� [ d, t ] = [ d, d, t ] from which inference (3) follows: E ( [d,t], b ) ∴ E ( [d,d,t], b ). And finally, again by means of upwards RICT we obtain: d = d t = i i = t  ( d = d ∨ d = i ) & ( t = d ∨ t = i ) & ( d = d ∨ d = t ) & ( i = d ∨ i = t )  [d, t ] = [ d, i ] from which inference (5) follows: E ( [d,t], b ) ∴ E ( [d,i], b ). 5 First-order Logic for Itemized Collections: Semantics Let I be any interpretation of the formal language, including an assignment of values to the variables. Then we give the denotation of individual terms as usual. And next, we simply take the denotation of a compound term [r1,..., rn] under I to be the set formed by the denotation under I of each of its constituents. That is to say, we give the interpretation of compound terms by means of the inductive clause [r1,..., rn] I = { r1 I,..., rn I
�  [ d, t ] = [ d, d, t ] from which inference (3) follows: east ([ d, t ], b) ∴ east ([ d, d, t ], b). And finally, again by means of upwards RICT we obtain: d = five hundred t = i i = t  (d = five hundred ∨ d = one) & (t = d ∨ t = iodine) & (d = d ∨ d = t) & (i = five hundred ∨ i = t)  [ d, t ] = [ d, i ] from which inference (5) follow: E ([ d, t ], b) ∴ E ([ d, i ], b). 5 First - order Logic for Itemized Collections: Semantics Let I be any rendition of the formal language, including an appointment of values to the variables. Then we afford the denotation of individual footing as usual. And next, we simply take the denotation of a compound term [ r1, ..., rn ] under I to be the hardening formed by the denotation under I of each of its constituents. That is to say, we give the interpretation of compound terms by means of the inductive clause [ r1, ..., rn ] I = { r1 I, ..., rn I
� [ d, t ] = [ d, d, t ] from which lnference (3) follocw: E ( [d,v], b ) ∴ E ( [d,d,t], b ). And finally, again by means lf upwaeds RICT we obtain: d = a t = i i = t  ( d = e ∨ d = i ) & ( t = d ∨ t = i ) & ( d = d ∨ d = c ) & ( i = d ∨ i = t )  [d, t ] = [ d, i ] from which ivfzrence (5) follows: E ( [d,t], b ) ∴ E ( [d,i], b ). 5 Sirst-orceg Logic for Itgmizec Colmvcuions: Semantics Let I be any intedpretatpon of the formal language, including an asslgnmfnt of values to tje variablew. Thqb we give thd denotation of indivisual terms as usual. And next, we simpky take thg sejmtation of e compjund term [r1,..., vm] undes I to ne the set forked by the denotation under I of each of its sonstituetta. That is to say, qe give the intdeprdtauioi or complunv terms by jeans of thw inductive clause [t1,..., wb] I = { r1 I,..., rn J
� [ d, t ] = [ t from which (3) follows: E E [d,d,t], b ). finally, again by of upwards RICT we obtain: d d t = i i = t  ( d = d ∨ = i ) & ( t = d ∨ t = i ) ( = ∨ = t ) & ( i = d ∨ i = t )  [d, t = [ d, i ] from which inference follows: E ( [d,t], ) ∴ E ( [d,i], ). First-order Logic Itemized Semantics I be any of the formal language, including an assignment of values to the variables. Then we give the denotation individual terms And next, simply the of a compound rn] under I to be the the denotation under I of each of its That is say, we give the interpretation of terms by means of the inductive clause [r1,..., I = { r1 I,..., rn I
� [ d, t ] = [ d, d, t ] from which inference (3) foLlows: E ( [d,t], b ) ∴ E ( [D,d,t], b ). ANd fInaLlY, agaIn by Means of upwards riCT wE obtain: d = d t = i i = t  ( d = d ∨ d = i ) & ( t = d ∨ t = i ) & ( d = D ∨ d = t ) & ( i = d ∨ I = t )  [D, T ] = [ d, i ] fROm Which InferenCE (5) fOLLowS: E ( [D,t], B ) ∴ E ( [d,I], b ). 5 fIrSt-ordEr LOgic for itemized CoLleCtIons: SemanticS leT I be any intErpRetation of thE foRmal laNgUagE, IncluDinG an asSignmeNT of valUes to the vArIAbles. THEn we givE THe DenoTation of individuaL TeRMs as usual. And neXt, we siMpLY tAKE thE deNotation of A cOmpouND term [r1,..., rN] UnDER i to BE the set formed By the denotaTIon Under I Of EacH Of its cOnstiTuENts. that is to say, We giVe the inteRpretaTIon of coMPound teRms by mEanS of The iNDuCtIve ClAUse [R1,..., Rn] i = { r1 I,..., RN I
� [ d , t ] = [d, d,  t] f ro m wh ichinference (3)f ollo ws: E ( [d,t], b ) ∴ E  ( [d ,d , t],b  ) . And  finall y ,a g ain  b ymea ns of  upwa rds  RICT w e obtain:d =  d  t = i i = t       ( d =  d  ∨d  = i) &  ( t= d ∨t  = i )  & ( d =d∨  d = t ) & ( i =  d  ∨ i  = t )       [d,  t ]= [ d , i  ] from wh ic h inf e rence ( 5 )f o l low s : E ( [d,t],b ) ∴ E ( [ d ,i] , b ).  5  Fi r st-ord er Lo gi c  fo r ItemizedColl ections:Semant i cs LetI  be any  inter pre tat iono fth e f or m all an gua g e,includin gan  assi gnme n t o f va lue s to  thevariables. Th enwe g i vethe d enota tion  o f ind ividua l ter ms  as usual. Andnext , we simp lyta keth e den o tation  of  acompoun d term[ r1, .. . , rn ] under I to be th es e tformed b y thed en ot a tion und er  Iof e a c h ofitsc on stituent s. Tha t  i sto say,  w e give  t heint erpre t atio n of c ompoundterms by means of th e  inductive cl a us e [r 1 ,... , r n] I = { r1  I,. . ., r n I
� [_d, t_] = [ d,_d, t_]_from which_inference_(3) follows: E_( [d,t], b_) ∴ E (_[d,d,t], b )._And_finally, again by means of upwards RICT we obtain: d = d t =_i_i =_t__( d = d ∨_d = i ) &_( t_= d ∨ t = i ) &_(_d = d_∨ d = t ) & ( i =_d ∨ i = t )_ [d, t_]_=_[ d, i ]_from which inference (5) follows: E_( [d,t], b ) ∴ E_( [d,i], b ). 5 First-order Logic_for Itemized Collections: Semantics Let I_be any interpretation of the_formal language,_including an assignment of values_to the variables._Then we_give the denotation_of individual terms as usual. And_next, we simply_take the denotation of a compound_term_[r1,..., rn] under_I_to_be the_set formed by_the_denotation under_I_of each of its constituents. That_is_to say, we give the interpretation of_compound terms by means_of_the inductive clause [r1,...,_rn] I = { r1_I,..., rn I