Denying a Conjunct

The train only had one more seat available, so Ava and Min couldn’t have both gotten on. Ava didn’t get on, so Min did.

Denying a conjunct takes this form:
Not both P and Q.
Not P.
Therefore, Q.

In the previous example,
P: Ava got on the train.
Q: Min got on the train.

It is a fallacy because someone else might have gotten the last seat.

To explain better, it’s worth introducing some logical symbols:

∧ means logical conjunction.
It’s similar to AND.
Sometimes it’s written as &.

P∧Q means both Ava and Min got on the train.

¬ means negation.
It’s similar to NOT.
Sometimes it’s written as ~.

¬(P∧Q) means it is not the case that both Ava and Min got on the train.

Note: Parentheses are used to group terms just like in math. ¬P∧Q means Ava did not get on the train and Min did!

Now we can rewrite the fallacy with symbols:

¬(P∧Q)
¬P
Therefore, Q.

¬(P∧Q) tells us that P and Q can’t both be true, but it doesn’t tell us that one of them must be true. They can both be false.

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