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Denying the Antecedent

Any argument of the following form is invalid: (1) If A then B (2) Not A (3) Therefore, Not B


“Some non-philosophers who first come across the cogito attempt to refute it in the following way. ‘I think, therefore I exist’, they argue, can be reversed as ‘I do not think, therefore I do not exist.’ They argue that a rock does not think, but it still exists, which disproves Descartes’ argument. However, this is the logical fallacy of ‘denying the antecedent’. The correct corollary by modus tollens is ‘I do not exist, therefore I do not think’. This fallacy and its prevalence is illustrated by the popular joke: Descartes is sitting in a bar, having a drink. The bartender asks him if he would like another. ‘I think not’, he says, and vanishes in a puff of logic.” (New World Encyclopedia)

If you get hit by a car when you are six then you will die young. But you were not hit by a car when you were six. Thus you will not die young. (Of course, you could be hit by a train at age seven, in which case you still die young.)

If I am in Calgary then I am in Alberta. I am not in Calgary, thus, I am not in Alberta.


Show that even though the premises are true, the conclusion may be false. In particular, show that the consequence B may occur even though A does not occur.