This mistake occurs when the conversion rule from categorical logic, which is a type of deductive logic, is used improperly.
In deductive logic, conversion is a rule that allows the subject and predicate claims of a categorical claim to be exchanged. As with most rules, it has correct and incorrect applications. In the case of conversion, the correctness of the application depends on what sort of claim is subjected to the rule.
In categorical logic, there are four sentence types: All S are P, No S are P, Some S are P, and Some S are not P. C applies correctly to two of them: No S are P and Some S are P. A conversion is legitimate when the converted claim logically follows from the original (and vice versa). Put another way, the rule is applied correctly when its application does not change the truth value of the claim.
For example, “No cats are hamsters” converts legitimately to “no hamsters are cats.” Interestingly, “some dogs are huskies” converts correctly to “some huskies are dogs”, at least in categorical logic.
In categorical logic, “some” means “at least one.” Hence, “at least one dog is a husky” is converted to “at least one husky is a dog.” In this case, the inference from one to the other is legitimate because it is made in the context of categorical logic.
The illicit use of conversion is an error. This error occurs in two ways. The first is when the rule is applied incorrectly in the context of categorical logic: if conversion is applied to an All S are P or Some S are not P claim, then the rule has been applied improperly. This can be easily shown by the following examples.
The first example is that while it is true that all dogs are mammals, the conversion of this claim, all mammals are dogs, is not true. As another example, the claim that some dogs are not huskies is true while its conversion, some huskies are not dogs, is false. This sort of mistaken application of the conversion rule can also be presented as a fallacious line of reasoning, as shown by the following flawed inference patterns:
Fallacious Pattern #1
Premise: All S are P
Conclusion: All P are S
Fallacious Pattern #2
Premise: Some S are not P
Conclusion: Some P are not S
The second type of error occurs when the conversion rule is applied outside of the context of categorical logic as if it were being applied within such a context. That is, it occurs in contexts in which “some” does not mean “at least one.” The mistake, which is sometimes known as an illicit inductive conversion, is as follows:
Fallacious Pattern #3
Premise: P% (or “some”, “few”, “most”, “many”, etc.) of Xs are Ys.
Conclusion: Therefore P% (or “some”, etc.) of Ys are Xs.
For example, to infer that most people who speak English are from Maine because most people from Maine speak English would be an obvious error. This is because “most” in this context is not taken to mean “at least one” but is instead taken to refer to a majority.
While people usually do not make such obvious errors, they can fall victim to conversions that seem plausible. For example, when people hear that a medical test for a heart condition is 80% accurate, they might be tempted to infer that 80% of those who test positive have the condition. However, to convert “80% of those who have the condition will test positive” (that is what it means for a test to be 80% accurate) to “80% of those who test positive have the condition” is an illicit use of conversion.
“Very few white men have been President of the United States. Therefore, very few Presidents have been white men.”
“A small percentage of automobile accidents involve drivers over 70. Therefore, a small percentage of drivers over 70 are involved in automobile accidents.”
“Most conservatives are not media personalities on Fox News. Therefore, most of the media personalities on Fox News are not conservative.”
“Most wealthy people are men, so most men are wealthy.”
“Most modern terrorists are Muslims, therefore most Muslims are modern terrorists.”
“Most modern terrorists are religious people, therefore most religious people are terrorists.”
Originally appeared on A Philosopher’s Blog Read More