7.3 lecture 1 PDF

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Lecture Notes for Chapter 7.3...

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7.3 rules are different from your previous two sections. These are called “rules of replacement”. There are two differences:  They go in both directions.  You can use them anywhere in a sentence, even in parentheses. For example, remember from 7.2, you learned the rule Simplification: p·q p  This rule only goes in one direction – from the top down. So, if you’ve got p · q on a line, you can use Simp to get p. However, you cannot start with the p and try to derive p · q.  You also cannot use Simp on a part of the sentence trapped in parentheses. You can only use it if the dot is the main connective. Neither of these restrictions holds in 7.3 rules. What we have in this chapter are five pairs of logically equivalent sentences. So, you can switch out one sentence with the one that it’s logically equivalent to, even if that sentence occurs as part of a larger sentence. You’ll get the hang of it once we have a few to practice on, so let’s learn the five rules. Double Negation (DN) p :: ~~p This rule tells us what your English teacher has told you all along: a double negative equals a positive. What this means is that if we have two tildes next to each other, we can drop both tildes. It also means that we can add two tildes if we need them. And, we can add them anywhere we want to. So, let’s practice adding two tildes – I’ll give you a sentence, then show you where you can apply DN.

1. E ⊃ (A · C)

2. ~~[E ⊃ (A · C)] 3. ~~E ⊃ (A · C) 4. E ⊃ ~~(A · C) 5. E ⊃ (~~A · C) 6. E ⊃ (A · ~~C)

1, DN 1, DN 1, DN 1, DN 1, DN

(applied to the whole sentence) (applied to just the E) (applied to A · C) (applied to just the A) (applied to just the C)

I can do any or all of these. (Though if I do all of them, I have to do them one at a time, each on a separate line.) Make sure that you are only adding two tildes, and that you’re adding them to the same thing. Here are some common mistakes when applying DN: 7. E ⊃ ~(~A · C)

NOT an accurate use of DN.

Don’t make this mistake. Notice that the parentheses interrupts the two tildes. This means they’re not even T to the same thing – the first one attaches to the compound sentence A and C, and the second one attaches just to the A. 8. E ⊃ (~A · ~C)

NOT an accurate use of DN.

This is also invalid. The two tildes don’t attach to the same thing. Getting rid of two tildes is easy – anywhere I see two tildes right next to each other, I can drop them, using DN. 1. (S v ~~T) ⊃ M 2. (S v T) ⊃ M 1, DN

Dropping the 2 tildes on the T.

Just make sure they really do attach to the same thing. They cannot be attached to different letters, and they cannot have parentheses coming between them.

It is also important to remember that you can only add or drop tildes in pairs. Never, ever drop or add just one tilde. That’s like saying

“My brother is tall” means the same thing as “My brother is not tall.” They mean exactly the opposite from each other! Double negation, on the other hand, turns “It is false that my brother is not tall” into just “My brother is tall.” I dropped both of the negations. Time for another rule! Commutativity (com) p · q :: q · p p v q :: q v p Commutativity tells me that order does not matter when I have an “and” or an “or” statement. “My brother and sister are tall” means the same thing as “My sister and brother are tall.” So, in a proof, I can rotate things around a dot or a wedge if I need to. Like all of the rules in this chapter, we can work it even inside parentheses. So, here are places I can use Com: 1. (S v T) v (L v M) 2. (L v M) v (S v T) 3. (T v S) v (L v M) 4. (S v T) v (M v L)

1, com (rotating around the main connective) 1, com (rotating S v T, inside parentheses) 1, com (rotating L v M, inside parentheses)

Notice that whether I rotate around a main connective, or rotate inside parentheses, the rest of the sentence has to be left exactly the same. Important: only use this with dot and wedge. Never, never use it with the horseshoe. Remember, order matters with a horseshoe – with a conditional, there’s a difference between the antecedent and the consequent.

DeMorgan’s Law (DM)

~(p · q) :: ~p v ~q ~(p v q) :: ~p · ~q This rule tells me that “neither my brother nor sister are tall” means the same thing as “my brother is not tall, and my sister is not tall.” Both of these phrases mean that I have zero tall siblings. It also tells me that “It’s false that my brother and sister are both tall” means the same thing as “Either my brother is not tall, or my sister is not tall.” Both mean that I do not have two tall siblings, someone is short. The short one could be my brother, could be my sister, or could be both of them. Symbolically, this means that I can push a tilde through parentheses, if there is a dot or wedge inside – I just have to remember to change the connective. Also, if I have two things with tildes on them, and they’re connected by a dot or a wedge, I can pull the tildes off of both sides and stick them to parentheses, as long as I change the connective. Associativity (assoc) p · (q · r) :: (p · q) · r p v (q v r) :: (p v q) v r This is a “regrouping” rule. If you have dots all the way through, or wedges all the way through, then you can move the parentheses. Make sure that ALL you change is the parentheses, and make sure it’s dots both times or wedges both times. Distribution (dist) p · (q v r) :: (p · q) v (p · r) p v (q · r) :: (p v q) · (p v r) So, what if you don’t have two dots, or two wedges – but a mix of them both? In that case, you can distribute. Think of distribution as sort of like DeMorgan’s Law, only instead of pushing a tilde through the parentheses, you’re pushing a “p ·” or a “p v”. On the top row of the rule, notice that

both the q and the r get their own “p-dot”, and the wedge becomes the main connective. Don’t forget that like all of these rules, they go in both directions. So if you start with two conjunctions, connected by a wedge, and they have a shared term, you can pull that shared term out and put it on the front. A couple of places this is helpful: Suppose I’ve got (A · B) v (A · C) and I just want the A by itself. I can’t do simplification inside parentheses. But if I do a reverse distribution, I can pull that A out of parentheses, and end up with: A · (B v C). Now I can simplify, and end up just with my A. Also, suppose I’ve got M v (B · S) – either my mom is tall, or both my brother and sister are. I want to get rid of my brother, and focus just on how either my mom or my sister is tall. Distribution is the answer here too. Distribute the M-wedge: (M v B) · (M v S). Use commutativity to switch the two statements around the dot: (M v S) · (M v B). Then we use Simp to get M v S by itself.

In the next lecture, we’ll apply these rules into proofs. Remember, you will really only get the hang of how to use these rules by using them – and the more you practice, the more you’ll understand what’s going on....