Construct formal proofs for the validity of 3 arguments, below. You will have to add lines and justify them according to the 19 Rules of inference on pages 339, 355, and 356. You will have to read the sections that correspond to these rules to make sense of them. Also, read the Module 7 Lecture and view the videos at the bottom of that "lecture." Further, work on the sample proof given in chapter 9 whose answer is supplied. Otherwise, you will not know how to construct the proofs. Here is a sample argument and the proof (pp. 333-334): Sample Argument 1. A - B 2. B - C 3. C - D 4. ~ D 5. AVE : E Sample Proof 1. A - B 2. B - C 3. C - D 4. ~ D 5. AVE 6. A= C 1, 2, H.S. 7. AND 6, 3, H.S. 8. ~A 7, 4, M.T. . E 5, 8, D.S.Construct formal proofs for the validity of the following 3 arguments: Argument 1: 1. A = (B - C) 2. ~C . ~Av ~B [Hint: use "Exportation," as well as other rules for your proof.] Argument 2: 1. (C ) D) VE 2. C : DVE Argument 3 : 1. (B . C) V ~ A 2. B - D [Hint : You will use Com., Dist., Simp., Imp., and H.S.] Use the following symbols. That is "copy" and "paste" them for your symbolic sentences and tables, as needed: . And V Or 3 If then If and only if ~ Not(p = q) = 1(p >9) . (9> p)] These 356 we kn chapter 9 Methods of Deduction Equiv. ( p = 9 ) = [( p . 9 ) V ( - p . ~ 9 )] clump Equiv 17. Material Equivalence [ (p . q) > r] = [p >(9>r)] group Exp. p = ( PVP ) is true are fall 18. Exportation Taut p = (p . p) 19. Tautology 13. I Of all tautold Let us now examine each of these ten logical equivalences. We will use them fre. junction quently and will rely on them in constructing formal proofs of validity, and therefore we lent to must grasp their force as deeply, and control them as fully, as we do the nine elementary second valid argument forms. We take these ten in order, giving for each the name, the abbreviz In oth tion commonly used for it, and its exact logical form(s). r is tru true an 10. De Morgan's Theorems (De M.) ~ (p . q) = (~ pv ~ q) if p is t ~ ( P V q ) = ( ~ p . ~ 9 ) is true) conjun This logical equivalence was explained in detail in Section 8.10. De Morgan's theorems the firs have two variants. One variant asserts that when we deny that two propositions are both disjund true, that is logically equivalent to asserting that either one of them is false, or the other conjun one is false, or they are both false. (The negation of a conjunction is logically equivalent or both to the disjunction of the negations of the conjuncts.) The second variant of De Morgan's true. T theorems asserts that when we deny that either of two propositions is true, that is log exhibit cally equivalent to asserting that both of them are false. (The negation of a disjunction is logically equivalent to the conjunction of the negations of the disjuncts.) 14. D These two biconditionals are tautologies, of course. That is, the expression of the material equivalence of the two sides of each is always true, and thus can have no fals Intuitiv substitution instance. All ten of the logical equivalences now being recognized as rule alent to of inference are tautological biconditionals in exactly this sense. 15. T 11. Commutation (Com.) (p V q) = (q V p) This log ( p . 9) = (9. P) that if These two equivalences simply assert that the order of statement of the elements must a conjunction, or of a disjunction, does not matter. We are always per conditi remain exactly the same. around, to commute them, because, whichever order happens to appear, the mean of its ar ays permitted to turn the the logi Recall that Rule 7, Simplification, permitted us to pull p from the conjunction but not q. Now, with Commutation, we can always replace p . q with q . p-of the 16. M conjuncts from a conjunction. with Simplification and Commutation both at hand, we can readily infer each of This log tion exp 12. Association (Assoc.) [p V (q Vr)it saw the false or355 9.6 Expanding the Rules of Inference: Replacement Rules least one of the premises to effect the link that makes the deduction possible. De Morgan pointed out that this is not correct if we know that "Most Ps are As" and "Most Ps are Bs." With some quantitative premises we can deduce a connection between As and Bs. Suppose, for example, a ship had been sunk on which there were 1,000 pas- sengers, of whom 700 drowned. If we know that 500 passengers were in their cabins at the time of the tragedy, it follows of necessity that ne at least 200 passengers were drowned in their cabins. This he called the numerically definite syllogism. De Morgan also advanced the field called the logic of relatives. Identity and difference are relations to which logicians have given her great attention, but there are other relations, such as equality, affinity, and especially equivalence, that also deserve the logician's attention, as De Morgan showed. Two logical equivalences, widely useful and intuitively clear, received from De Morgan their time-honored formulation and carry his name: De Morgan's Theorems (explained in Chapter 8 and in this vill chapter), which remain a permanent and prominent instrument in es, deductive reasoning. set overview The Rules of Replacement: Logically Equivalent Truth-Functional Statement Forms Any of the following logically equivalent statement forms may replace each other wherever they occur . Name Abbreviation Form 10. De Morgan's Theorems De M. ~ (p . q) = (- PV - q) ~ ( p V q ) = ( - p . ~ q ) (pv q) = (9 vp) 11. Commutation Com. (p . q) = (9. p) [p V (q V r)] = [(p V q) vr] 12. Association Assoc. [p . (q . r)] = [(p . q) . r] 13. Distribution Dist. [p . (q v r)] = [(p . q) V (p.r)] [p V (q . ")] = [(pv q) . (p vr)] 14. Double Negation D.N. P = ~ ~ p (p > q) = (-q5-p) 15. Transposition Trans 16. Material Implication Impl. ( b / d - ) = ( bcd )339 9.2 The Elementary Valid Argument Forms cause with he readily proved to be valid using a truth table. Each of them is simple and intuitively clear; as a set we will find them powerful as we go on to construct formal proofs for the validity of more extended arguments . the on is is be inc. overview Rules of Inference: Elementary Valid lidly Argument Forms Ow Name Abbreviation Form 1. Modus Ponens M.P. p >q P .. q 2. Modus Tollens M.T. paq can 3. Hypothetical Syllogism H.S hey der it as 4. Disjunctive Syllogism D.S. p V q ion - p on- 5. Constructive Dilemma C. D. (p > q) . (r>s) p Vr ..q Vs 6. Absorption Abs. p q :. p> (p . q) 7. Simplification Simp. p . q is sly 8. Conjunction Conj P hat er a p . q it 9. Addition Add. on. :. pvq nc- wn Two features of these elementary argument forms must be emphasized. First, they must be applied with exactitude. An argument that one proves valid using Modus Ponens must have that exact form: p > q, p, therefore q. Each statement variable must be replaced by some statement (simple or compound) consistently and accurately. Thus, for example, if we are given (CV D) 3 (J V K) and (C V D), we may infer (J V K) by Modus Ponens. But we may not infer (K V )) by Modus Ponens, even though it may be true. The elementary argument form must be fitted precisely to the argument with which we are working. No shortcut-no fudging of any kind-is permitted, because we seek to know with certainty that the outcome of our reasoning is valid, and that can be known only if we can dem- onstrate that every link in the chain of our reasoning is absolutely solid