A Lesson in Logic
The final thing worth discussing before moving into substitution rules is contradictions. What they are, how they’re formulated, and why they’re a problem in most instances of logic.
Contradiction came up way back when we talked about conjunction, as one of the laws of logic. The Law of Non-Contradiction says that A and ~A can’t both be true. This seems intuitive, if we think about A as “It is raining”. It can’t be both raining and not raining in the same place, at the same time. Another example of a proposition is “Cats are mammals”. If it’s true that cats are mammals, then it must necessarily be false that cats are not mammals. The law says that this is true for all propositions. Now, Professor Graham Priest has done some work on picking out whether or not this is actually true, and argues for a view that there might be some propositions which can be both true and not true, but for our purposes, it’s safe to say the law of non-contradiction applies.
Contradictions present a problem for what we’ve learned so far not just because they involve insisting that two contradictory things are true, but because, using the rules of induction, you can literally prove anything to be true from a contradiction. Watch.
1. (C ~C) Cats are mammals and cats are not mammals.
2. (C D) If cats are mammals, then dogs are mammals.
3. (D) Therefore, dogs are mammals.
Looks pretty straightforward. Let’s prove it.
4. C – Simplification from 1
5. ~C – Simplification from 1
6. C F – Addition to 4.
7. F – Disjunctive Syllogism from 5 and 6.
Wait, F? I thought we were going for D? But there it is, entering by addition. Through addition and the disjunctive syllogism, you can prove anything from a contradiction and have it be a completely valid argument. It will never be sound (or rarely, according to Priest), but but follows all of the right rules. From that contradiction, we can prove other contradictions. We can prove things until the cows come home. So in order to preserve the rules and have them be useful, we can’t let contradictions participate in legitimate proofs. And with contradictions under wraps, you’re ready to learn some substitution rules. See you next week!