Formal Logic 101 – Part 14: Replacement Rules (Part 5)

Our last two replacement rules will be Distribution, and Exportation. I will freely admit that these are the two rules that I’ve found the least use for, but they do exist, and they are occasionally handy.

Distribution

Distribution is useful with three-proposition sentences that use a mixture of “And” and “Or”. For instance the sentence:

Joe Biden is the President, and his eldest son is either named Joseph or Beau.

We can symbolize that as “P^(JvB)”. Now looking at it, the part where Biden is president isn’t a point of contention. That’s just stated as fact. The uncertainty comes in regarding what his eldest son’s name is. Distribution allows us to split the sentence up into (P^J) v (P^B) (Either Biden is the President and his son is named Joseph, or Biden is the President and his son is named Beau). Again, since this is a replacement rule, it also works in reverse.

The other way that Distribution works is a bit less intuitive (to me at least), and states that P v (J^B) is equivalent to (PvJ) ^ (PvB). If you’re like me, you’re thinking “I’m not sure about that one” but if we test it quick, we see that it is correct.

For P v (J^B) to be true, only one side of the “Or” statement to be true. So even if it turned out the Biden’s son was named Thomas, that statement is still true because Biden’s the president. For the (PvJ) ^ (PvB) version, both sides of the “And” statement must be true, but both sides of the “And” statement include “Biden’s the president”, so again, that’s all it takes to make the whole statement true.
On the flip side, Biden’s son’s given name is Joseph, but the name he answers to is Beau, so I think it would be fair to say that “J^B” is true. That’s clearly enough to make P v (J^B) a true statement, and if we check again, it’s enough to make (PvJ) ^ (PvB) true as well.

Since you can use Distribution in this way, and it is truth preserving, it is a valid use of the rule.
(I’m not going to do a formal proof here because – again, I’ve found this to be among the least-used rules in Logic. If you want to try to construct a proof for extra credit, go right ahead.)

Exportation

Finally, we come to Exportation. I think this one will also be best explained by example, rather than definition. Let’s say you’re in the woods. We’re going to have a scavenger hunt. We establish: If you bring back an Egg and a Spoon, then you Win. (symbolized: (E^S)→W)

Let’s say you’re out there and you see an egg high up in a nest. You exclaim “If we get that Egg, then if we find the Spoon then we Win!” Congratulations, you have just done Exportation! Exportation states that “(E^S)→W” is equivalent to “E→(S→W)” (or vice versa). That is the only thing that rule does. Again, I’ve found it to be very niche in its application so I’m not going to dig out the formal proof of the rule, but you can try to craft one yourself for extra credit.



If you’ve made it thus far, congrats, you have now been acquainted with the entirety of Propositional Logic! Next time we’ll start in on Predicate Logic. Don’t worry: We’re not starting from scratch with a whole new system of logic or anything absurd like that. Predicate Logic basically takes everything that we learned in Propositional Logic, but then asks “what if we need to do logic on PARTS of a proposition?” If you’re feeling comfortable with what we’ve been doing so far, it should be a breeze. If you’re not feeling confident with your current understanding… the comments sections is your oyster! 

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