Logic 16 Implication b - Grade A+ PDF

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NATURAL DEDUCTION: RULES OF IMPLICATION 2 So far, we’ve learned four rules of implication: modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism. In this lecture, we’ll cover four additional rules. Name Simplification

Abbreviat ion Simp

Conjunction

Conj

Addition

Add

Constructive Dilemma

CD

Form

Sample Proof 1

p∙ q ∴p p q ∴ p∙q p ∴ p∨q

1. (A ≡ B) ∙ C / (A ≡ B) 2. A ≡ B 1, Simp

( p⊃ q )∙ ( r ⊃ s )

1. 2. 3. 4. 5.

p∨ r ∴q∨s

1. C ∨ D 2. A ⊃ B /(A ⊃ B) ∙ (C ∨ D) 3. (A ⊃ B) ∙ (C ∨ D) 2,1 Conj 1. E / E ∨ (D ≡ F) 2. E ∨ (D ≡ F) 1, Add C ∨ ~D C⊃A ~D ⊃ ~B / A ∨ ~B (C ⊃ A) ∙ (~D ⊃ ~B) 2,3 Conj A ∨ ~B 4,1 CD

Here’s the basic “idea” behind each rule:    

Simplification. Your premise is that both p and q are true. Your conclusion is that p is true. Easy enough, right? (Later on, we’ll learn a rule that allows you to turn “p ∙ q” into “q ∙ p.” This will allow you to get q as well.) Conjunction. You have two premises: one says that p is true and the other says that q is true. Your conclusion say that p ∙ q is true. Addition. In logic, a disjunction is true if at least one (and maybe both) disjuncts are true. So, if we already know that p is true, then so is p ∨ (WHATEVER YOU WANT!) Constructive dilemma. We talked about this earlier. You’ll often need to use conjunction to “set this up” by putting both of the conditional statements on the same line.

MORE STRATEGIES When working with these new rules, you should always keep in mind the first, most basic: start with the conclusion, and work your way back. Again, it is often helpful to think of the “big” proof as being made up of a number of smaller proofs, each with its own subconclusion. In any case, here are some additional strategies to keep in mind when using the new rules: 1. If (part of) the conclusion appears as part of a conjunction in the premises, consider using simplification to obtain it. See example 1 below. 2. It the conclusion is a conjunction, consider using conjunction to “put together” two premises to produce it. See example 1 below. 3. If (part of) the conclusion is a disjunction, consider using rules like addition or constructive dilemma. See example 2 below. a. Reminder: You’ll often need to use conjunction to set up constructive dilemma. 4. (IMPORTANT) If there is a letter in the conclusion which does not appear in any premise, then you MUST use addition to obtain that letter (at least until you learn conditional and indirect proof). See example 2 below. Example 1. Prove the validity of the following argument: A ∙ B / C // C ∙ B 1. A ∙ B 2. C Conj) 3. A 4. A ∙ C conclusion)

/A∙C 1, Simp 3,2 Conj

(The conclusion is a conjunction. Our plan: get each part, then use (Simplification allows us to get the first conjunct of line 1) (Conjunction allows to put together lines 3 and 2 to get the

Example 2. Prove the validity of the following argument: D ⊃ A / ~E ⊃ ~C / D ∨ ~E // (A ∨ ~C) ∨ B

1. D ⊃ A 2. ~E ⊃ ~C 3. D ∨ ~E / (A ∨ ~C) ∨ B (The conclusion is a disjunction. Plan: Get the first part, then use Add) 4. (D ⊃ A) ∙ (~E ⊃ ~C) 1,2 Conj (We are doing this to set up a constructive dilemma in the next step) 5. A ∨ ~C 4,3 CD (We’ve got the first part of the conclusion!) 6. (A ∨ ~C) ∨ B 5, Add (Because the conclusion is a disjunction, we can use Add to get the 2nd part) COMMON ERRORS When you are first using the rules, it’s easy to make some mistakes. Take a look at the following proof for a few of the most common: 1. 2. 3. 4. 5. 6.

A⊃B ~A (B ⊃ C) ⊃ D ~B ~B ∙ ~C A ⊃ (C ⊃ D)

/ Whatever 1, 2 MP (No! This is the fallacy of denying the antecedent) 4, Add (No! Add only works with disjunction, not conjunction) 1, 3 HS (No! HS doesn’t allow you to rearrange the parentheses)

And here are a few more: 1. 2. 3. 4. 5. 6.

~(A ∙ B) (A ∨ B) ⊃ C ~C ∨ ~D ~C ~A B⊃C lines)

/ Whatever 3, Simp 1, Simp 2, 5 DS

(No! Simp only works with conjunctions, not disjunctions) (No! Simp requires getting rid of the parentheses) (No! Rules of implication never work on parts of lines, only on whole

The last error is especially important, because it represents a key difference between rules of implication (which only on whole lines), and rules of replacement (which can operate on individual parts of lines). REVIEW QUESTIONS Please complete the following proofs by identifying the correct line numbers and rules. Proof 1: 1. F ⊃ (~T ∙ A) 2. (~T ∨ G) ⊃ (H ⊃ T) 3. F ∙ O / ~H ∙ ~T 4. F 5. ~T ∙ A 6. ~T 7. ~T ∨ G 8. H ⊃ T 9. ~H 10. ~H ∙ ~T

(Premise) (Premise) (Premise/Conclusion) 3, Simp (Sample—you do the rest!)

Proof 2: 1. 2. 3. 4. 5. 6. 7. 8. 9.

(F ∙ M) ⊃ (S ∨ T) (~S ∨ A) ⊃ F (~S ∨ B) ⊃ M ~S ∙ G /T ~S ~S ∨ A F ~S ∨ B M

(Premise) (Premise) (Premise) (Premise/Conclusion) 4, Simp (Sample—you do the rest!)

10. F ∙ M 11. S ∨ T 12. T...