LESSON #27
Rules of Implication II
Reading Assignment: 7.2 (pp. 374-380)
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Constructive Dilemma:
To do constructive dilemma you need two "É " statements connected by an "· ". You also need an "v" statement made up of the antecedents of both the "É " statements. Then you can conclude with an "v" statement made up of the consequents of the "É " statements. This is a pair if modus ponens arguments, linked by the fact that you don’t know which of the antecedents is true, so you don’t know which of the consequents is. (p É q) · ( r É s)
p v r
q v s
Simplification:
This one is simple. Simplification is like a freebie. You should always begin with simplification if you have it. All you need is a "· " statement. You can then break it apart into separate parts. This works because the only way for an "· " to be true is when both sides are true. So anytime you have an "· " statement you already know that each side of the "· " is true. As with DS, though, by convention, we only get to derive the left conjunct by this rule, to get the right one we will use a rule we will learn in the next lesson. p · q
p
Conjunction:
This one is also simple. Anytime you have two things that you know are true, you can put them together with an "· ." p
q
p · q
Addition: "the black magic rule"
This is simple to do, but it feels like cheating. When you have something you know is true, then you can add absolutely anything else to it with an "v". This is because all it takes for an "v" to be true is that at least one side is true. Well, if you are starting with something you already know is true, then the "v" statement will be true no matter what you add to it. This one is often the one to use if it looks like you can't do anything. p
p v x
Don't get the names of conjunction and addition confused. Memorize them.
NOTE: Notice that you only require one line number for the justification with simplification and addition. All the others, so far, need two line numbers
Logic Coach Assignment: 7.2 I all, II 1-15, III 1-4
Assignment: (20 points each)
Use the first eight rules of inference to derive the conclusions of the following symbolized arguments.
NOTE: Be sure to copy the problems down neatly and correctly.
| A) | 1. | (L v T) É (B ·G) | |
| 2. | L ·(K º R) | // L· B | |
| B) | 1. | F É (D É E) | |
| 2. | F v (H ·G ) | ||
| 3. | ~(D É E) | // H | |
| C) | 1. | (~A v X) É (D v E) | |
| 2. | A É D | ||
| 3. | ~D | // E | |
| D) | 1. | (A É D) · B | |
| 2. | C v (D É C) | ||
| 3. | ~C | // ~A v X |
E) If half the nation suffers from depression, then if either the insurance companies have their way or the psychiatrists have their way, then everyone will be taking antidepressant drugs. If either half the nation suffers from depression or sufferers want a real cure, then it is not the case that everyone will be taking antidepressant drugs. Half the nation suffers from depression. Therefore, it is not the case that either the insurance companies or the psychiatrists will have their way. (H, I, P, E, W)