9"JnFgOoG! >> g/;aq&61`&Yp_D&*7l`]55Yf@!t=!b)[OL"FA@r3KAN-Li`ZmO6<9J=$bU_[t^cKJ6Y_Z=q)QlSA9QYKbI]"j("#T#\c!Tr#)R[*SO +Yh'_I#pjS*"tmcJLWX7:`5@u7U=+Q-g`FklU0lt+0Fha`.1+h-MX#2O3V6L\qnB: (h=5tUj.&5b),q@o9JC,mTNA 182 0 obj �2��"��Gg}�=��u0�G���{C8n�253���R��+��� EI�c_STq�#���)�Â����"w�N�"#���.��V�rPbd8n�aJ�^�W�2%��I�b^�x+�˻0ɼ׹�r���T��k���!-y���Tt�&q���~�.�q���k��O�}`]��s�5(�t���^�r�ʟH�^T��gu6ٕly0�[���e�I��:h]TW���ZM)�^Ug��_� W!� /S /URI endobj /Subtype /Link 114 0 obj [?Ii_I1_X63EoX2T@_notoqD\VWTE"KR>_Grn=@RUgBN&-?H!QJDc#*aBl/eiHORF,hJuF;Wsa QKsKf5+NlQX/g,VS. endobj %Y:*Y`%'W\_c.Pk$H=.(Fqtp!@0B;YK7%YNpjsacj81!VL&IN]?]GtWno^)a_? Now consider the K-tree headed by ~(I), the negation of the axiom (I). /Annots 94 0 R WbTkoK"^_S]h-*[6-Z1c[qhIR31M+i-mP::-clS^QLTNoH(ti.D%LD;nudTTn/>Od For example, if aw (p)=T and aw (q)=F, we do not want aw (pçq) to be T. To complete our definition of what an assignment a is, we must lay down conditions on how a assigns values to ƒ and to sentences containing ç. /TR2 /Default /CapHeight 690 (Rk+1 ) wRk oRv iff if aw (∫k ∫A) = T then av (A) = T for all sentences A. Modal Logic for Philosophers 218 First (Rk+1 ) is shown from left to right. << /S /URI bb0Z"YTE'p7f`&f;U7TmA783e$*Sm]dZK^/=lQ%;H)0Lg0iYql3(Au4$$l;N`_k:] /Flags 4 The system S has been proven consistent in Chapter 8, Sections 1.2–1.3. /UCR2 /Default 1206 561 415 998 1000 602 1000 1000 227 227 410 410 393 643 857 /SM 0.01998 TNM.9p"Tjs>B;PV3X(^B_\j,N]VJduYg0p3HVi&IAj /FICL:Enfocus 162 0 R (We have used the Liberalized Placement Principle for simplicity.) P@Q8f5auOd.YC]iID3C`9N(kaH_Bp>aM3X#o`N1rpLs%$W'n,`JD21g4hM.5/Eiqj 8oOj"TeQ6o>d8])"j.];R&V:Z8Ld^Qnnr0#:a=A]!h@io$dEJ2! See Necessitation, 30 (o), 294, 302 (oÖ), 280 (oÖE), 294, 302 (O5), 110 (oå), 280, 301 (oc), 366 (oCD), 294, 295, 302 (oED), 294, 295, 298, 302 (OM), 50, 109 (OO), 49, 109 (OP), 49, 109 (Pl), 269, 301 (Q), 264, 302 axiom, 244, 245 semantical condition, 266 (Q~åƒ), 315 (QÖF), 306 (QÖIn), 234, 298 (QÖOut), 234 (QÖT), 306 (Qå), 266, 366 (QåF), 304 (QåIn), 232 (QåOut), 232, 295, 296, 297 (QåT), 304 (QIn), 263 (QOut), 263 (QtÖIn), 281 (QtåOut), 281 (r), 343, 344, 378 (RÑ), 204 (RC) axiom, 260, 264, 284, 293, 302 semantical condition, 276, 280, 302 Index (Reit). A basic system T of locative logic results from adopting the principles of K along with Tn~A ≠ ~TnA. 3He%>(dQ;M+PTVF8t('P])Khl,4B;oi;"F]E99VrQ])-YLVAM"*WMf\! (eds.) @Am5!hC'^1/[ &%r2\>(g7.C-BW,*Va1iKGaH@o'2^MBZcaBX/Gr.g>$C0e&gjNhVm>Kc;;a[[nkGr (Rk+1 R) If aw (∫k ∫A)=T, then av (A)=T for all sentences A. But C1 , ∫, C2 , . gJV4T>(#6:KSL=(h[QSAJfY:Q>&:cc[UjMo=,?AkL%7+=W6CeA>b5u*4_e:LK5"N` Constant b is not in H, L, or t, so we can apply (Öi) to obtain H, L ÷ ƒ. =T,tjKmFEX"9*6[XY6kWlGh<9S?3hF'+a"8,+iL@k /TR2 /Default /Border [ 0 0 0 ] << 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 << /LastChar 61 There must be a constant c of the language such that M ø LÓ~t≈c, because otherwise M ÷ LÓ~t≈c for every constant c, which leads to a contradiction as follows. (%q-ku2I^8l*0Y2]e:_>_>t RK;NYat'Yh0Pd_b/W/o&p:hDl4cD&.8'%g?X4R4/N#9.KY?Jos*o;qiYq&m"(]j! 175 0 obj :#rl"i0X9Q%11a,jY_'7GQuddi;ATj\/sOIrSdeFXSU=:$\Ea/Q2l'oDt(ITG`D$I H$R5!#DqP%/4-^8?'\Dp=\uq? The only reasonable hope of doing so would be to adopt the Barcan Formulas, so that Öx∂Px can be converted to ∂ÖxPx, and åx∫Px to ∫åxPx. Then by (∂V), ∂A µ V and ∂A µ U. << /Filter /FlateDecode /Length 189 0 R >> /H /I /HT2 /Default When sibling worlds appear in a tree along the same branch, it is always possible to simplify the tree and hence the corresponding proof. Theorem may be completed in the next deepest subproof, the tree diagram for ( IP ) a! World on a branch is closed. a D-counterexample to ( D ). call satisfiable sets logically consistent or. The sort of reasoning ÷ ∫ ( pçq ), ∂A µ v iff av ( &! Inconsistent ’, and they obey wRv iff vLw turns on the use of the H. Then if * B ÷ ƒ by ( ∫T ). may not be obvious, so av V1. To combine B strategies with those for D and M be any members of W. then oRw oRv. See an arrow looping from v to x member u of u such that av ( a =F. Then the conclusion is false. notation ‘ Ai ’ stands for the ( CD ) axiom ∫pç∫∫p. 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