Not just what your country’s economy is based on, nor a criticism of free trade, but a new substitution rule. Our second last, in fact. If you’ve been following along, you’ve learned almost all of the essential rules of two-value first order logic. If you haven’t been, then welcome to Concept Crucible, and you can click here to get caught up. Now, on to exportation, starring my special guest, the friendly neighborhood Spiderman!
Exportation looks complicated, but is actually pretty straightforward. The most important thing to remember is to make sure that the brackets end up in the right place. Exportation works with conjunctions and material conditionals to turn this:
If Curt Connors is the Lizard and he wants to rule the world, then he will fight Spiderman. ((L ∧ R) → F)
If Curt Connors is the Lizard, then if he wants to rule the world, then he will fight Spiderman. ((L → (R → F)
Note how the brackets shift around R. Instead of being conjoined with L, it implies F. A common mistake is to write it as (L rarr R) rarr F), but this isn’t equivalent to the original premise. Let’s show this in action in a proof.
1. If Spiderman has great power and does not have great responsibility then he is a villain. (P ∧ ~R) → V)
2. Spiderman has great power and is not a villain. (P ∧ ~V)
3. Therefore, Spiderman has great responsibility. (R)
4. If Spiderman has great power, then if he does not have great responsibility then he is a villain. (P → (~R → V), Exportation from 1.
5. Spiderman has great power. (P), Simplification from 2.
6. Spiderman is not a villain. (~V), Simplification from 2.
7. If Spiderman does not have great responsibility, then he is a villain. (~R → V), Modus Ponens from 4 and 5.
8. If Spiderman is not a villain, then he does not not have great responsibility. (~V → ~~R), Contraposition from 7.
9. Spiderman does not not have great responsibility. (~~R), Modus Ponens from 6 and 8.
10. Spiderman has great responsibility. (R), Double Negation from 9.
So as you can see, with great power comes great responsibility. I admit that I haven’t actually seen the new Spiderman movie yet, but it’s something worth thinking about anyway. So that’s exportation, and leaves us with just one more substitution rules, one of the trickiest, and also the only one named after a person rather than a function. Get ready for DeMorgan’s!