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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‘rr =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 {Cfr, 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, ellEL Sinoe we have instead {e1 u‘erL and (ti elmEL 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 uELZ ﬁr!” L[G1,.U n Mir} —2 L[_t,t31) [gé] = T iff for all uELZ either whim LtuLﬂ a Mir} [35] = F or 134:!“ L{_t,o1) [33$] = T iff for all uELZ either sh!“ Lto1,_t) [33] = F or ﬁrst?” Mir) [35] = F or “.34an L[_t,C1) [35] = T
iff for all uELZ either Colitisg {3:} )EL“ or 3; (I) EM“ or (33” (it) ,uﬁﬁEL“ iff for all uELZ either Efﬁl'lﬂ or (IiU 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‘If {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) [35] = 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) [35] = T
iff for all nEU. either E132“, 335 {.t) ):ELcr 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. {Intuitivelsi 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.') [33;] = F or s4sz L{C3,.r} [33] = T
iff for all nE U. either {3;§{1.'),cflTIELE'“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 assistsis for f: {ﬁg’lEL fEM {d,ﬂEL
for 3: {3.sUEL 3Ei‘Itr [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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