation between variables. A few go further and calculate the likelihood of a true causal relationship; examples are the Granger causality test and convergent cross mapping.
The opposite assumption, that correlation proves causation, is one of several questionable cause logical fallacies by which two events that occur together are taken to have a cause-and-effect relationship. This fallacy is also known as cum hoc ergo propter hoc, Latin for “with this, therefore because of this”, and “false cause”. A similar fallacy, that an event that follows another was necessarily a consequence of the first event, is sometimes described as post hoc ergo propter hoc (Latin for “after this, therefore because of this”).
In a widely studied example, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD. Re-analysis of the data from the epidemiological studies showed that women undertaking HRT were more likely to be from higher socio-economic groups (ABC1), with better-than-average diet and exercise regimens. The use of HRT and decreased incidence of coronary heart disease were coincident effects of a common cause (i.e. the benefits associated with a higher socioeconomic status), rather than cause and effect, as had been supposed.
As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not imply that the resulting conclusion is false. In the instance above, if the trials had found that hormone replacement therapy caused a decrease in coronary heart disease, but not to the degree suggested by the epidemiological studies, the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed.
Usage
In logic, the technical use of the word “implies” means “to be a sufficient circumstance.” This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of logical implication: if p then q symbolized as p → q. That is “if circumstance p is true, then q necessarily follows.” In this sense, it is always correct to say “Correlation does not imply causation.”
However, in casual use, the word “imply” loosely means suggests rather than requires. The idea that correlation and causation are connected is certainly true; where there is causation, there is likely to be correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are made after correlations are confirmed to be real and all causational relationship are systematically explored using large enough data sets.
Edward Tufte, in a criticism of the brevity of “correlation does not imply causation,” deprecates the use of “is” to relate correlation and causation (as in “Correlation is not causation”), citing its inaccuracy as incomplete. While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following:
“Empirically observed covariation is a necessary but not sufficient condition for causality.”
“Correlation is not causation but it sure is a hint.”
General pattern
For any two correlated events A and B, the following relationships are possible:
A causes B;
B causes A;
A and B are consequences of a common cause, but do not cause each other;
There is no connection between A and B, the correlation is coincidental.
Less clear-cut correlations are also possible. For example, causality is not necessarily one-way; in a predator-prey relationship, predator numbers affect prey, but prey numbers, i.e. food supply, also affect predators.
The cum hoc ergo propter hoc logical fallacy can be expressed as follows:
A occurs in correlation with B.
Therefore, A causes B.
In this type of logical fallacy, one makes a premature conclusion about causality after observing only a correlation between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B, even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:
A may be the cause of B.
B may be the cause of A.
some unknown third factor C may actually be the cause of both A and B.
there may be a combination of the above three relationships. For example, B may be the cause of A at the same time as A is the cause of B (contradicting that the only relationship between A and B is that A causes B). This describes a self-reinforcing system.
the “relationship” is a coincidence or so complex or indirect that it is more effectively called a coincidence (i.e. two events occurring at the same time that have no direct relationship to each other besides the fact that they are occurring at the same time). A larger sample size helps to reduce the chance of a coincidence, unless there is a systematic error in the experiment.
In other words, there can be no conclusion made regarding the existence or the direction of a cause and effect relationship only from the fact that A and B are correlated. Determining whether there is an actual cause and effect relationship requires further investigation, even when the relationship between A and B is statistically significant, a large effect size is observed, or a large part of the variance is explained.
Examples of illogically inferring causation from correlation
B causes A (reverse causation)
The more firemen fighting a fire, the bigger the fire is observed to be.
Therefore firemen cause an increase in the size of a fire.
In this example, the correlation between the number of firemen at a scene and the size of the fire does not imply that the firemen cause the fire. Firemen are sent according to the severity of the fire and if there is a large fire, a greater number of firemen are sent; therefore, it is rather that fire causes firemen to arrive at the scene. So the above conclusion is false.
A causes B and B causes A (bidirectional causation)
Increased pressure is associated with increased temperature.
Therefore pressure causes temperature.
The ideal gas law, PV=nRT, describes the direct relationship between pressure and temperature (along with other factors) to show that there is a direct correlation between the two properties. For a fixed volume and mass of gas, an increase in temperature will cause an increase in pressure; likewise, increased pressure will cause an increase in temperature. This demonstrates bidirectional causation. The conclusion that pressure causes temperature is true but is not logically guaranteed by the premise.
Third factor C (the common-causal variable) causes both A and B
All these examples deal with a lurking variable, which is simply a hidden third variable that affects both causes of the correlation; for example, the fact that it is summer in Example 3. A difficulty often also arises where the third factor, though fundamentally different from A and B, is so closely related to A and/or B as to be confused with them or very difficult to scientifically disentangle from them (see Example 4).
;Example 1
Sleeping with one’s shoes on is strongly correlated with waking up with a headache.
Therefore, sleeping with one’s shoes on causes headache.
The above example commits the correlation-implies-causation fallacy, as it prematurely concludes that sleeping with one’s shoes on causes headache. A more plausible explanation is that both are caused by a third factor, in this case going to bed drunk, which thereby gives rise to a correlation. So the conclusion is false.
;Example 2
Young children who sleep with the light on are much more likely to develop myopia in later life.
Therefore, sleeping with the light on causes myopia.
This is a scientific example that resulted from a study at the University of Pennsylvania Medical Center. Published in the May 13, 1999 issue of Nature, the study received much coverage at the time in the popular press.CNN, May 13, 1999. Night-light may lead to nearsightedness However, a later study at Ohio State University did not find that infants sleeping with the light on caused the development of myopia. It did find a strong link between parental myopia and the development of child myopia, also noting that myopic parents were more likely to leave a light on in their children’s bedroom.Ohio State University Research News, March 9, 2000. Night lights don’t lead to nearsightedness, study suggests In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.
;Example 3
As ice cream sales increase, the rate of drowning deaths increases sharply.
Therefore, ice cream consumption causes drowning.
The aforementioned example fails to recognize the importance of time and temperature in relationship to ice cream sales. Ice cream is sold during the hot summer months at a much greater rate than during colder times, and it is during these hot summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water-based activities, not ice cream. The stated conclusion is false.
;Example 4
A hypothetical study shows a relationship between test anxiety scores and shyness scores, with a statistical r value (strength of correlation) of +.59.The Psychology of Personality: Viewpoints, Research, and Applications. Carducci, Bernard J. 2nd Edition. Wiley-Blackwell: UK, 2009.
Therefore, it may be simply concluded that shyness, in some part, causally influences test anxiety.
However, as encountered in many psychological studies, another variable, a “self-consciousness score,” is discovered which has a sharper correlation (+.73) with shyness. This suggests a possible “third variable” problem, however, when three such closely related measures are found, it further suggests that each may have bidirectional tendencies (see “bidirectional variable,” above), being a cluster of correlated values each influencing one another to some extent. Therefore, the simple conclusion above may be false.
;Example 5
Since the 1950s, both the atmospheric CO2 level and obesity levels have increased sharply.
Hence, atmospheric CO2 causes obesity.
Richer populations tend to eat more food and consume more energy
;Example 6
HDL (“good”) cholesterol is negatively correlated with incidence of heart attack.
Therefore, taking medication to raise HDL will decrease the chance of having a heart attack.
Further researchOrnish, Dean. “Cholesterol: The good, the bad, and the truth” (retrieved 3 June 2011) has called this conclusion into question. Instead, it may be that other underlying factors, like genes, diet and exercise, affect both HDL levels and the likelihood of having a heart attack; it is possible that medicines may affect the directly measurable factor, HDL levels, without affecting the chance of heart attack.
Coincidence
With a decrease in the wearing of hats, there has been an increase in global warming over the same period.
Therefore, global warming is caused by people abandoning the practice of wearing hats.
A similar example is used by the parody religion Pastafarianism to illustrate the logical fallacy of assuming that correlation equals causation.
Relation to the Ecological fallacy
There is a relation between this subject-matter and the ecological fallacy, described in a 1950 paper by William S. Robinson. Robinson shows that ecological correlations, where the statistical object is a group of persons (i.e. an ethnic group), does not show the same behaviour as individual correlations, where the objects of inquiry are individuals: “The relation between ecological and individual correlations which is discussed in this paper provides a definite answer as to whether ecological correlations can validly be used as substitutes for individual correlations. They cannot.” (…) “(a)n ecological correlation is almost certainly not equal to its corresponding individual correlation.”
Determining causation
David Hume argued that causality is based on experience, and experience similarly based on the assumption that the future models the past, which in turn can only be based on experience – leading to circular logic. In conclusion, he asserted that causality is not based on actual reasoning: only correlation can actually be perceived. David Hume (Stanford Encyclopedia of Philosophy)
In order for a correlation to be established as causal, the cause and the effect must be connected through an impact mechanism in accordance with known laws of nature.
Intuitively, causation seems to require not just a correlation, but a counterfactual dependence. Suppose that a student performed poorly on a test and guesses that the cause was his not studying. To prove this, one thinks of the counterfactual – the same student writing the same test under the same circumstances but having studied the night before. If one could rewind history, and change only one small thing (making the student study for the exam), then causation could be observed (by comparing version 1 to version 2). Because one cannot rewind history and replay events after making small controlled changes, causation can only be inferred, never exactly known. This is referred to as the Fundamental Problem of Causal Inference – it is impossible to directly observe causal effects.Paul W. Holland. 1986. “Statistics and Causal Inference” Journal of the American Statistical Association, Vol. 81, No. 396. (Dec., 1986), pp. 945-960.
A major goal of scientific experiments and statistical methods is to approximate as best as possible the counterfactual state of the world.Judea Pearl. 2000. Causality: Models, Reasoning, and Inference, Cambridge University Press. For example, one could run an experiment on identical twins who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation.
Well-designed experimental studies replace equality of individuals as in the previous example by equality of groups. This is achieved by randomization of the subjects to two or more groups. Although not a perfect system, the likeliness of being equal in all aspects rises with the number of subjects placed randomly in the treatment/placebo groups. From the significance of the difference of the effect of the treatment vs. the placebo, one can conclude the likeliness of the treatment having a causal effect on the disease. This likeliness can be quantified in statistical terms by the P-value .
When experimental studies are impossible and only pre-existing data are available, as is usually the case for example in economics, regression analysis can be used. Factors other than the potential causative variable of interest are controlled for by including them as regressors in addition to the regressor representing the variable of interest. False inferences of causation due to reverse causation (or wrong estimates of the magnitude of causation due the presence of bidirectional causation) can be avoided by using explanators (regressors) that are necessarily exogenous, such as physical explanators like rainfall amount (as a determinant of, say, futures prices), lagged variables whose values were determined before the dependent variable’s value was determined, instrumental variables for the explanators (chosen based on their known exogeneity), etc. See Causality#Statistics and Economics. Spurious correlation due to mutual influence from a third, common, causative variable, is harder to avoid: the model must be specified such that there is a theoretical reason to believe that no such underlying causative variable has been omitted from the model; in particular, underlying time trends of both the dependent variable and the independent (potentially causative) variable must be controlled for by including time as another independent variable.
Use of correlation as scientific evidence
Much of scientific evidence is based upon a correlation of variables – they tend to occur together. Scientists are careful to point out that correlation does not necessarily mean causation. The assumption that A causes B simply because A correlates with B is a logical fallacy – it is not a legitimate form of argument. However, sometimes people commit the opposite fallacy – dismissing correlation entirely, as if it does not imply causation. This would dismiss a large swath of important scientific evidence.
In conclusion, correlation is an extremely valuable type of scientific evidence in fields such as medicine, psychology, and sociology. But first correlations must be confirmed as real, and then every possible causational relationship must be systematically explored. In the end correlation can be used as powerful evidence for a cause and effect relationship between a treatment and benefit, a risk factor and a disease, or a social or economic factor and various outcomes. But it is also one of the most abused types of evidence, because it is easy and even tempting to come to premature conclusions based upon the preliminary appearance of a correlation.