Now that we’ve done the contradictory of a conjunction, the contradictory of a disjunction will be no problem.
“Dan will either go to law school or become a priest”.
This is a disjunction. What am I saying when I say that this disjunction is false?
The disjunction says that either one, or the other, or both of these are true — Dan will either go to law school, or he’ll become a priest, or both.
If this is false, that means that Dan doesn’t do any of these things. He doesn’t go to law school, and he doesn’t become a priest.
So the contradictory looks like this:
“Dan will not go to law school AND Dan will not become a priest.”
The disjunction has become a conjunction, with each of the conjuncts negated. This is structurally identical to the rule we saw in the previous video, with the “OR” and the “AND” switched.
In English we have a natural construction that is equivalent to this conjunction:
“Dan will neither go to law school nor become a priest.”
Remember this translation rule for “neither … nor …”:
“(not-A) and (not-B)” is the same as “neither A nor B”.
Don’t be fooled by the “or” in “nor” — this not a disjunction, it’s a conjunction.
Let’s put the rules for the contradictory of the conjunction and the disjunction side-by-side, so we can appreciate the formal similarity:
not-(A and B) = (not-A) or (not-B)
not-(A or B) = (not-A) and (not-B)
In propositional logic these together are known as DeMorgan’s Rules or DeMorgan’s Laws, named after Augustus DeMorgan who formalized these rules in propositional logic in the 19th century.
They’re also part of Boolean logic, and are used all the time in computer science and electrical engineering in the design of digital circuits.
That’s it. These are the rules you need to know to construct the contradictories of conjunctions and disjunctions.