Abbre viated ruth ab les The full tr uth tab le method of Section

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4 Abbre viated ruth ab les The full tr uth tab le method of Section 73 is xtremely cumbersome or xample an argument with only our statement letters requires tr uth tab le with 32 ro ws One with 57346v requires tr ID: 27887 Download Pdf

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Abbre viated ruth ab les The full tr uth tab le method of Section

4 Abbre viated ruth ab les The full tr uth tab le method of Section 73 is xtremely cumbersome or xample an argument with only our statement letters requires tr uth tab le with 32 ro ws One with 57346v requires tr

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Abbre viated ruth ab les The full tr uth tab le method of Section




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7.4 Abbre viated ruth ab les The full tr uth tab le method of Section 7.3 is xtremely cumbersome or xample an argument with only our statement letters requires tr uth tab le with 32 ro ws One with v requires tr uth tab le with 64 ro ws Ob viously tr uth tab les of these siz es are simply impr actical to constr uct. Ab bre viated tr uth tab les pro vide uch more efcient method or deter mining alidity The method The ke insight behind the method If can constr uct just one ro of tr uth tab le or an argu- ment that mak es the premises tr ue and the conclusion alse

then will ha sho wn the argument to be in alid. If ail at such an attempt, will ha sho wn the argument to be alid. Recall the argument from the lecture or 7.3: Abor tion is per missib le only if etuses are not innocent human beings or it is not alw ys wrong to kill innocent human beings But it is alw ys wrong to kill innocent human beings So abor tion is not per missib le (A: Abor tion is per missib le; B: etuses are innocent human beings; W: It is alw ys wrong to kill innocent human beings .)
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symboliz ed this sentence as ollo ws: [A W)]. The method applied to an in alid ar

gument 1. Wr ite do wn the symboliz ed argument thus (note that e'v dis- pensed with per iods; the just get in the y): W), 2. Hypothesiz that the premises are tr ue and the conclusion alse: W), 3. Calculate the immediate consequences of this ypothesis: Cop the tr uth alue assigned to to its other occurrence: W), Calculate tr uth alues of the compound statements whene er ou kno the tr uth alues of their component par ts Thus can calculate that is alse in vir tue of our assumption that is tr ue:
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W), Calculate tr uth alues of the component par ts of compound statements when- er

ou kno the tr uth alues of the compound statements (and cop an tr uth alues assigned to statement letters to all other occurrences). Thus can calculate that ust be tr ue giv en that is alse: W), Fur ther more since (i) e'v ypothesiz ed that [A W)] is tr ue and (ii) ha calculated that the antecedent of [A W)] ust be tr ue (giv en our initial ypothesis about A), it no ollo ws that W) ust be tr ue as ell, lest alsify our ypothesis that [A W)] is tr ue (Recall that conditional is alse if its antecedent is tr ue and its consequent is alse .) Hence: W), But no that ha deduced that the disjunction W)

ust be tr ue (giv en our initial ypotheses), kno that, because its ight disjunct is alse its left disjunct ust be tr ue or disjunction is tr ue if and only if at least one of its disjuncts is Thus: W),
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And this of course enab les us no to deduce that is alse which completes the ro w: W), So ha identied ro that mak es the premises of our argument tr ue and the conclusion alse so ha thereb demonstr ated that the argument is in alid when is tr ue is alse and is tr ue complete the ab bre viated tr uth tab le recording this in alidating tr uth alue assignment into the tab

le: W), Much shor ter than the full tr uth tab le! remind ou: W), Note that ro is xactly the ro that just constr ucted using the ab bre viated tr uth tab le method.
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The method applied to alid ar gument What happens if the argument in question is alid demonstr ate with fur ther xample e'll cut ight to the chase with symboliz ed argument without orr ying about the English argument it symboliz es begin with the usual ypothesis that the premises are tr ue and the conclusion alse: (W Z) (J Z), (W J) Since the conclusion (W J) is alse its component statement (W J) ust be tr ue: (W

Z) (J Z), (W J) rom the tr uth tab le schema or the only or (W J) to be tr ue is if both and are tr ue So record this inf or mation belo the conclusion, and cop it into the rest of the tab le: (W Z) (J Z), (W J) So ar so good. No from the act that is tr ue inf er that is alse and record this inf or mation into the tab le: (W Z) (J Z), (W J)
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This then enab les us to ll in the tr uth alues or the tw disjuncts in the second premise in accordance with the tr uth tab le schema or conditions: (W Z) (J Z), (W J) But no there is prob lem: our ypothesis as that the three

premises ere all tr ue But the second premise is disjunction with tw alse disjuncts Hence it is itself alse But this conicts with our initial ypothesis Hence ha sho wn that it is not possib le after all or the premises of the argument to be tr ue and the conclusion alse indicate the contr adiction that ar ises when assume otherwise putting `T/F' under neath the main logical oper ator of the prob lematic premise (all of them, if there are more than one): (W Z) (J Z), (W J) T/F When there are se veral wa ys the conc lusion can be false If the conclusion can be made alse in more than one

then er ust be tr ied until either an in alidating assignment is ound or all the ys of making the conclusion alse are xhausted. Again, star with the usual ypothesis: A, But note that, the tr uth tab le schema or there are three ys to mak the conclusion alse So simply pic one of the three to star with, viz., the one where both and are alse:
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A, Cop ying these to the rest of the ro and calculating tr uth alues get: A, Because of the tr uth tab le schema or and our ypothesis that (C D) is tr ue it ollo ws that has to be alse as ell: A, Cop ying this alue to (A C) get A, But no see

that there will be no to mak (A C) tr ue since one of its conjuncts is alse Consequently this par ticular of making the conclusion alse ails to yield an in alidating assignment, and so indicate: A, T/F
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Note: The eb utor will require ou to complete the ro do so just assign an arbitr ar alue to (it on't matter because (A C) will be alse either just because is); `F' is easiest in this case: A, T/F No need to tr the tw remaining ys of making the conclusion alse; will (arbitr ar ily) tr the one where is tr ue and is alse rst: A, T/F But, when calculated out, this

possibility ares no better than the rst: A, T/F T/F So ust tr the nal case where is alse and is tr ue: A, T/F T/F
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But, when calculated out, this possibility too ails to yield an in alidating assign- ment: A, T/F T/F T/F Only no that ha xhausted the possib le ys of making the conclusion alse are per mitted to sa that the argument is alid; or on all the possi- le ys of making the conclusion alse ound ere unab le to mak the premises tr ue If ho er an of these ys of making the conclusion alse had yielded ro with tr ue premises the argument ould ha been sho wn to be

in alid. emphasiz e: to sho in alidity all it tak es is one ro one tr uth alue assignment to the statement letters on which the premises are tr ue and the conclusion alse