Formal Fallacies

Question 1

A formal fallacy is

Answer 1

an error in logic that can be seen in the argument’s form. All formal fallacies are specific types of non sequiturs.

Question 2

using a personal experience or an isolated example instead of sound reasoning or compelling evidence.

Answer 2

Anecdotal fallacy/Misleading vividness

leads to: hasty generalizations about the occurrence

but there’s a very strong psychological effect because of a cognitive heuristic called the availability heuristic.

Question 3

a statement that takes something for granted because it would probably be the case (or might be the case)

Answer 3

Appeal to probability

Question 4

assumes that if an argument for some conclusion is fallacious, then the conclusion is false.

Answer 4

Argument from fallacy [argument to logic (argumentum ad logicam)]

Question 5

If P, then Q.
P is a fallacious argument.
Therefore, Q is false.

Answer 5

argumentum ad logicam

Question 6

argumentum ad logicam

Answer 6

Objection

Tom: All cats are animals. Ginger is an animal. Therefore, Ginger is a cat.
Bill: You have just fallaciously affirmed the consequent. You are incorrect. Therefore, Ginger is not a cat.
Tom: I speak English. Therefore, I am English.
Bill: Americans and Canadians, among others, speak English too. By assuming that speaking English and being English always go together, you have just committed the package-deal fallacy. You are incorrect. Therefore, you are not English.

Both of Bill’s rebuttals are arguments from fallacy. Ginger may or may not be a cat, and Tom may or may not be English. The fact that Tom’s argument was fallacious is not, in itself, sufficient proof that his conclusion is false.

Question 7

argumentum ad logicam

Answer 7

Counterargument

Joe: Bill’s assumption that Ginger is not a cat uses the argument from fallacy. Therefore, Ginger absolutely must be a cat.

That one can invoke the argument from fallacy against a position does not prove one’s own position either, as this would be an argument from fallacy itself, as is the case in Joe’s argument.

Question 8

Argumentum ad logicam

Answer 8

can be used as an ad hominem appeal: by impugning the opponent’s credibility or good faith it can be used to sway the audience by undermining the speaker, rather than addressing the speaker’s argument.

Question 9

assumes that if an argument for some conclusion is fallacious, then the conclusion is false.

Answer 9

Argument from fallacy

Question 10

Something can go wrong (premise).
Therefore, something will go wrong (invalid conclusion).
If I do not bring my umbrella (premise)
It will rain. (invalid conclusion).

Answer 10

A fallacious appeal to possibility (appeal to probability):

Question 11

making a probability judgment based on conditional probabilities, without taking into account the effect of prior probabilities.

Answer 11

Base rate fallacy

If presented with related base rate information (i.e. generic, general information) and specific information (information only pertaining to a certain case), the mind tends to ignore the former and focus on the latter.

https://en.wikipedia.org/wiki/Base_rate_fallacy

Question 12

Base rate fallacy Example:

In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. To simplify the example, it is assumed that all people present 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(T | 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 he/she is a terrorist are actually near 1%, not near 99%.

Answer 12

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 actually is 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

Question 13

assumption that an outcome simultaneously satisfying multiple conditions is more probable than an outcome satisfying a single one of them.

Answer 13

Conjunction fallacy

Question 14

conjunction fallacy example:

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?

1) Linda is a bank teller.
2) Linda is a bank teller and is active in the feminist movement.

Answer 14

The majority 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 ^ B)

Question 15

conjunction fallacy example:

Which of the following events is most likely to occur within the next year?

1. The United States will withdraw all troops from Iraq.
2. The United States will withdraw all troops from Iraq and bomb Iranian nuclear facilities.

Answer 15

The probability of the conjunctions is never greater than that of its conjuncts. Therefore, the first choice is more probable. No matter how unlikely it is that America will withdraw troops within the year from Iraq, it is even less likely they will do so and bomb nuclear facilities

Question 16

the substitution of identical designators in a true statement can lead to a false one.

Answer 16

Masked man fallacy (illicit substitution of identicals)

In philosophical logic, the masked man fallacy (also known as the intensional fallacy and the epistemic fallacy[1]) is committed when one makes an illicit use of Leibniz’s law in an argument. Leibniz’s law states that, if one object has a certain property, while another object does not have the same property, the two objects cannot be identical.

Question 17

Masked man fallacy (illicit substitution of identicals)

The name of the fallacy comes from the example:

Premise 1: I know who Bob is.
Premise 2: I do not know who the masked man is
Conclusion: Therefore, Bob is not the masked man.

Answer 17

The premises may be true and the conclusion false if Bob is the masked man and the speaker does not know that. Thus the argument is a fallacious one.

Question 18

Masked man fallacy (illicit substitution of identicals) Example:

Answer 18

Lois Lane believes that Superman can fly.

Lois Lane does not believe that Clark Kent can fly.

Therefore Superman and Clark Kent are not the same person.

In symbolic form, the above arguments are

Premise 1: I know who X is.

Premise 2: I do not know who Y is.

Conclusion: Therefore, X is not Y.

Question 19

Masked man fallacy (illicit substitution of identicals) VALID Example:

• X is Z
• Y is not Z
• Therefore, X is not Y

Answer 19

This is valid because being something is different from knowing (or believing, etc.) something.

The valid and invalid inferences can be compared when looking at the invalid formal inference:

• X is Z
• Y is Z, or Y is not Z.
• Therefore, X is not Y.

Question 20

an error in logic that concerns compound propositions.

Answer 20

Propositional fallacies

For a compound proposition to be true, the truth values of its constituent parts must satisfy the relevant logical connectives that occur in it

Question 21

concluding that one disjunct of a logical disjunction must be false because the other disjunct is true; A or B; A, therefore not B.

Answer 21

Types of Propositional fallacies:

Affirming a disjunct

Question 22

the antecedent in an indicative conditional is claimed to be true because the consequent is true; if A, then B; B, therefore A.

Answer 22

Types of Propositional fallacies:

Affirming the consequent

Question 23

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.

Answer 23

Types of Propositional fallacies:

Denying the antecedent

Question 24

an error in logic where the quantifiers of the premises are in contradiction to the quantifier of the conclusion.

Answer 24

Quantification fallacies

Type: Existential fallacy – an argument that 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.

One example would be: “Every unicorn definitely has a horn on its forehead”. It does not imply that there are any unicorns at all in the world, and thus it cannot be assumed that, if the statement is true, somewhere there is a unicorn in the world (with a horn on its forehead). The statement, if assumed true, only implies that if there were any unicorns, each would definitely have a horn on its forehead.

Question 25

logical fallacies that occur in syllogisms.

Answer 25

Formal Syllogistic fallacies (TYPES):

1. Affirmative conclusion from a negative premise (illicit negative) – when a categorical syllogism has a positive conclusion, but at least one negative premise.
2. Fallacy of exclusive premises – a categorical syllogism that is invalid because both of its premises are negative.
3. Fallacy of four terms (quaternio terminorum) – a categorical syllogism that has four terms.
4. Illicit major – a categorical syllogism that is invalid because its major term is not distributed in the major premise but distributed in the conclusion.
5. Illicit minor – a categorical syllogism that is invalid because its minor term is not distributed in the minor premise but distributed in the conclusion.
6. Negative conclusion from affirmative premises (illicit affirmative) – when a categorical syllogism has a negative conclusion but affirmative premises.
7. Fallacy of the undistributed middle – the middle term in a categorical syllogism is not distributed.

Question 26

Or:

We don’t read that trash. People who read that trash don’t appreciate real literature. Therefore, we appreciate real literature.

Answer 26

Affirmative conclusion from a negative premise (illicit negative):

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:

It is a fallacy because any valid forms of categorical syllogism that assert a negative premise must have a negative conclusion.

Question 27

Example of an EOO-4 invalid syllogism

E Proposition: No cats are dogs.
O Proposition: Some dogs are not pets.
O Proposition: Therefore, some pets are not cats.

Answer 27

Explanation of Example 1:

This may seem like a logical conclusion, as it appears to be logically derived that if Some dogs are not pets, then surely some are pets, otherwise, the premise would have stated “No Dogs are pets”, and if some pets are dogs, then not all pets can be cats, thus, some pets are not cats. However, this breaks down when you apply the same logic to the conclusion: If some pets are not cats then it would seem logical to state that some pets are cats. But this is not supported by either premise. Cats not being dogs, and the state of dogs as either pets or not, has nothing to do with whether cats are pets. Two negative premises cannot give a logical foundation for a conclusion, as they will invariably be independent statements that cannot be directly related, thus the name ‘Exclusive Premises’. It is made more clear when the subjects in the argument are more clearly unrelated such as the following:

Question 28

Additional Example of an EOO-4 invalid syllogism

E Proposition: No planets are dogs.
O Proposition: Some dogs are not pets.
O Proposition: Therefore, some pets are not planets.

Answer 28

Explanation of Example 2:

In this example we can more clearly see that the physical difference between a dog and a planet has no correlation to the domestication of dogs. The two premises are exclusive and the subsequent conclusion is nonsense, as the transpose would imply that some pets are planets.

Conclusion:

It is important to note that the truthfulness of the final statement is not relevant in this fallacy. The conclusion of the first example is true, while the final statement in the second is clearly ridiculous; however, both are argued on fallacious logic and would not hold up as valid arguments.

Question 29

a categorical syllogism that has four terms.

Answer 29

Fallacy of four terms (quaternio terminorum) 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.

Question 30

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.

Answer 30

Fallacy of four terms (quaternio terminorum)

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.

Question 31

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.

Answer 31

Fallacy of four terms (quaternio terminorum)

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.

Question 32

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.

Answer 32

Fallacy of four terms (quaternio terminorum)

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. [note: this argument is now valid but unsound because the major premise is untrue]

The fallacy of four terms also applies to syllogisms that contain five or six terms.

Question 33

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.

Answer 33

Fallacy of four terms (quaternio terminorum)

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.”).

Question 34

a categorical syllogism that is invalid because its major term is not distributed in the major premise but distributed in the conclusion.

Answer 34

Illicit major: a formal fallacy committed in a categorical syllogism that is invalid because its major term is undistributed in the major premise but distributed in the conclusion.

Question 35

This fallacy has the following argument form:

All A are B
No C are A
Therefore, no C are B

Answer 35

Illicit major

Question 36

Example:

All dogs are mammals
No cats are dogs
Therefore, no cats are mammals

Answer 36

Illicit major:

In this argument, the major term is “mammals”. This is distributed in the conclusion (the last statement) because we are making a claim about a property of all mammals: that they are not cats. However, it is not distributed in the major premise (the first statement) where we are only talking about a property of some mammals: Only some mammals are dogs.

The error is in assuming that the converse of the first statement (that all mammals are dogs) is also true.

However, an argument in the following form differs from the above, and is valid (Camestres):

All A are B
No B are C
Therefore, no C are A

Question 37

a categorical syllogism that is invalid because its minor term is not distributed in the minor premise but distributed in the conclusion.

Answer 37

Illicit minor: 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.

Question 38

This fallacy has the following argument form:

All A are B.
All A are C.
Therefore, all C are B.

Answer 38

Illicit minor:

Question 39

Example:

All cats are felines.
All cats are mammals.
Therefore, all mammals are felines.

Answer 39

Illicit minor:

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.

Question 40

– when a categorical syllogism has a negative conclusion but affirmative premises.

Answer 40

Negative conclusion from affirmative premises (illicit affirmative):

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.

Question 41

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)

Answer 41

Negative conclusion from affirmative premises (illicit affirmative):

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.)

Question 42

Example (invalid aae form):

Premise: All colonels are officers.
Premise: All officers are soldiers.
Conclusion: Therefore, no colonels are soldiers.

Answer 42

Negative conclusion from affirmative premises (illicit affirmative):

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.

Question 43

Invalid aao-4 form:

All A is B.
All B is C.
Therefore, some C is not A.

Answer 43

Negative conclusion from affirmative premises (illicit affirmative):

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.

Question 44

– the middle term in a categorical syllogism is not distributed.

Answer 44

Fallacy of the undistributed middle:

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.

Question 45

In this example, distribution is marked in capital:

All Z is b
All Y is b
Therefore, all Y is Z

Answer 45

Fallacy of the undistributed middle:

The fallacy of the undistributed middle occurs when the term that links the two premises is never distributed.

b is the common term between the two premises (the middle term) but is never distributed, so this syllogism is invalid.

Question 46

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

Answer 46

Fallacy of the undistributed middle:

The middle term, Z ,is distributed, but Y is distributed in the conclusion and not in any premise, so this syllogism is invalid.

Question 47

1. All students carry backpacks.
2. My grandfather carries a backpack.
3. Therefore, my grandfather is a student.

or:

1. All students carry backpacks.
2. My grandfather carries a backpack.
3. Everyone who carries a backpack is a student.
4. Therefore, my grandfather is a student.

Answer 47

Fallacy of the undistributed middle:

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:

1. All students carry backpacks.
2. My grandfather is a student.
3. 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