Teller's Logic Manual Answer Key (Book 1)

8/13/2019 Teller's Logic Manual Answer Key (Book 1)

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CONTENTS

.................................................r e f a c e iiiA n s w e r s to E x e r c i s e s i n Volume I

C h a p t e r 1 ........................................... 1C h a p t e r 2 ............................................. 4C h a p t e r 3 ............................................ 5C h a p t e r 4 ............................................ 11C h a p t e r 5 ........................................... 17

.............................................h a p t e r 6 23

............................................h a p t e r 7 29C h a p t e r 8 ............................................ 47C h a p t e r 9 ............................................ 55

A n s w e r s toC h a p t e rC h a p t e rC h a p t e rC h a p t e rC h a p t e rC h a p t e rC h a p t e rC h a p t e rC h a p t e rC h a p t e r

E x e r c i s e s i n Volume I 1............................................ 75............................................ 76

3 .......................................... 79............................................ 82............................................ 8 7............................................ 99............................................ 118

8 ............................................ 1329 ............................................ 152

...........................................0 172

1 1. n argument is a bslch of sentences. kre of them is called theconclusion aml the rest are called the praaises. Zhe idea is that i f ycubelievethatthepmnisesaretrue, t h e n t h a t s h a i l d g i v e y o u s a r e ~for beliar- that the &usion is true also.

l3xx-e are a great many different a a q l e s you cculd give of bothindudive aml deductivee ere is are exanples which are a l i t t l edifferent fnmn the ones given m the text:m i v e :Highgavenmnentspendingcausesinterestra testogoup.e garenmnent is spending a lot.

nterests rates will go up.BdwtiveEither the garerr Rent w i l l not spend a lot or interest rates will go up.T h e qavennnent will a lot.

Intere rates w i l l go up.In the indudive Lf you really believe the you are goto think that the d u s i o n has a good chanoe of b e i r q true. But even l fth r is thecausal o m n e & i o n ~ g y ~ s p e n d i n g , a m lnterest ratesclaimed in the f i r s t p m m b , the connection u not r e f m . my factorsother than garerrPnent spendin M m interest rates, for v l e hedemmi for loans by fusbesses aml home amxs. In the case of the deductiveargument, i f the pmnis s are the conclusion has got to be t rue also.F o r w t h a t t h e f i r s t v i s t r u e . I f thesecondpremiseistruealso the f l r s t disfurct of the f l r s t prerpise is false. So the d y ay l ef tfor the f i r s t premise to be true is for ts s d isjurct to be true, whichis the d u s l o n .

1 3. A non-truth-fumticmal sentm e is a lcp-r3er sentence buil t up fmm omor m r e shorter carponent sen- inwhich the truth value of the whole isOP determined JUST by the tnxth of the ~cnpo3lents In otheri n order to figurecut thetruthvalue of thewhole youwould have to knowm r e t h a n j u s t t h e t n x t h v a l u e o f t h e ~ a r c c a p o n e n t S . H e r e a mexanples: \Smithbelieves that Tdq has a large r @ation than New York.l e expression \Smith believes that* can be regarded as a caplective. Put itin frwt of a sentence aml ycu get a new 1-er sentmce But thet m t h v a l u e o f t h e s h o r t e r s e n t m c e i s n o t e n a l g h t o ~ t h e t r u t hvalue of the lorqer sentmce ~u ulwirq sther Tdq has a larger @ ationthan New York w i l l not tell you khether Smith believes that Tdq has alarger @ ation than New York. Here is another exauple. Su~pose e aref l i ~ p - a penny. mi : \ Ihe pmny canes up heads is more pmbable thanthe penny caresup tails. Knowingthetruthvalue of \ pennyheads. aml he pamy cares up tails. is W t enalgh to know whether


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up heads s ( o r w a s ) mrre likely th n mh g up tails lt~nw bhe t b r this true ycu med to knw her or not the coin s wighted to make headsmore likely th n tails

=ID, -GGGI D(DV-B) )h(M) I DJBDV-B, Dl -B-BBD, BLl ( W -N Mv-L)M -N (Mv-L)-Nl Hv-LNM -LL

1-5. b is not a pxpr s e n t e n c e logic se nte nc e . There is noway t o g e t t k~ i o n b y h u i l ~ i t u p b y t h e f ~ l l ~ t i o n n r l e s f m m s h o r t e r s e n t e n zlogic sentenzs. - I plies only to a e n t e n c e , rrl no sentencebegins w i t' I . f ) i s a l s o n o t a mteme logic se nte nc e . In this casethepxblem jthatparenthesesaremiss ingwhichwouldte l l~whetbrthesentence~s q p e to be a mjunction of two shorter sentences or a isjunction of tshorter seartenmshe ther expressions are a l l pxpr s e n t e n c e logic sentenoes

a ) b )1 6. A B -B -BVA BVA - BVA) C Q T W q + + T ( W ) ( + J ~ )

t t f t t f t t t f f f ft f t t t f t f t f t t tf t f f t f f t t t f t tf f t t f t f f f t t t f

e A B -Bt t t ft t f ft f t tt f f tf t t ff t f ff f t tf f f t

f ) K P M - Pt t t ft t f ft f t tt f f tf t t ff t f ff f t tf f f t

q D B -B -B DV-B DV-B (DV-B) (DV-B) DV-B) (Dv-B) ] (Dm)t t f t t t t t tt f t f t t t t tf t f t t f t f ff f t f f t f f fh LM N* -N -L +l v- L- N (+ lv -L ) Hv[-N (*-L)] L (Hv[-N (*-L)])t t t f f f f f t t

t t f f t f f f tt f t t f f t f ft f f t t f t t tf t t f f t t f tf t f f t t t t tf f t t f t t f ff f f t t t t t t

1-7. Quotationmarks function to farm the NAME of a etter or expression.' A 1 i s a ~ o f A . nrs ,whenIwanttotalkabcutasentenxlet ter , m ac op nd I prt polxs a d he expression. CYI he oth r hard, Iuse no plrrtes when I IEE an expression. Also I am wing b a l d face capitalsto talk bcut sentenzs generally.F f d l o s q h e r s ~ e r to t h i s d i s ti n C t i o n a s t h e d is t i n c t io n b e t w e e n U s errl Mention.

d ) D G G W G D W )v(G D )~ t t f f tt f t t f t

f t f f f ff f f f f f


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2-1. To say tha t Adam is not both ugly and duub is to say that he is notugly, or he is not duub (or possibly not eit he r ugly or d u b ) : \uv-D*.Qual ly , this o sayirq that he is not both ugly and du : -(Urn). I fycuthinkabcutthesetwoexpessions, I h c p e y c u wi l l s e e t h a t t h e y c a n e t othe same thing. We sha l l prcne t h a t they are logically equivalent in thenSrt dmpter. But '-U -D' is an h a m e ttanscripticm. T r a n ~ ~ i b et backinto qglish, and ycu get: Adam is not ugly and not duub,' d is sayirqscmmmgs t ronge r than tha tAdamisno tbo thug lyandduub .

2-3. a ) Eit her Adam @ b l d or notb) M a m l a v e s E v e a n d i s n o t b l d .c) Neither is Adamin lavewithEvenor isEveclever.d) Mam is b l d , or Eve is dark eyed nd not clever.e) Eve is e i the r in lave w i t h Adam or not clever, and either Adam is notb l a n d o r h e i s i n l a v e w i t h E v e .f ) E i the r it is the case t h a t both Adam laves Eve or Eve laves Adam andE v e i s c l e v e r , o r i ti s t h e c a s e t h a t E v e i s c lw e r a n d n o t d a r k q r e d .g Eve is clever. Rnthemore, e i the r it is not both t rue th at Eve1 o v e s P d a m a r d A d a m i s b l d; o r E v e i s d a r k ey e d a n d A d a m e i t h e r is n o tbland or is in lavewithEve.2-4. Y c u r choice of sentence letters may of course vary ran mine But ycushculdhavesentenoeletterssymbolizirqthe~meataaicsentencesasintheanswers belcw.a) RvT (R: Roses are red. T: W ler w i l l e a t his hat.)b) P -R (P: Ik y Pythcn is funny. R: khr t W c a d is funny.)c) +X-N (C: Chicago is bigger th n New York. N: New York is thelargest c i ty . )d) W D F: I w i l l finish this ogic ccurse. D: I w i l l die t r y i r q tof in ish th is logic cause.e) -(I &),r +ly, 4W-S (H: W.C Fields is handsame. S: W.C.Fields is snart.)

f ) - GUU) or equally, 4 - U (G: Uncle Scrooge was genemus. U: UncleSQooge--.)g T O (T: Mirnesata Fats t r i e d to diet. 0: Wmemix Fats was veryoverweight.)h) (P I) -E P: F e t e r likes pickles. I: Peter l ikes ice cream E: Peterlikes to ea t pickles and ice crpam together.)i ) (Rf ) T (R: Roses are red. B: Violets re blue. T: T J X U I S C ~ ~ ~ ~ ~this jirqle is not hard to do.j m)(-s -N) or +ly, (w)- SvN) (0: Cblmhs sa i led the cce nblue in 1491 . T: 031rmjxls sailed the ccean blue in 1492. S: 03 1m hsdkccmxd the Sarth mle. N: 031- discrrvered the Nnrth mle . )k) [ (C P)v( W) D (C:Iulre w i l l c a M up with IhI UIVader. P: Iulrew i l l p t a r r l e x d t o D a r t h V a d e r G: aar thvaderwi l lge taway . T: IhI UIVader w i l l cause mre tmuble. D: Everrtudlly the mire w i l l be stmy .)

3-1. We are goirq topruve the DeMolqan lawwhidl says t ha t - ( M ) islogically equivalent to -X -Y. We wil l do this by calculath3 the VennDiag?.am rea fo r the f i r s t sentence and then for the entence:

All points h i d e All points artside ofof X or inside of Y ( M I

All points artside All points artside All points h i d eof X of Y both -x and -YW ec an se et ha tt h ea re as a re th es a w. S i n c e t h e a r e a s ~ t h e c d s e sinwhidlthesentencesaretrue, thetwosentencesaretrueinthesamecases h a t is they are logically equivalent~awweusethesamemethodtop~~~ethelogicalequivalenmof(YvZ)and (X Y)V(X Z)

Yvz

All points h i d e All points inside of bath Xof Y o r h i d e of Z and b i d e of YvZ7 :

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A l l p o b h i d e A ll po in ts h i d e A l l points h i d e of0fbothXandY OfbothXandZ either X Y or X Z.sirm the coincide th smt m s are logically equi~err t .

Finally we do the same for Xv(Y Z) and ]rvY) ( W )

All points h i d e0 f b a t h Y a n d Z A l l p o i n t s h i d e o f xor h i d e of (Y Z)

A l l points inside All points inside A l l points inside ofof X or inside of Y of X ar inside of Z of both ]rv and of ma

BV-A-AvB CM-AV-B EN, SIE-( B) PI

(c wv (= )v [= - (-=-A)( W ) V c B)V[C - (B A ] at SIE[ (C Wv (- ) Iv[C -(-=-A) A(A*) IV[a -=-A) Dr[aAVB) ] V [a i (-BV-A) ] PI, SIE( A m ) I v P BVA) EN, SIEP ( A W v [C ( AW CM, SIE( A m ) R

h) N o t e th t in proving two sentences X and Y to be logicallyequivalent one can just as well start w i t h the secard and pmve it logicallyequivalent to the first as start with the first and pmve it logicallyequivalent to th seccdC -AV- (*A) ]C [-Av (+A) ] PI. SIEC -Av (=-A) ] EN, SIE(C -A)V [C (*A) D(C -A)v[ (=) -A] A, SIE(=-A) V (-A) R, SIEC -A R

PI, SIEEN, SIEA, SIEDA,aA,R, SIECM, SIEA, SIER ~


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3-3. For any Sentenoes X, Y, and Z, -(X Y Z) is l o g i c a l l y equivalent to-m-w-Z. 2u d- m)s l o g i c a l l y equivalent to -x -Y -z.3-4. SqpceewehaveadisjuncticmMrY, a n d s t q y e t h a t x i s a l o g i c a ltruth. T h a n ~ t h a t i n e v e r y p c s s i b l e c a s e X ~ s t r u e . utthe~unCticmMrYistrueinanycaseinwhi eitheraneofitsdisj l lnctiaclfis tzue Sinm X is a l w a y s true, rY is always true, wh i is to s a y thatrY is a logical truth

S q p s e , rrw, tha twe have a m jlmcticpl, X Y, and that X is a cclltradiction lhat is to say, in every possible case, X is false. But ther m j u n c t i c m X Y is false in any case in whi either of its rmjuncts isfalse. Sinm X is false in every possible case, X Y is false in everypossible case, i i means that X Y is a r m t r a d i c t i c m .3-5. By \A Bv-B) is l o g i c a l l y equivalent to \A'.

Av(-A B)(Av-A) (AvB) DAVB E E ( I a w of l o g i c a l l y true rrJrriun=t)

(AVB) (Av-B)Av(B -B) DA m

h) (A B)V(-Ah-B)(Av-A) ( -A) (AV-B) (Bv-B) D ( P m b l e m 3-2 f )(Bv-A) (Av-B) rn(-Am) (- A) a4, SIE

i ) - -Am) v-AvC(-A6-B) V-AvC PI, Em3(A -B)v-AvC DN, SIE[ A6-B) v-A]v C A, SIE[ Av-A) (- -A) ]V D, SIE(- -A) vC rn, =Av-B) v C t =-Av-BvC A3-7. a ) Neither b) O n t r a d i c t i cm . c ) Neither d ) L o g i c a l Ruthe ) ~ m t x d i c t i c m . f ) C a rt ra d ic ti cm . g ) Neither h ) L o g i c a l Ruthi ) L o g ic a l Ruth j) meal Ruth k) Neither 1 ) L o g i c d l Ruth3 -8 . a ) We first work the p r c b l e m w i t h a truth table:

A BA B-(A B) ' - ( A B ) ' i s t r u e i f c a s e 2 i s t r u e I f c a s e 3t t t f is true or i f case 4 is true. 'A k B ' says thatt f f t o s e 2 i s t r u e t ' - A B ' s a y s t h a t o s e 3 i s t r u ef t f t and 'A B says that 4 is true. SO, thef f f t d i s j l m c t i a c m of thse three Qseilescribingsentences is l o g i c a l l y equivalent to '-(A B)

Ihe dis un=tive normal form of - A B) s A -B)v(-A B)v (-A -B)Ncm let s try to work the p r c b l e m b y usir a s psme of laws of logicalequivalence. Ib work the prcblem given what ycu kmw, ycu have to a ~ p l yC xgan's l a w and then fiddle aranrd until ycu get the Serrtence into theright form. Ihe followirg does the j d :

--Av-B PI[(-A B)v(-A -B)]v[(-B A)v(-B -A)] Zaw of e x p m i c m ( p r c bl e m 3-6 c ) , SIE(-A B)v(-A -B)v(-B A) A, Rb ) F i r s t w i t h truth ables:

A B C -A A B -AM: (A B)v(-W) -[ (A B)v(-AM:) ]t t t f t f t ft t f f t f t ft f t f f f f tt f f f f f f tf t t t f t t ff t f t f f f tf f t t f t t ff f f t f f f t

' I h e r e a r e f w ~ i n w h i t h e f i n a l s e n t e n c e i s t r u e , Q s Q - i b e d b y t h esentences 'A -B C', 'A6-B C', '-AE -C', a n d '-A -B6-C'. Ihe w h o l ei s t r u e i n a n y a n e o f t h s e c a s e s , a n d i n m a t h e r s ; s o w e g e t a l o g i c a l l yequivalent sentace b y lacing the d i s j l p l c t i c m : Ihe disjunctive mnnal formof' -[ ( ~ B ) v ( - A K ) is A6-B C)v(-AE -C)V(A -~ -C)V(-A -~ -C)Ncm let s use l a w s of logicdl equivalm. first a ~ p l y eMorgants a w9


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twice. lbk gives a conjurcticm of disjunctions, an3 want the qpsi tenamely a disjunction of omj unc t i a . But can rmvert the cne to the therby using the disbibut ive l a w twice. Actually we already did the work inpmblem 3-4. e ) , so just cit e that pmblem far the e s u l t . The reai l t ir~~conjunctions only irnrO1ve two of the three letters in the pmblen, so I aveto apply the a w of expnsicm to ea of them to get the final disjunctivemxmd farm:-[ (A=)v(-AW I- A=) 6- -=I M(-Av-B) (Av


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e ) A B C C A B A6 C Bv-C (A B)v(A -C)t t t f t f t t *t t f t t t t t *t f t f f f f ft f f t f t t t *f t t f f f t ff t f t f f t ff f t f f f f ff f f t f f t fSirce tbxe are n, lest the argument s VAIJD.

4-3. First, sq p c e t h a t X s logically equivalent to Y. That mans tha t ina l l possible cases X have the same trut h value. In particular , in a l lc a s e s i n w h i c h X i s tr u e Y i s t r u e a ls o , w h i c h i s t o s a y th a t t h e a r g u m e n tX /Y s valid Similarly, in a l l possible cases in which Y is true X is truedlso, whichis tosaythattheargumentY/Xisva l id .N c w ~ t h a t t h e a r g u m e n t X / Y d Y / X a r e b o t h v a l i d . Wehavetoshow tha t on this a s t q t i o n X have the same truth value in a l l possiblecases. W e l l , srppose t t y don't. That is, suppose there is a possible-n w h i X i s t r u e d Y ~ s f a l s e o r a p o s s i b l e o s e i n w h i c h Y i s t r u e dX s false . A possible- f the f i r s t kind would be a v l e othe argumoent X /Y possible case of the second kind would be aw l e o the argument Y/X. Hmever, re su~posing ha t theset m rguments are valid. So there can't be any wfi pcssible c seshave the same tr uth value in a l l possible cases4-4. Fbll~~ationules:i ) hrery capital letter 'A', 'B', C ... s a sentence of sentencelogic. Such a sentence s called an Atonic Sentence or a entencex' er.ii) f X is a sentence of sentence logic, so is (-X) , h a t is, thesentence farmed by taking X, w r i t i n g a \- n ront of it,smmrdkqthewholeby-. Suchasentenceiscalleda-Negated Se;lteKE.iii) f X, Y, . Z are sentences of sentence logic, so is (X Y . . Z) ,t h a t i s t h e s e n t e n c e f a r n r e d b y w r i t i r q a l l o f t h e s e n b r a sX,Y, ..., separated by ' 's -he whole with

parentheses. Such a sentence is called a Canjurcticn, heiv)

v)

v i

sentences X, Y, ... are called its Canjuncts.I f X, Y,. . ,Z are s nt nms of sentence logic, so is M..vZ)t h a t is thesentence fa rn redbyw ri t i rqa l l o f t h e s e n b r a sX,Y,. . Z separated by 'v's umam q the whole withparentheses. Such a sentence s called a Disjumticn, hesenten es X, Y, . Z are called its Disjuncts.I f X are serrtenoes of sentence logic, so is (X -> Y , tha t isthe sentence farnred by writirq XI follawed by -9,ollawed by Yling the whole with parentheses. Such a sentence iscalled a Cearditicmal. X s called the mrditicmal's ntecederrt atdY its C m s q w n t .I f X are s nt nms of sentence logic, so is (X .c-> Y , tha tis the sentence farnred by writirq X, follawed by , follawed byY d s t m m d b q t h e w h o l e w i t h p a r e n t h e s e s . Suchasentenceiscalled a Bimrditicmal. X are called the bimrditicmal'sCcnponents.

vi i) Qlly those ormed us- r u l e s ( i ) - ( v i ) are sentences ofsentence logic.Rules of valuation:i) I h e t r u t h v a l u e o f a n e g a t e d s e n t e n c e i s ' t ' i f t h e m E p c n e n t ( t h e

serrte zwhich has been negated) is 'f ' . 'Ihe truth value of anegatedsentenceis ' f ' i f thet ruthvalueof theccaponent i s ' t ' .ii) he truth value of a amjunction is t i f a l l an]uncts have truthvalue t . Otherwise the tru thva lue of the an junc t ion is 'f ' .i i i ) I h e truth value of a d jumt icp .1 is t i f at least one of thediju ts have trut h value t . Otherwise the trut value of thedid- s f .iv) Ihe truth value of a mrditicmal s ' f ' i f its anboxk is truets is false . Otherwise the trut value of themrditicmal is t .

v) Ihe trut h value of a b iwrd i t iona l s t i f both anpenents havethesametruthvalue. Otherwisethetruthvalueofthebiamkiticmal s ' .4-5.a ) A B A->Bt t t *t f f Si nx there s a axmterewnple,f t t*C E t h e a r g u m e n t i s m .

f f tb) A B -B A->-B -At t f f ft f t t f Si nx there are n, lestf t f t t * theargumerrtisWD.f f t t tC) A B -B AB AV-Bt t f t t *t f t f t Si nx there is a -letf t f f f the argument is l l w u u I D .f f t t t*CE

e) A B C AvB AM:t t t t tt t f t ft f t t tt f f t ff t t t ff t f t ff f t f ff f f f f

AV-B AVBt tt t * Si nx there am2 n, ccllmtersranples,f t t h e a r g u m e n t i s w .t f

CVA (AVB)-> AM:) Ct t f * C Et f tt t f * C E Sinxtheream2carnter-t f t -1- to the argument,t f f t h e i 5 m p E a i s m .f f tt t f *CEf t t

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f ) A B C -B -C AVB AV-C (A m) (Av-C) -BVC AvCt t t f f t t t t tt t f f t t t t f tt f t t f t t t t t S b t h e r e a r e n ot f f t t t t t t t W l e s t o t h ef t t f f t f f t t ~ f t h e ~f t f f t t t t f f is VALID.f f t t f f f t t tf f f t t f t f t f

4-6 .a ) (A->B) (A)(A->B) (A->B) 8 LEThis is a logical truth, s b ->B s l o g i c a l l y equivalent to itself, andbi-tiandl is a logical truth i f its carponentsare lggically equivalent.

But 'EK->B* is a l o g i c a l truth ( a s in prcblem a ) , and a d i s j u n c t i c m with alogical truth as a disj is a logical truth. So the original sentence ~a logicdl truth.c ) A-A(A->-A) (-A->A) B(-Av-A) (-AvA) C

AhA t N , R , S I E

d ) A->-B-Av-B C (neither a l o g i c a l truth nor a a m t r a d i c i t i c m . )e ) (A->B) v A->-B)-AvBv-Av-B C, A, S I E-Av W-B) Rt A,A d i s j u c t i c m w i t h a logically w isjunct is logically w exercise3-4). So this is a logical truth.f ) - AVB) (-A->B)- ( A m ) A B C* S I E- AVB) (AVB) PI, t N , S I EIhis is a c o n j u n c t i c m of a sentenz w i t h i t 's n e g a t i c m and so isa c m t r a d i c t i c m .9 ) (AA) ->(B-B) Pmblm (c) that the is ac m t r a d i c t i c m . S b A* is logically e q u i v a l e r r t to itself, the antecedentis a logical truth. S b he antecedent is always w nd theis always false, this is a c o n t r a d i d i c m .

h) [- (H A ) (C->A) ] > (C->B)-[-@->A) (C->A) ]V (C->B) C@->A) V- (C->A) V (C->B) PI, SLE(-WA) v C k A ) v a) t SLE(-BvA) v e B ) A) A, S I E-BvB) Av-C)v a - A ) CM, A , S IE

i ) [A-> B-X) ]-> [ (A->B) -> (A-X) ]- -AV-BVC) V[- (- Am ) V-AvC] DIAt =(A B -C)v Ak B) v-AvC PI, t N , A , S IE[ C v ( m - C ) lv[-Av(*B) A,{ [Cv(A B) ] (Cv-C) )V [ (-AvA) (-Av-B) ] D, SIE[Cv(A B) ]v-Av-B L E , A , S IEW[ 1 v- ( A m ) PI, A* =

This is a logical truth because one of its disj is a logical truth4-7.a ) ' Yo u w i l l sleep too la te unless set y ~larm * This senten emakesa causal claim, to the effect that setting y ~larm w i l l keep fransleeping too late. Whether or m t the causal cwmcticn holds is a m a t t e raboveandbeycadthetnl thvalueofthecarpenerr ts ' Y o u w i l l s l e e p t o olate, * and 'You set yw alarm. *b ) Argumerrt 4-5.a) is valid unless it has a c c u n t e r e x a n p l e * . S b , byd e f i n i t i c m , an aqm?nt is valid i f and only i f it has IKI a x l r r t e r e x a n p l e s ,only the t ruth value of the c rpenerrts is relevant to the truth value of thise x a m p l e .c ) i ) Youm* t g e t a nA i n t h i s c a u s e u n l e s s y s t u d y h a r d . * histranscribes as 'SV-A*, (S: You hard. A: You get an A in this )i i ) Ihe cr i t ica l ly ill p t i e r r t w i l l die unless the doctors cperate.Ihis transcribes as '-0->D* 0: Ihe doctors cperate. D: Ihe cr i t ica l ly illp t i e r r t w i l l d i e . )i i i ) 'Y ou may drive thnxlgh the irRersecticn unless there is a redl igh t . * Dc->-R (D: You may drive the htexezticn. R: Them is ared l i g h t . )These ewnples, and the w h o l e b i n e s s of transmibirq ' u n l e s s * requiresane Garment. First, most logic textswill tell to t r a n s c h ' u n l e s s *as a Ccpditiandl or a disjuncticn. By the l a w of the corditiandl, these twoapproaches are -le sanetimes the one b e i n g rmre mtural,sanetimes the other. W wen menticn the p o s s i b i l i t y of transcribirqwith the bicorditiandl. I, in a m i n o r i t y , think that in moet cases thebicanaitiandl -lies a mor faithful transcriptim.N o t i c e that there is a s lzaqpara l l e l betwen the difference intranscribing ' u n l e s s * with a canaitiandl versus a bicorditiandl and thedifference in hansmbing ' o r * as irrclusive versus exculsive disjunction.Ihe parellel berxmPs rmre StriJCirKJ when note that 'X-Y* theaclusive d i s j u n c t i c m of X and Y. Pws transcribing X unless Y as -Y->XIequivalently as XvY, corresporrls o transcribirq ' u n l e s s * as irrclusive ' o r * ;while transcribing it as X-Y mmspds to txanscribirq ' u n l e s s * asexclusive or*. .


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I h e ~ l e i s s u e i s r w i l l y v e r y ~ e a r ,ni thereascni s tha t indLmostall usagss, \unless8 s not tnrth f l n r c t i d Thus, like a gmat marry usesof \If...ther~...~, tranScipticms t ~ogic are faulty and -te atbest.

4 9. The follcuirqwm can be used in varicus ways to build up xn-tmthf u n c t l d amnectives in m g l h . Fbr exauple, \believes8, k n cfmbinedw i t a pemon8s name ma ces sucfi a ocanective: 'Jchn believes that.. .~ 8 i s t h e ~ i a l w a r d i n t h e a n n e c t i v e I t c u p l t t o b e t h e c a s ethat...' nd so CPIi) ~maltes8, causes8, because8, rirqsatcut8 \since8,'mans8, 'in3icates8.ii) 3wst8, \possible8, w l e 8 , mare pmb ble (likely) than8,'a2ght8 'may'.iii) %elieve8, 'hope8, \hark8, i sh8 , \fear8, exp zt8 , \like8,' v b o a m g e 8 .iv) \&fare8, soosler than8, \after8, later than8, \at the same timeas8, ' s h 8 .

lh r are, of axuse arry my other sucfi axmectives in English


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Q> s 1,R3,4,>EQ>(SVF) 2,RSVF 3,6,>EF 5,7,-Q>F 3-8,>1

456789

A>K 1 3K 3,4,>ERVP 5,-(RVP)>L 2,RL 6,7,>EA>L 3-8, I

1 VB P

5-2. k1234

I

67891011l1314151617

P P(-mq>B P(Ftr-D)+K PD(W-F) PP'-(Ftr-D)+K 3,R-K 5,6,>EP 1,RD(Mr-F) 4 3Mr-F 8,9,>E-F 7,10,vED 5,11,vE-DvK l vI(->B 2,RB l3,14,>EBV-P 15,vI(Ftr-D) (BV-P) 5-16, I


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2-A 1 2 >E

4 I-A 3 -E


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4-7, I-F AFx 1 3I , 9,10,>E8 R-F 9 l2 1F l3,-E6. far most valid a qmmk sane truth assigpmwts to the senten el e t t e r s w h i l l l a k e o n e o r m x e p r e m i s e f a l s e w i l l m a k e t h e c a d u s i o n hand sane su h assigpmwts will make the c a d u s i o n false. mr ewmple,the argument, A, A->B. Pleref B. is M i d Making A true arrl Bfalse makes one premise \A->Bf,arrl he c a d u s i c m s \Bf alse. Making\Af alse and \B' true makes one premise, \Af alse arrl the c a d u s i o n \B'

true. Sane M i d rguneits do not have so rmcfi freedcm. B. -foreB. is valid O b v i ~ ~ ~ l yn this exanple the arrl d u s i o n alwayshave the- ruth value. B. Therefore Av-A. ~slso M i d ~ hiscase the canclusion s a logical truth a d o lw ys true ut these respecial cases In general anythhq can hagpen, depenlirg on the detailsof the case.


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WS) (QD) PSAX>( [D (r n >(NW P ShB) K PQP)&(GvP) Pw->-Pa) P

S A-B A-Bc-> (-Pa) 3 R-=(-Pa) 6,OE-Pa 5,7,>E-P a =G a =(QP) (GVP) 2 RQP =,aP 1O,l2,>E

-B 5-U,-IB 14, EShB 4r15ra

>K 1,RK 16,17,>Ej>K 4-18, >I

I2,R3,4,-m 3-5, >IA DG 3 RI , 5,6,>E4,R- i 5-8, IH 9t-E

4-10. >IAI 11,VIlw 11-l2, >INM; lO,U,>EG 14,&EDG +=,>I

AC3 H 2,RI, 13,14,l2,R< U-16, -I

I G 12-17, I*-Xi 11,18,01

QD 4,RD 17,18,>ESV 19,vI(rn)>((D(PVK))>(NW) 2 3(D(rn) >(NW 20,21,>EA

mK 23,vIF (rn 23-24,>INM; 22,25,>EN 26, E

17-27, IK-X 16,28,0I,vI3 R-D 9-11, 1

CV-B 2,l2,>E(kvB)&(CV-B) 8,U,&I


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IfBV 4 ,vIBV 1,2-3,4-5,AC

AM7,a

C 7,=I =) 8,10, 1vIAh =) 1,2-6,7-l1,AC

B3 ,R6,7,>E8,vI

VC 2,4-5,6-9,AC


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rv(=) PA2 vI2 , v I(AvB) (AW ) 3,4, Ipi A

B 68=C 6 1 aAvB 7 vIA v C 8 , v I( AvB ) ( AW ) 9 , 1 0 , IAVB) (AW) 1 2-5,6-11,AC

3 , RA 4 1 5 , aA D 21RD 6 , 7 , > ERVD 8 ,v I(AW) M) 4 - 9 , > I

l l , RU VI( AW ) m ) 1 2- 14 , ICAW) M)1 3 - l O , l l - l ~ , A C


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IJ 2,Ram u8U8s* 3 RD 14,I5,W3RrD 16,VI

RrD 5,6-ll,u-17,kC-(MI 48Rrn 5-19 IWD >- RU) 4-20, >I

12 X+>Y prqlut for erive ruleX -

1 XY lnp for ,rived ruleY


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-Irqut or derived rule-

A Irprt or derived rule23 I

MI Irprt or eriwruleX A

1 2,-1,R-X 2-4 -IA6,VII Y M ) 1 3-Y 6-8 -I-W-Y 5,91=

-x&-Y Irprt oreriw ruleA

(=y)P=-XC--Yx-YXYXLYXLY)-(-WY)-MrY

-f=v e d ruleA

2,M3 , s3 , s4,-E5,-E6,7,=1,R2-9, I10, E

- W Y Wyt orerive rule

12

D Y Irprt orrived rule-Y A

12

456789

- D Y Irprt orderived rule-Y Ar

-X A-DY 1 rRI 3,4,>E2 ,R-X 3-6,-IX 7t-E-Y>x 2%,>I


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PB A

L

L Rmm 2,3,>EVRI

4 316,--?DR 5-7, >Im-?DR) 2-8,>1

12

Inprtf=derived nile

- my) InpR ferive nile-X A

It' '?t ornved nile

12

M Inprt orderived nileX

123456789

W IbP) PEM PM AEM 2,RD 3,4,>EW IbP) 1rRbP 3,6,>EP 5,7,>EW P 3-8,>1


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7-5. we have a derivation w i t X as its only premise and Y and -Y asomzlus?ons. Relabel X as an a s m p t i o n , and make the le derivat ion thesub-denvation of a e s s rterderivation. The su b 4 e r i v a t i o nlice- dmwing X as final Ocndlusion of its OUterQrivation by lyingRD ~ a n y ~ o f t h e n e w t e s t f o r c c P r t s a d i c t i o n ~ n b e ~ t oan instance of the old test

[ I W ) P PM PJlu P3 P

rn 2 3T 3,4,>ED B 3B 3,6,>EB T 5,7 , ID( =) 3-8, >I( M ) > ( D ( B W ) 2- 9,> 1( B ) ( ) D ) ) 1-10, >I

F Af W AK 2,a~ 3,4,>E

(W )> F 2-5, > I-=- =a 6,-[IOF) -F>- W ) 1-7, >I-u

NN>(C K)c KK

KP)IOPPBP6lB- ( p a )


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- W-R) - (N-R) 1- W-R) 21=- M->-R) 21=* -R 3 1 m.I 5 ,-R 5 1 aR 7.-E


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V LID

V LID

V LID


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8-2. Since a omjunction, X U , is true whenever both X and Y are true whenwe have a omjunction on a tree both omjmzts shauld be w r i t t e n on mbrandl directly bel w th ccnjuncticm: X YXYbr a negated omjunction, -(X Y) , w first u s e - Law to get -Xv-Y.Then, we sinply use the mle for disjunctions.Similarly, t o get th mle for ccrditicndLs, X->Y, w use th OmriiticndLLaw to get MInd then us the mle for ~ u n c t i o n s .Again for negatedccrditionals -(X:>Y), w use th cxrditiavil Law to get X -Y, and then thmle for anjumtlons.Biaalitianals are a b i t more tricky. Firs t , w m i d e r wha t thbiccrditicnal XY is logically equivalent to: (X Y)v(-X -Y). Sinm this isi tself a disjunction w will have w brarrfres eacfi of wh i will have thdlsocapositim pxcdwts from a omjunction on it th d t s:xY

X -xY -YFor negated biccrditiavils, w note that -(XY) is logically ecprivalent t o-[(X->Y) (Y->X)]. By DeMCqan s Law, this is equivalent to -(X->Y)v-(Y->x);and, by the ccrditicmal law w get: (X -Y)v(Y -X). Again w use th rulesfor disjunctions and amjunctions to get w rarrfiea similar to that abave.

3 I-F I-xXINVAU D. C.E.: (X

1 * -(-S T)2 S3 I* -s4 S

XRWALD. C.E.: -5ti-T)

INVAIJD. C.E .: ( F P

1 * K v H P2 -x P3 * - H D) C4 Ix IH 1vX I5 i i ID 3 -RWALD. C.E.: (-K H -D) X

l X V A ID . C.E.: (F -X)


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INVALID. C.E. (+I -N)

I* I A- * v -A)IA*- I-AA

IT 4I IX

C.E.: ( I M )KI -K I-HX

INVWD C.E.: (K -H S)


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1 * -L)2 * -(L -C3 * -(F L)4

I-F I* -L5 LI I I6 -L -L

IT *I-- I X7 F -F8 -L L9 X X C CI-- I I--10 F -F F I-F11 -L L -L LX X

IWALXD C.E.: (-FKXL)

* C > N* -(I > C)* -(N V -I)* - 4-I)I-c

4* -I4* -IIII-c IN

C.E.: (I6dX-N) X

1213INKUZD C.E.: (Q -D B R)8 4. 01 R -S2 -(R T)3 -(R -S,.IR I-R5 s s

I6 R I-R4I-- I X8 -R * -S

9 S10 S-------11 R I-R4 TX I--

13 -R IsIWNXD C.E.: (-R S T)


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8-5. Indoingtruthtrees,wearecmcemedwithtbetruthvaluesofatauicwhich areminimilly sufficient to make tbe original entence true.Thus, i f a nile tells us to w r i t e : X-Y we could ju s t as well write tbestackintbereversearder ~ i s b e c a u s e i t i s t b e t r u t h v a l u e s o f t b eatauicsentencesthatwearecancernedwith, soarderdDesnotll~ltter[ N o t i c e t h a t X a n d - Y o n t h e s a m e s t a c k o a r r e s p c P d s t o (X -Y),whichislogically equivalent to (-Y X) .] Similarly, tbe order of braKfies does notmtter s h vY is logically equivalent toM

B* -[(B v C (B v D)]I* -(B v C * -(B v D)-B -B-C -DX X

8-6. By using DeMoqanls Law, w can ch rye a negated omjunction -(X Y) toa disjunction -Xv-Y, and tAen o x p e this using tbe rule for disjunctiarrs.In thisway, we can away with the rule for negated omjunctions.Similarly, we can &ange any negated disjunction -(XvY) to a omjunction,-X -Y, ard then use the nile for omjunctiarrs.We c n also do away with tbe rules for disjunctions and omjunctions, arduse cmly tbe rule for negated disjunctions ard negated -junctions. n thiscase, l mmnxw had a omjunction, X Y, on a tree we wxld use l I e M o ~ sLaw to get - -Xv-Y), ard use the rule for negated aisjunctions. Likewiseany disjlnrctim, XvY, could ke mnverted to a negated omjunction, -(-X -Y) ,using IxMxqan s Law. * D v FK > -F* -[-F > (D v K)]-F

* -(D v K-D8-7.

a) S h omjunction X Y Z is true wlwever X and Y and Z are a l ltrue whenever we have a omjuncticn with three omjuncts in a tree wesinply mite: XYZ um emeath the 01cigjna.l omjunction.Similarly, any disjunction MvZ is t r u e wbmver either X or or Z is true.So, on a tree we write three bmmhs urder tbe disjunction:

b) In general for a omjunction of any length, X Y . . Z, in a tree wewri te a l l o f the omjuxztx in one br rrch directly belay the originalomjunction. For adis junc t ionof any length, MrYv...vZ, i n a tree wewriteone branch for e cfi of tbe disjuxztx belay the original disjunction.8-8. By the truth table definition of tbe Sheffer Sbmk, we saw t ha t X Y islogica lly equivalent to -(XvY); a d by DeMxganls Law, it is logically@vale to -X -Y. So, in a truth tree -we see xIY, we write -Xarrd -Y o one bmmh directly belay it We also saw t ha t X ~ Xs logicallyequivalent to -X; so, wherwer XIX ~pe rs n a truth tree w siPply write-X dj rec t ly belay it


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[ ( F L - B ) v Q l > [ A L ( S v T ) lL - (S LA )* S V A- [ B > ( A > S ) ]F-(S L A)B-(A > SA*I- [ ( F L -B v Q 1 I* A ( S V T)-(F -8) AQ * s v T

s IA Is II- I* -A I-* I-A11 x I- -- I-P -B S TX B X

INVAUD C.E.:


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- { ( I 5 ) v [ ( J K v - (K I ) ] )- ( I L 4- [ ( J L K v - K I ) ]- ( J K)

1 ( P V G)(-P L 6 )*I* P v G I- ( P v G)3 * + L C -(-P L 6 )4 -P5 66 IP G7 -F89

I-P

IG1 P GT c r Q J X X

-([(C (A v D ) ) V - (C F ) ] V - (A L 6 ) )* - [ (C L ( A v D ) ) V - ( C L P ) ]-(A 6 )- [ C ( A v D ) ]-(C L F* A +* C FA

CP 1 * [ I v ( J K ) ] > [ ( I v J ) L K ]I[Iv ( J L K ) ] I( I V J ) L KI * I v J

4 - ( J K KI5 I-K II IJW IT A Q3MRAMCTRW. C.E. : -1- -1- XU) KU)


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1 * -([B (Mv -P)] > [ P > (-M v -B)])2 * B L ( Mv -P )3 * - ( P > (-M v -B))4 B5 * M v - P6 P7 * -(-M v -B)8 *+49 * -B10 M11 B12 M I-PXWT A CX3NIRADICmoN. C.E.: m

1 * [K > (D > P) ] [ ( - K v D) - (K > P ) ]2 * K > ( D > P )3 * (-K v D) -(K > P)4 * - K V D5 * -(K > P)6 K7 -PI I8 -K * D > PX9

I-K IDX I10 -D IPaNmADICmoN X X

1 * [ ( G V Q ) (K > G ) ] - ( P v -K2 * ( G v Q) (K > G)3 * - (P v -K)4 * G v Q5 * K > G6 -P7 * -K8 K

A9 IG IQ10 I I I-------K -K GX X XWT A aNlX4DICI'IoN. C.E.: (-F KKW)9-5. Flpm e ~ e ~ ~ : i s e-3 ve saw that two sentences X an dY are logicallyequivalentifamicnly if tbetwoarguments . lhereforeParrl Y.M a r e XI m ath valid S h e c n t st be validity of gumrbusing t ru th treps ve c n formilate a new way of t sting logical @valencewithtrepsbyusingthisinfappatim Tb ' w k t h e r X a n d Y a r elogically @vale , it is sufficient t o t st bath of tbe ars X y andY/Xforvalidity. N a t e ~ w i t h t h i s n e w m e t h o d eneedtwotruthtrees.


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OTOT

< Z< Z9

n2 ITfOT


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9-7.a Yes. caxd ig to the definition (Cl), a senten e of SL iswnsistent if ard only if it is not a rmtadic t ion . But, no logical trutham be a contradictian; so, a l l logical truths are amsistent.b S ~ a r m t a d i c t i i c n i s f i n e d a s a s m t e m x w h i d s n n o t b e t r u eun er any possi le assigment of truth values to ac senten e letters anysenten e w h i d has a t least one assignwrt of truth values to ac senbncel e t t e r swh i d mak es i t t me i smt acan t r ad i c t i o n . But, ifthesentemeist a colrtradiction, then it is consistent (Cl). Likewise, i f a senteme iswnsistentaazxdqto (Cl) , then it i sm taamt rad ic t i on ; Urns there isa t least one assigm mt of truth values to atcmic senten es which makes theserrtenoe true.C (C2) ssentially states the fact that the set (X,Y,Z) s consistent i fard only if there is an assigm mt of truth value8 to ac ser mes w h i dma ke sXa r dYa r dZa l l t r ue . But,whereverth is isthecase, thereisanassignwrt whid makes X Y Z true. lpi mrdirq to (Cl) , any Sentence o SLis amsistent i f it is not a -on I f WY Z s true in anassignwrt, then it ismt a cantradicticn. 'Iherefore, a set of sentences isconsistent amrdirq to C2) if ard only if the a m j d o n of its manbers isconsistat a to a).d) No, the given set ~sot wnsistent. According to a),set ofsentences is amsistent if ard only if there s an assigmmt of truth valuestosentence le t t e r swhidmakesa l l -of these t t rue . But, anyassignment which makes -A tru w i l l Mke -a false, ard vice versa.e ) ny infinite set of sentences whid dces mt include a sentence X ardits negation -X or some sentence or sentenxs whid imply -X) is awnsistent set. Wre is onevie: Al, A2 A3, .

-([L (S vT)][(L S) v (L T)I* L S VT ) 1- (L (S v T ))-[(L S) V (L T) ] (L S) V (L T)L* S v T I-L I-(S v T)-(L S)-(L T) I--I* L S * L T

9-7. f l )2

IS ITI---L I-S I---L I-SX X X I l-L L

X X L TS XX -S

I I* L S * L T


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Teller's Logic Manual Answer Key (Book 1)