Mistaken Logical Implication

One of the most common logical inferences uses logical implication. For example, you know that if it rains then the grass will be wet. If you look outside and see that it rains, you do not have to look at the grass to know that it is wet. This inference is called modus ponens: if A implies B and A is true, then B is true. Formally, the implication can be written as:

  \text{it rains} \to \text{the grass is wet}

The modus ponens belonging to this implication can be written as:

  \frac{\text{it rains} \to \text{the grass is wet},\; \text{it rains}}{\text{the grass is wet}}

A commonly made mistake is to erroneously also assume the opposite: if the grass is wet it, is raining. This is called the converse:

  \text{the grass is wet} \to \text{it rains}

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