Formal Logic 101 – Part 10: Replacement Rules (Part 1)

I mentioned previously (clear back in Part 3) that one of the benefits of symbolization is that plain English has several different ways of stating the same thing, and this can make it difficult to trace the line of reasoning without doing something to burn off the dead wood. While symbolization certainly helps with this, it does not create a “perfect” system where there is only one correct way to symbolize any given statement. There are still a few instances where one symbolization is logically equivalent to another. Replacement rules help us identify these instances where one phrasing can be replaced with another equivalent phrasing, thereby making proofs much shorter. As we go through these, do keep in mind that replacement rules are reversible. Since they are essentially different ways of phrasing the same thing, you can always swap back and forth. Also, because we are not reaching “new” conclusions, but merely re-stating old ones, replacement rules can be used on parts of lines (this will be covered in more detail in the next post, for now just keep it in the back of your mind somewhere.)

Double Negation (DN)

Let’s start with the simplest one. From time to time, you’ll find yourself in a position where you find yourself faced with a double negative such as ~~A. As you might expect, in logic two negatives makes a positive. So Double Negation simply states “~~A” is equivalent to A”
1. ~~A        Premise (P)
2. A             1 DN
Of course, since it’s reversible, you could equally say:
1. A             P
2. ~~A        1 DN
Both applications of the rule are correct.

Do note however that Double Negation does not cross parentheses. Consider the phrase:
“It’s not true that I will either not get paid or I will do the job.”

If P is “I will get paid” and J is “I will do the job”, this is symbolized as
1. ~(~PvJ)        P

We cannot conclude:
2. PvJ                1 DN (Invalid)

Remember from our previous post that when a “Not” is outside the parentheses, it is negating the operator. The first “Not” is not saying anything about whether I’m getting paid, it’s denying that this is an “Or” situation. Meanwhile, the second “Not” is concerned with whether I’m getting paid. So this is technically not a double negative, so Double Negation does not apply.

Duplication (Dup)

Duplication is a rule that typically only gets used in the middle of a proof to tidy up a bit. Duplication states that “A^A” is equivalent to “A”, and also states that “AvA” is equivalent to “A”.

1. AvA            P
2. A                1 Dup

or

1. A^A            P
2. A                1 Dup

Consider the phrases:
1. Abe Lincoln was president during the American Civil War, and also Abe Lincoln was president during the American Civil War.
2. Either Abe Lincoln was president during the American Civil War, or Abe Lincoln was president during the American Civil War.

In both cases, I know I read those sentences and think “Ok, apparently you needed ten more words to meet your word count so the professor didn’t mark you down for it… got it.” Knowing that “And” says “both of these are true” and “Or” says “at least one of these are true”, having the same thing listed twice is superfluous and needlessly messy. So you can just shrink it down to just listing that thing once.

Again, it is reversible, you can go the other way, but I recommend having a plan for why you feel like you need to do that before doing it.

Association (Assoc.)

Association is the last of our “tidying up” rules. With Association we can have a three-proposition sentence, where the propositions are either both separated by "Or", or both separated by "And", and we can move the parentheses around.

1. (AvB) v C         P
2. A v (BvC)         1 Assoc.

or

1. (A^B) ^ C         P
2. A ^ (B^C)         1 Assoc.

If we look at the phrase “Palpatine was a Senator, and the Chancellor, and a Dark lord of the Sith” it should be apparent that it doesn’t really matter if we symbolize that as (S^C)^D or as S^(C^D), the meaning is the same.

To show an example of where this doesn’t work, let’s look at the phrase “Either Palpatine was the Chancellor and a Dark Lord of the Sith, or Count Dooku engineered the whole civil war.” Let’s use C for “Palpatine was the Chancellor”, L for “Palpatine was a Dark Lord of the Sith”, and D for “Dooku engineered the war.”

We would thus symbolize as:
1. (C^L) v D         P

Now let’s say we use association wrongly to conclude:
2. C ^ (LvD)         1 Assoc. (Invalid)

If we translate back to plain English, we should note that our premise stated “either these two things are true, or this third thing is true, or maybe all three, we’re not sure.” While our conclusion states “This one thing is absolutely true, and one or both of these other two things might be true,” which is quite different from where we started. Remember that replacement rules should leave us with a phrase with identical meaning. So if you use a replacement rule and end up with a totally different meaning, something was done wrong.

That’s the end of the “tidying up” rules. The remainder of the replacement rules will be short cuts to make our lives easier. I’m not including practice problems with this one, just because most problems that require use of these rules will also require other replacement rules, and I’d rather not jump the gun. I'll resume practice problems later on once we have enough replacement rules for it to be feasible

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