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Why is any implication true when the antecedent is false?


The question is misconceived.

First, some terminology. A conditional statement is one of the form ‘if p, then q’, e.g. ‘if JM has three cars, then JM has an odd number of cars.’

Second, the ‘antecedent’ of a conditional statement is the ‘if’ part, and the ‘consequent’ is the ‘then’ part. So in ‘if JM has three cars, then JM has an odd number of cars’, the antecedent is “JM has three cars’ and the consequent is “JM has an odd number of cars.”

That particular conditional is true, even though the antecedent is false. It is true that if JM has three cars, then JM has an odd number of cars. It would be true if I had 0 cars, 1 car, 2 cars, 3 cars, etc. All that matters is the relationship between antecedent and consequent.

The truth-values of antecedent and consequent are irrelevant, with the qualification that, if the conditional is true, then the consequent cannot be false if the antecedent is true.

Suppose I had zero cars. In that case, both antecedent and consequent would be false, but the conditional would still be true.

Suppose I had one car. In that case, the antecedent would be false and the consequent would be true, and the statement as a whole would be true.

Suppose I had two cars. In that case, both antecedent and consequent would be false, but the conditional would still be true.

Suppose I had three cars. In that case, both antecedent and consequent would be true and the conditional would be true.

By the same token, even if both antecedent and consequent are true, the conditional may be false. For example, “if Warren Buffet is from Nebraska, then Warren Buffet is a billionaire” is a false conditional with a true antecedent and a true consequent.

As the term “if” is used in elementary logic classes, any conditional is true so long as it doesn’t have a false consequent and a true antecedent. But that is an artifact of a neologism and is of no philosophical substance.

Your question seems to be motivated in part by a misunderstanding of this last fact.

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