# Fallacy

In philosophical logic, a fallacy or fallacy - in Latin fallacia - is a conclusion in which the derived statement does not follow from the explicitly stated or implicitly assumed assumptions.

This does not necessarily mean that the derived statement is also false: a fallacy just does not provide any information about the actual truth content of the derived statement.

In the argument scheme of the conceptual logic , the expression paralogism is also used for false conclusions. A fallacy is based on an error in the application of inference rules ; it is not correct according to the rules of formal logic . Occasionally, however, formally valid conclusions from false assumptions are also referred to as false conclusions.

## Classification of fallacies

A deliberately brought about fallacy is also referred to as a catch , pseudo-argument or sophism , an unintentional fallacy is also called paralogism. Unintentional false conclusions can be explained psychologically as a consequence of certain cognitive distortions (English bias ) or through judgment heuristics , which in special cases do not lead to the correct result. In addition to logic , social psychology and thought and cognitive psychology deal with fallacies. Extensive lists of false conclusions and sham arguments are also known from rhetoric .

Various types of fallacy have been investigated since ancient times, for example in Aristotle's Sophistic Refutations or in the texts of the older Stoa . Modern classics that consider fallacies rhetorically and philosophically are Schopenhauer's Eristic Dialectic and John Stuart Mills' system of deductive and inductive logic, which is decisive for the English-speaking world .

Since the development of classical formal logic , philosophical logic has been much less concerned with listing, classifying and systemizing fallacies. Instead, especially in analytical philosophy, individual fallacies are reconstructed using formal language. The translation into a formal language should show where the fallacy makes a derivation step that violates the formal inference.

A famous fallacy in philosophy is the naturalistic fallacy , which, according to one reading, is about the question of whether one can infer from descriptive to normative statements, from being to what is ought. The intentional fallacy is also philosophically significant , which is based on the fact that in so-called intensional contexts (A believes, thinks, knows, etc.) expressions, although they denote the same thing, may not simply be exchanged.

The following list of types and examples can neither claim completeness nor a clear systematics, since such a claim would presuppose a specific theory of logic.

## Mathematical bogus evidence

In mathematics , false conclusions arise from incorrect application of calculation rules. Are known z. B. Joke evidence based on hidden division by zero .

Examples

The equation is a true statement. If you divide both sides and ignore the fact that this does not provide a result that can be used in conventional computing systems, you get the wrong statement . Often the division by zero is obscured by instead of e.g. B. the factor is used. A fallacy can also arise when applying the laws of the roots , which apply to positive real numbers, to complex numbers . For example, the equation doesn't always hold true . One therefore always considers both square roots at the same time. ${\ displaystyle 2 \ cdot 0 = 1 \ cdot 0}$${\ displaystyle 0}$${\ displaystyle 2 = 1}$${\ displaystyle 0}$${\ displaystyle (xx)}$
${\ displaystyle {\ sqrt {a}} \ cdot {\ sqrt {b}} = {\ sqrt {a \ cdot b}}}$

## Confusion of reason and correlation

From a correlation , the common occurrence of two facts (e.g. temporal or statistical, within a sample), it is wrongly concluded that there is a justification. Such inferences are neither deductive nor inferring conclusions and no truth-conserving logical operations: no false conclusion can be derived from true premises. In the best case, the combination of two facts can provide a hypothesis about a connection that needs to be further tested using scientific methods. If one regards the establishment of such a hypothesis as a logical conclusion, one speaks of an abduction . For the usual logic, however, the concept of inference is restricted to truth-conserving operations.

### Temporal connection of events

Cum hoc, ergo propter hoc (Latin for 'with this, so therefore') describes a logical error in which two events that always occurred together are explained as cause and effect. Even the special case post hoc, ergo propter hoc (Latin: 'afterwards, so therefore') - the event A occurs before B - does not prove causality : the day always follows the night, but not because this is the cause for that is. A is possibly a necessary but not a sufficient condition of B. The extent to which post hoc, ergo propter hoc applies, was the subject of studies by David Hume and Immanuel Kant . While Hume denies that per post hoc, ergo propter hoc rules can be obtained from experience that express more than a repeated observation and the resulting intellectual association, according to Kant, natural causality can be asserted a priori with reference to general laws . A well-known example of this problem concerns the transfer of kinetic energy: if, for example, a billiard ball hits another, an observer sees nothing but the temporal relationship between the impact of the first and the change in speed of the second. The transfer of kinetic energy from one ball to the other cannot be observed, only the speeds of the balls before and after the collision. With the inclusion of Newton 's laws of motion, however , a causality must even be assumed here.

### Spatial connection of events

This is the spatial variant of Cum hoc ergo propter hoc : From the spatial proximity of two events (logically incorrect, but plausible) a cause for this proximity is deduced. So Peirce chose the following example of an abduction ( see also Evidence in the Weak Sense ): someone finds some white beans and next to them a sack full of white beans. He concludes:

 These beans are white. All the beans in this sack are white. Abduction: These beans come from that sack.

The inferred hypothesis is plausible but not compelling.

In the Gestalt psychology which is law of proximity known: elements with small distances from each other, are perceived as belonging together. If this togetherness is interpreted as a subordination or causal relationship, a false conclusion is drawn.

### Irrelevant reference value

Cannabis and heroin use (Venn diagram)

A widespread fallacy concludes from the frequency of occurrence of a property F under condition G in a statistical survey that the presence of F is a relevant indicator for G in individual cases . In doing so, however, an incorrect or unsuitable reference value is often chosen, so that a prevalence error occurs. If G lies after F and the statistical correlation is interpreted as a causal relationship (e.g. in the sense of a sufficient but not necessary condition), then it is a special case of post hoc, ergo propter hoc .

For example, there are occasional warnings against cannabis as a gateway drug for heroin . There is an actual overlap between the two groups of users: According to statistical surveys, most heroin users ( H ) were previously cannabis users ( C ). However, it does not follow that cannabis use leads to heroin addiction: in fact, the majority of cannabis users are not interested in heroin. So points decision theorist Gerd Gigerenzer that it is also true if the statement were "Most heroin users earlier cannabis users" would be wrong to conclude "Most cannabis users will be heroin users."

Since in this example a conditional probability is equated with its inversion, one speaks here of the confusion of the inverse .

### Examples

#### Rattle stork

 The storks return to Europe in spring. In spring, the number of births rises in Europe. Fallacy: The return of the storks causes an increase in the number of births.

Given the real cause of births, it is clearly a fallacy. If it is met naively, it is a paralogism; however, if it is brought up to convince someone that the children will be brought by the stork, it is a matter of sophism .

#### Quick diagnosis

 Patient XY has back pain . Patient XY has a herniated disc . Fallacy: The herniated disc is the cause of the back pain.

Even if the conclusion is true, it is a fallacy: medically, there is a good chance that the back pain has other causes, and further research should be done to rule it out. Because of the obvious possible explanation, there is a misuse of Occam 's razor . This fallacy can also be understood as the simplest special case of a false disjunction .

## Syllogistic fallacies

In the tradition of logic, not only were the valid conclusions examined, but also logical fallacies within the framework of the syllogistics customary up to the end of the 19th century were considered and categorized.

### Quaternio Terminorum

( Lat. Four of concepts ) in the categorical syllogism exactly have three different terms occur: The generic term as a predicate in the major premise and conclusion, the middle term as a subject in the major premise and as a predicate in the pedestal and the lower term as a subject in pedestal and conclusion. In the quaternio terminorum , however, two different middle terms appear, whereby the conclusion becomes invalid regardless of the truth of the premises and the conclusion:

All dogs (middle term) are animals (generic term). All cats (sub-term) are mammals (middle term). So all cats (sub-term) are animals (generic term).

Quaternio-terminorum fallacies are seldom as obvious as in the example, since the difference between the terms is often hidden by a real homonymy or a homonymy created by formalization . A quaternio terminorum through homonymy violates the form of the syllogism by replacing the middle term in the upper and lower clauses with an ambiguous expression that makes the major clause a true statement in one meaning and the minor clause in another. The result is a fallacy, since a fourth term was introduced with the alternative meaning of the term in the position of the middle term. In addition to homonymy, amphibolism can also be the cause of the deception, or a meta-base eis állo génos , i.e. a grammatical ambiguity or a change in the reference system of the terms. The conscious use is also referred to as trickery or subreption ( see also: Fallacia falsi medii false disjunction ).

#### Real homonymy

Homonymy is easy to discover in the following example:

 Anything with a beard can be shaved. Keys have a beard. Fallacy: Keys can be shaved.

( Key beard and whisker )

#### Homonymy through formalization

The following example is more complex:

 All parents love their children. All children love chocolate. Fallacy: All parents love chocolate.

However, if one admits the truth of the premises, a fallacy is made because the relation “x loves y” is mistakenly taken for the predicate. For syllogistic inferences, however, only single-digit terms are permitted as predicates (“x loves chocolate”, “x loves her children”). In this syllogism there are four terms when the premises are syllogistically formalized.

 The more good you do, the better it is. Taking the medicine is good for the patient. Fallacy: The more medicine you take, the better it is.

The problem with this conclusion is not just the questionability of the first premise. In fact, the remedy is only helpful under the condition that one is sick, and the second premise does not even quantify the amount of remedy, as the conclusion does. In fact, there are drugs that are harmful if overdosed. A wiser conclusion, which avoids confusing a good deed with a single dose of medicine, might come to the conclusion: "The more sick people are given a medicine, the better".

### Classic fallacies according to Eubulides

The sophisms of Eubulides aim to shake a conversation partner in his certainty or to discredit it in front of an audience by making him admit something paradoxical .

 What you haven't lost, you still have. You haven't lost any horns. Fallacy: So you have horns.

 Do you know who this veiled one is? - no! It is your father! Fallacy: You don't know who your father is.

### Logical distribution error

If the syllogistic subject is distributed in the final clause of a syllogism , that is, if a judgment is made about all members of the class designated by the subject (e.g. "All S are P", "None S are P"), then must also the minor premise (2nd premise, in which the subject of the conclusion is introduced) be a judgment about all class members. In these judgments, the predicate or the middle concept applies to each individual object that falls under the subject concept, the predicate is distributed (distributed).

#### Not distributed subject

False conclusions arise when the second premise only refers to a subset of the subject, but the conclusion relates to all elements of the category. Two examples:

 All vegetarians are healthy. Some people are vegetarians. Fallacy: All people are healthy.

The fallacy is easy to spot here. The following example is more complicated:

 Omnivores eat meat. Humans are omnivores. Fallacy: All people eat meat.

While in the final sentence every single living person is specifically meant (distributive), the "people" in the lower sentence are generally the representatives of the biological generic term (collective). Likewise in the major principle: As a collective, it is the omnivores' responsibility to eat meat. For the individual omnivore, this does not mean that he has ever eaten meat, but that he belongs to a species that is “predisposed” to it or can digest meat.

#### Not distributed middle term

Also: sophism of the collective middle term ( Latin non distributivi, sed collectivi medii )

In a valid syllogism, the middle term is distributed in at least one premise. If it is not, a fallacy like the following can occur:

 All humans are two-legged. Some bipeds are birds. Fallacy: Some people are birds.

Here in the subsection “ x is a bird” is not distributed over all bipeds.

### Wrong disjunction

In disjunctive syllogism , too, false conclusions in the sense of surreptitiousness are possible, in which the relationship between the middle term and the other terms does not meet the requirements of syllogistic inference on closer inspection. This is the so-called false disjunction (see there).

## False conclusions in criminology

False conclusions in the DNA evidence

The result of a DNA analysis , a fingerprint or any other trace alone cannot determine whether a suspect is guilty or not. It is only taken as an indication that must be supplemented by further. However, many suspects make a confession when confronted with the result. If this is not the case, the result must be interpreted, whereby false conclusions cannot be ruled out. In the following, the special case of a DNA test is considered, but the mechanisms can also be transferred to other types of traces . An invalid step invalidates the entire final chain:

1. DNA test does not show a match.
2. Incorrect agreement due to false positive test results.
3. Random match. The suspect does not have to be the originator of the trail just because there is a match.
4. The biological material may have been deposited by someone else.
5. The biological material does not have to have been deposited at the time of the crime.

The fallacy of the prosecutor (engl. Prosecutor's fallacy ) consists of a double prevalence errors and a resulting reference fallacy (s. O.). The prevalence error is based on the confusion of two probabilities: The probability that the author of a DNA trace will be tested positive in a DNA comparison test, with the probability that someone who tests positive in a DNA search is the author of the Must be trace (for a calculation example see prevalence errors ). The a priori probability of false positive test results is ignored, as is the natural prevalence of a certain genetic fingerprint in a sufficiently large population.

A DNA raster search alone is unsuitable for incriminating an otherwise unsuspecting person. If there is already a suspicion on the basis of other circumstances that are independent of the trace, the test can confirm or dispel the suspicion. Its significance increases the smaller the population of possible authors becomes - in the example it is very large - but only as long as it can be ensured that the author of the track comes from the population. In order to be able to infer authorship from a match (“3” in the Fig. “False conclusions in DNA evidence”), a group of people must first be found that is objectively eligible - a subjectively suspected A- priori probability cannot be accepted. This fundamental problem arises both in a judicial investigation (“group of perpetrators”) and in a paternity test. If you read there “The probability that the blood (at the crime scene) came from someone other than the suspect is 1 in a million”, then that is a fallacy.

In the copyright fallacy , the chain of conditions is skipped and concluded that the author of a DNA trace must also be the culprit. The apparent inference, "Since both samples have a chance of one in a million coincidentally, the probability of innocence is also one in a million, or the probability of guilt is a million to one," links the accuser's fallacy with a copyright fallacy. Not only is the author's probability wrongly stated far too high (“copyright fallacy”), the trace could also have been “laid” ( 4 in the figure) or in a context other than the crime ( 5 in the figure).

## List of individual fallacies (selection)

Fallacy definition example Remarks
A dicto simpliciter ad dictum secundum quid The incorrect application of a general rule to uncontested exceptions. “I believe that you should never hurt anyone. That's why I couldn't be a surgeon. " Informal fallacy
Affirming a Disjunct Construction of a disjunctive syllogism with respect to two sets that are not at all disjoint “To have a girlfriend like Tom's, you have to be either rich or famous. Tom is rich, so he can't be famous. "
Appeal to Probability False assumption that a probable or possible event will in any case actually occur “In the universe there are billions of galaxies with billions of stars. So there has to be another planet with intelligent life. " Special case of non sequitur
Affirming the consequent Inadmissible reversal of antecedent and consequence “When the lamp is broken, it is dark. It is dark. So the lamp is broken. " Formal logic error
Confusion of the Inverse Confusing a conditional probability with its inverse “Most accidents happen at home. To be sure, you should stay at home as little as possible. " Misjudgment of probability
Conjunction Fallacy False assumption that a more specific case is more likely than a more general case “Linda studied philosophy and takes a great interest in social issues such as discrimination and social justice. Which is more likely? 1. Linda is a bank clerk. 2. Linda is a bank clerk and active in the women's movement. "-" Answer 2 of course. " Formal fallacy
Ignoratio elenchi
Illicit major Conclusion in which the generic term has a distribution that it does not have in the first premise “All dogs are animals. Cats are not dogs. So cats are not animals. " Special case of non sequitur
Intentional fallacy
Moralistic fallacy
Non sequitur Collective term for conclusions that cannot be derived from the premises. “The universe had a beginning. So it also has an end. " Formal fallacy
Ecological fallacy
Petitio principii Special case of a circular argument
Quaternio Terminorum Fallacy that occurs when not three but four terms are used in a syllogism. “All trees are plants. All birds are animals. That is why all trees are animals. " Formal fallacy; see above
Player fallacy The false assumption that a random event, if it hasn't happened for a long time, is more likely to happen than if it happened recently. “I haven't rolled a six in twenty throws. One of the next throws must be a six. " Misjudgment of probability
Circular reasoning A flaw in evidence that consists in the fact that what is to be proven is used as evidence. "M. is a great mediator because he has a knack for bringing disputes together. " Formal fallacy