Converse Errors
A converse error treats a rule as equivalent to its reversal. For general claims, the original rule may hold while its converse fails, or vice versa; both may also hold or both fail. A necessary condition is not automatically sufficient.
THE IDEA
CONVERSE ERRORS
Converse errors occur when someone makes the mistake of thinking the converse of a statement is equivalent to the statement. It's an understandable mistake to make in a way, given that there are eight ways of saying "A implies B" and eight ways of saying the converse. This happens when you tell students that they have to work hard in order to do well, and then they think that if they work hard they should automatically do well. Working hard is a necessary but not sufficient condition for doing well. It is not sufficient because you also have to work hard in the right sort of way, and if you think otherwise you are making a converse error.
For general rules such as the examples below, knowing whether the original rule holds does not settle whether its converse holds. Across such rules, all combinations of truth and falsity are possible:
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If you are a US citizen then you can legally live in the US. This is a true logical implication. The converse is "if you can legally live in the US then you are a US citizen". This is not true, as you can legally live in the US by being a permanent resident or having a visa.
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If you have a university degree then you are intelligent. I do not believe this is true. I think there are some degrees awarded to people who are not intelligent; sadly the pass mark is very low. The converse is "if you are intelligent then you have a university degree". This is also not true – I think there are intelligent people who do not have a university degree, especially in older generations when going to university was not such a standard next step in life.
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If you have experienced prejudice then you are a woman. This is not true; men and gender non-binary people can experience prejudice; gender non-binary people certainly do. The converse is "if you are a woman then you have experienced prejudice". I think this is true, whether or not you have ever acknowledged or complained about it.
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If you support Obamacare then you support the Affordable Care Act. This is true because Obamacare is simply an informal name for the Affordable Care Act. This means that the converse is also true, "If you support the Affordable Care Act then you support Obamacare". Unfortunately there are people who support ACA but refuse to support Obamacare, not realising that these are the same thing. They have such a strong objection to anything to do with Obama that rebranding something with his name in it was enough to put them off. This is eye-opening, and I think we can learn important lessons from it to do with how the way we present things really does matter and can override even very clear-cut logic.
We can sum up those conclusions in this table:
| Original | Converse | |
|---|---|---|
| Statement 1 | true | false |
| Statement 2 | false | false |
| Statement 3 | false | true |
| Statement 4 | true | true |
These examples cover all four combinations for a general rule and its converse. The reversal needs its own justification; the truth or falsity of the original rule alone does not settle it.
When to use it
When testing a hypothesis, avoid treating an observation it predicts as sufficient proof that the hypothesis is true.
In education, remind students that studying hard is necessary but not sufficient for good grades to prevent the converse error that studying guarantees success.
In business, do not assume that a competitor's successful strategy will work for your organization without considering different conditions.
When a qualification is required for a role, avoid assuming that holding it is enough to perform the role well.
FROM THEORY TO EVERYDAY LIFE
In practice
Example 1: US citizenship and legal residence
If you are a US citizen then you can legally live in the US. This is a true logical implication. The converse is "if you can legally live in the US then you are a US citizen". This is not true, as you can legally live in the US by being a permanent resident or having a visa.
Example 2: University degree and intelligence
If you have a university degree then you are intelligent. This is false because some degree holders are not intelligent. The converse is "if you are intelligent then you have a university degree". This is also false because many intelligent people lack a degree.
Example 3: Prejudice and gender
If you have experienced prejudice then you are a woman. False; men and non-binary people can experience prejudice. The converse is "if you are a woman then you have experienced prejudice". Cheng treats this converse as true; that is her judgment about experience of prejudice, rather than a truth established by the logical form.
Example 4: Obamacare and Affordable Care Act
If you support Obamacare then you support the Affordable Care Act. True because they are the same. The converse is also true. Yet some people support ACA but reject Obamacare due to emotional associations.
A MOMENT TO REFLECT
Put it to work.
Can you recall a situation where you assumed the converse of a statement was true? What was the outcome?
How does the distinction between necessary and sufficient conditions help avoid converse errors in decision-making?
In what ways can emotional bias cause someone to reject a logically valid converse, as in the Obamacare example?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)