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}

This statement is not necessarily true; it does not follow logically from the first implication. The grass could have become wet through other means, or it could have stopped raining. This can be seen easily by diligently constructing a truth table. The implication says nothing about the grass being wet if it is not raining, so the grass can be either wet or dry if it is not raining (the first two rows). If it is raining, then the grass cannot be dry, as that would contradict the implication (the implication would be false, third row). If it is raining, clearly the grass can, and should, be wet (the fourth row).

it rains the grass is wet it rains → the grass is wet
false false true
false true true
true false false
true true true

If you look at the rows in this table where the grass is wet is false and the implication is true, you see that the only possibility is that it isn’t raining. If you look at all the rows where both the grass is wet is true and the implication is true, you see that there are two possibilities. It can be raining, but it’s also possible that it isn’t raining. The mistake people sometimes make — forgetting about the negative option — is understandable; and certainly, the grass being wet makes it more probable that it actually is raining. However, one should take care not to make these kinds of mistakes. Logic is a basis of our reasoning, and we should use it with care.

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