hw_11_09 - Let the language contain the hinarj,P predicate...

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Unformatted text preview: Let the language contain the hinarj,P predicate L, the monadie predicates H, W, and M, and the individual constants o1 and C1. Let Elf be the model (U, L, H, W, M, e, d}, where: U = {0, d: e__ f: 31h} L = {(010}: CM")? {ref}, {dsln (with): '19.. e), {sf}, {eh}. {if}, (if), (its), {3191 {3:611 (319).. (sis), {halt {en} H = {-11 Isis} W = {e, at. e} M = {i 3, t1} [This means that M interprets the language so that Lfi‘r-r =L, H5”: H__ W“ = W, M“ =M, e1”: 0, or“: d.) We have the following truth value assignments for all assignments g of values to the variables: 1. L031, I32) v Lfflj, I31] [Intuitiver Either Claire likes Doris or Doris likes Claire] WI!“ L(C1,Gg} v L{C:I___C1} [g] = T iff either firfif L{e1,og} [3] = T or M395 L{c1,t:1jl [g] = T iff either [01”: Cf’QELM or {Cf-r, I31'""?JIEL3"’f iff either {e,d]EL or (d. t‘)EL Sinoe we have indeed [fltflEL it follows that My” L{G1__ I32} v Lte2,e1} [g] = 'I. Since the wff has no free variables: its truth value in M does not depend on 3. Hence it is true in M. {Intuitiver either Claire likes Doris or Doris likes Claire —whieh is true heeause Claire likes Doris.) 1 LEGLGz) —* Ltflzfiil [Intuitivelv If Claire likes Doris, then Doris likes Claire] WI!“ Ltflifisfl' w" LICLCi) [3] = T as either was! L{c1,og} [3] = F or “Pi-#5"! L{C1=C1:I'[g] = T iTT Elthfl' [C1311 131%ng DI (GEM, ClijLM iff either (e, cflEL or (d, ell-EL Sinoe we have instead {e1 u‘erL and (ti elm-EL it follows that “Mn?” LUZ], {:1} —3- L{Cj,fl1} [g] = F. Since the 1:Itff has no free variables, its truth value in Slrf does not depend on g. Henoe it is false mar. {Intuitivelv It is not the ease that if Claire likes Doris: then Doris likes Claire—bee ause Claire likes Doris hut Doris does not like Claire.) 3. 1'i.|".'tI:LI[Io11x} a Mfr) —:- L{.r:e1}} [Intuitivelv Every man whom Claire likes likes Claire] Mr!“ V_tI:L{o1,_tj a Mfr) —:- Llf.r1o1}} [3] = T iff for all uEL-Z fir!” L[G1,.U n Mir} —2- L[_t,t31) [gé] = T iff for all uEL-Z either whim LtuLfl a Mir} [3-5] = F or 134:!“ L{_t,o1) [33$] = T iff for all uEL-Z either sh!“ Lto1,_t) [3-3] = F or first?” Mir) [3-5] = F or “.34an L[_t,C1) [3-5] = T iff for all uEL-Z either Colitis-g {3:} )EL“ or 3-; (I) EM“ or (33” (it) ,ufi-fiEL“ iff for all uEL-Z either Effil'lfl or (Ii-U or {ELC‘J'EL Sinoe we have for e: {e1e}EL fix! to ma for u“: {e,d}EL fl“! Edejfl for e: jeleljfl get—M (admit. for f: {eutTrEL fEM [f egEL ssss 541 for 3: gagglEL 35H 3 c EL J for F1: gefilEL iiEi‘I-f {ii,c)EL I we have that every nEU satisfies at least one of the three desired conditions {underlined above). It therefore follows that sat-"3F 1'EI’.1.'EL{01..t:I a I'u'l{.t) —:- L{.t, C;)) [3] = T. Since the wff has no free variables, its truth value in M does not depend on 3. Hence it is true in 11f. {Intuitivele Ever}r man whom Claire likes likes Claire — which is true because the only man Claire likes is Frank. and he likes Claire.) V.ttL{Gg..t) a Mir) —:- L121; 133)) [Intuitiver Every; man whom Doris likes likes Doris, i.e.. Doris is liked by; the men she likes] We“ VA'IZLWLI) a M{.1.') —I- LEI. '31)] [3] = T iff for all nEU. diam: L{C3,.t) a Mir) E'- L{.t.Cg) [3 g] = T iff for all nEU. either sir!“ L{C3..1.') a M{.t) [397'] = F or We!“ L{.t,o3) [3-5] = T iff for all nEU. either 53W” L{Cg..t) [3 3%] = F or 'F/Ffiri'flf MD.) [3;] = F or atrial-’9“ L{.t,Cg) [3-5] = T iff for all nEU. either E132“, 3-35 {.t) ):ELc-r or 3£E.1.')EMM or (3;; {.r),c_v'-‘*-’)EL“-r iff for all nEU. either {dolfl or JEM or {n.sUEL Since we have for c: 5d: clEL (EM [c EL for d: 5d,;i1EL a”?! for e: 5d. elEL eEM Es,d)EL for f: 5d. DEL fE for 3: {d.3)EL 3Ei‘ltr :9 EL for F1: {ciliJEL i'i'EM {ELdJIEL ksaaaa we have that some nEU. namely; h {Harry}, does not satisfy; any of the three desired conditions. It therefore follows that Mrs“! V.ttL{cj,I) a MU) —1 LU. c;)) [3] = F. Since the wff has no free variables. its truth value in M does not depend on 3. Hence it is false in Elf. {Intuitivels-i It is not the case that every man whom Doris likes likes Doris— because Claire likes Harry; but he does not like Claire.) VIELUfijj a Mir) —:- L{fl:..t)) [Intuitivel'yi Doris likes every man who likes her] 5641f“ Vi'IiLLtfig) a M{.t) —:- LECQJLII} [3] = T iff for all nEU: SEW“ L{.1.',G_a) n Mir) —:v L{Cg,.1.') [3 3”] = T iff for all nEU. either see! L{.t,t33) n MU) [3;] : F or fire“ L{c3,.r) [33;] = T iff for all nEU: either fir?“ L{.t,c_i) [3 g] = F or Mrs“ M{.1.') [3-3;] = F or s4sz L{C3,.r} [3-3] = T iff for all nE U. either {3;§{1.'),cfl-TIELE'“r or 3-; {GEMM or {133553; {.t) )EL“ iff for all nEU: either {n.nfiEL or (JEEM or {d,n)EL Since we have for c: {c. c'IIEL ETM Ed,c)fl for d: 5d.g‘lEL "“ {d,d}EL for e: 5e,g‘lEL s’*M Ed,e)fl :1. la assists-is for f: {fig’lEL fEM {d,flEL for 3: {3.sUEL 3Ei‘I-tr [n‘ DIEL for Fa: thfilEL i'i'EM [if MEL we have that every nEU satisfies at least one of the three desired conditions {underlined above). It therefore follows Git!“ V.1'{L{.r,cg) a M{.r) —:~ L[33,.¥}] [3] = T. Since the wff has no free variables, its truth value in M does not depend on 3. Hence it is true in Elf. {Infuitl‘t’flljfi Doris likes every man who likes her—which is true because the onlv man who likes Doris is George. and she likes him.) ! J .Ui Ely‘fl'uS {11y} False in M] True in M1 (since the number 4 is indeed S ’d by all numbers) False in M3 Ely‘t'ruffiu :y —1 Stem) False in M] {since the number 4 is indeed S ’d by all cflier numbers) True in M1 If since the number 4 is indeed S ’d by all (other) numbers} False in M3 Vu‘fi'yufflsnu) —3- FWD False in M] True in M1 False in M3 VulfiSfaJi} —':~ F'flfl} True in M; [yacucus1y, since the number -1 doesn‘t 5 any number} False in M3 True in My (since the number 1 only 5’5 the numbers 1, 3, and 4} Vu‘fl'flfifanfl a Sims?) —3- Stan?) True in M; [yacucusly, since the number -1 doesn‘t 5 any number} True in M1 (since the number 1 3'3 every number} True in M5 Vu‘y'y‘U'wESfiuyj r» Sfuwj —:- Sfu,w}) True in M; {where S is indeed interpreted as a transitive relatian True in M1 Ifditrc} False in M3 {since 2 5’3 3 and 3 5’5 41 but 2 dces not 5 4) ‘rfu{P{uj —:- Elysl[u,y}) False in M] {since 4 is a P, but it doesn’t S anything) True in M1 (yacucuslyr since nctbing is P) True in 13".r'1_1 (since each of l 31 and 4 5’5 something) Vu‘fi'vVWImej A 51:10.11?) A Slimw} —2- v:w} Falge in M] {since 1, which is a P, 3:5 more than one nm11ber} True in M: (vacunuslyn since nothing '15 P) True in M5 (since each of 1 31 and 4 S ’5 exactly one number) HI 9. v: {114} $113.34; v = 3.2.3.4} _{1} v_ ...
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hw_11_09 - Let the language contain the hinarj,P predicate...

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