# In some arguments the attempt to fi nd the conclusion

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In some arguments the attempt to‘‘find’’the conclusion in the premises is notimmediately successful. When confronted with such an argument, it is often best tobegin by‘‘deconstructing’’the conclusion using the rules of replacement. In otherwords, one shouldfirst apply the rules of replacement to the conclusion to see howit is put together. After this is done it may be evident how the premises entailthe conclusion. This procedure is justified by the fact that the rules of replacementare two-way rules. As a result, after the conclusion is deconstructed, it can be de-rived by using the same rules in the reverse order. Here is an example of such anargument:1.K(FvB)2.GK/ (FG)v(BG)If immediate inspection does not reveal how the conclusion should be derived, wemay begin by applying the rules of replacement to the conclusion. The form of theconclusion suggests the distribution rule, butfirst we must use commutativity to movetheGs to the left-hand side. The deconstruction proceeds as follows:
Rules of Replacement I389SNL(FG)v(BG)(GF)v(GB)G(FvB)Com, ComDistNow we see that if we can obtainGon a line by itself, andFvBon a line by itself, wecan combine them on a single line via the conjunction rule. We can then obtain theconclusion via distribution and commutativity. Inspection of the premises reveals thatGcan be obtained from line 2 of the premises by simplification, andFvBcan beobtained from line 1 bymodus ponens.The completed proof is as follows:1.K(FvB)2.GK3.G4.KG5.K6.FvB7.G(FvB)8. (GF)v(GB)9. (FG)v(BG)/ (FG)v(BG)2, Simp2, Com4, Simp1, 5, MP3, 6, Conj7, Dist8, Com, ComHere are some strategies for applying thefirstfive rules of replacement. Most ofthem show how these rules may be used together with other rules.Strategy 10:Conjunction can be used to set up DeMorgans Rule:1.A2.B3.AB4.(AvB)1, 2, Conj3, DMStrategy 11:Constructive dilemma can be used to set up DeMorgans Rule:1. (AB)(CD)2.AvC3.BvD4.(BD)1, 2, CD3, DMStrategy 12:Addition can be used to set up DeMorgans Rule:1.A2.AvB3.(AB)1, Add2, DMStrategy 13:Distribution can be used in two ways to set up disjunctive syllogism:1. (AvB)(AvC)2.A3.Av(BC)4.BC1, Dist2, 3, DS
390Chapter 7: Natural Deduction in Propositional Logic1.A(BvC)2.(AB)3. (AB)v(AC)4.AC1, Dist2, 3, DSStrategy 14:Distribution can be used in two ways to set up simplification:1.Av(BC)2. (AvB)(AvC)3.AvB1, Dist2, Simp1. (AB)v(AC)2.A(BvC)3.A1, Dist2, SimpStrategy 15:If inspection of the premises does not reveal how the conclusionshouldbe derived, consider using the rules of replacement to‘‘deconstruct’’theconclusion. (See the previous example.)EXERCISE 7.3I. Supply the required justifications for the derived steps in the following proofs:(1)1. (JvF)vM2. (JvM)P3.F/(FvP)4. (FvJ)vM5.Fv(JvM)6.JvM7.P8.FP9.(FvP)(2)1. (KP)v(KQ)2.PK/QvT3.K(PvQ)4.K5.K6.P7. (PvQ)K8.PvQ9.Q10.QvT(3)1.Ev(DvC)2. (EvD)C/E

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Term
Fall
Professor
Yasir Gondal
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