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# Denying the Antecedent

_{ Jan 01 . 1970 }

Any argument of the following form is invalid: (1) If A then B (2) Not A (3) Therefore, Not B. (Also known as the “inverse error” or “inverse fallacy”.)

## Examples

### Non cogito, ergo non sum.

“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.”

“Cogito Ergo Sum”at New World Encyclopedia

The claim is: “x can get pregnant only if x is a woman.” This CNN author says this “conveys” that: “if x can’t get pregnant, then x is not a woman.” We call this the inverse of the original conditional. It definitely doesn’t follow! We cover that — no joke — in week 2 of Logic.

Tomas Bogardus at X (May 6, 2024)

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.

## Critique

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.

## Commentary

See, The Law of the Contrapositive

### On if an only if antecedents

[It’s] a formal fallacy with the form: If p then q not p

therefore not q Like the fallacy of affirming the consequent, it treats ‘if’ as if it means if and only if. For example, the following is an example of denying the antecedent: P) If the share prices rise, Q) then you’ll get rich. ∴ The share prices haven’t risen.

So you won’t get rich.In this example, it is possible for you to get rich despite the fact that the share prices haven’t risen. Rising share prices aren’t the only mechanism by which people get rich. Or consider another example: P) If you add horse manure to the soil Q} you will increase your yield of vegetables. ∴ You haven’t added horse manure to the soil.

So you won’t increase your yield of vegetables. Again, adding horse manure to the soil is not the only way of increasing the yield of vegetables: you can add compost, seaweed, pig manure and all kinds of inorganic fertilisers. So the conclusion

doesn’t follow logically from the premises: it is a non sequitur. In some cases, the context and subject matter of the argument make it clear that ‘if’ is to to be understood as meaning ‘if and only if. These are not cases of denying the antecedent. For example, in the following, ‘if’ can only mean ‘if and only if’:P) If you have a ticket for the national lottery you’ll stand a chance of winning. Q) You haven’t got a ticket. ∴ So you don’t stand a chance of winning. This is a valid argument (see validity) since the only way of standing a chance of winning the national lottery is by having a ticket.

Nigel Warburton,Thinking from A to Z(), pp. 47-8.