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Ӏ h x8E-8V@vd4 gӊ8  8 = 8^5;h9"D 8J=U8oE-85>8Ipf -!!P j M\A8  99+5ȀNf&8! 4z8oo!8oirǀ{m98˜I8#8N[7#of9r8>€98 s_[58N)91 8mzi>=+΀Vy4uUjc8 z*6O<ـ4B|]g4 e)88ٌ4VP4b8[E8 4NIF8ee_ ~ 9ʠ | 68̃B40;Lƹ<<8;04H8O68Bym:GF 4mRdF4s;u?k6 2{   Hockey is LiftLogic:  Hockey is Lifthypotheses 9K  Hockey is Lif`tNot$  Hockey is Lift a proposition, connective, or rule.jWa3alHj;aHPoX 4d%AY|:=8#Y;ľ>@Ā8In8fN8l:9;{E8HՁT4r={;>bA"80I8W_86.Nv4X:h8iQ 9q'u€G8d(b8y88E8!M4F8#M*8{i5 zBV8!^zl8M8Hl4em 4*?8::vt7{=d8 u!QV'dQ8%8pq8f8WY?8 4v8p1D8}bu=858$!N@cH6 4'8Ww?8 )8K8e!L8Q/R)S`UK" $8 8B8~k{8 Lb= Z3V8_dYV4 N 8sDl |ψ  88P4U3;kT /;8}>hK#5#E4G0#0ڀW>}Ѐ98g  V}84#2q8 `.|7M3T׀;W*ԀZ8zg?4>}8a}; ;Q -3a8;JjHes8E3 $8^58{^824hUuA#8eʦ 4Go 8oB~4 $[8 8xU. Ӏ h x8E-8V@vd4 gӊ8  8 = ;9"D 8J=U8oE-85>8Ipf -!!pN[4M\A8  99+5ȀNf&8! 4z8oo!8oirǀ{m98˜I8#*6O<ـ4B|]g4 e)88ٌ4VP4b8[E8 4NIF8ee_ ~ 95)Ef9r8>€98 s_[58N)91 8mzi>=+΀Vy4uUjc8 zG" | 68̃B408ʠ;~ƹ<<8;04H8O68Bym:CmRdF4s;uF ?f 2{   Hockey is LiftLogic:  Hockey is Lifthypotheses 9K'  Hockey is Lif3tNot a proposition, connective, or rule. "I  Hockey is LiftIt's a   Hockey is Liftjudgment@  Hockey is Lift we're making about some propositions, just like when we proved   Hockey is Lif`tB  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift.jWa3alHj;aHPoX 4d%AY|:=8#Y;ľ>@Ā8In8fN8l:9;8{E8HՁT4r={7>bA"80I8W_86.Nv4X:h8iQ 9q'u€G8d(b8y88E8!M4F8#M*8{i5 zBV8!^zl8M8Hl4em 4*?8::vt7{=d8 u!QV'dQ8%8pq8f8WY?8 4v8p1D8}bu=858$!N@cH6 4'8Ww?8 )8K8e!L8Q/R)S`UK" $8 8B8~k{8 Lb= Z3V8_dYV4 N 8sDl |ψ  88P4U3;kT /;8}>hK$45#E4G0#0ڀW>}Ѐ98g  V}84#2q8 `.|7M3T׀;W*ԀZ8zg?4>}8a}; ;Q -3a8;JjHes8E3 9"D 8J=U8oE-85>8Ipf -!!;c^58{^824hUuA#8eʦ 4Go 8oB~4 $[8 8xU. Ӏ h x8E-8V@vd4 gӊ8  8 =  M\A8  99+5ȀNf&8! 4z8oo!8oirǀ{m98˜I8#8N[7#of9r8>€98 s_[58N)91 8mzi>=+΀Vy4uUjc8 z*6O<ـ4B|]g4 e)88ٌ4VP4b8[E8 4NIF8ee_ ~ 9 | 68̃B408ʠ;~ƹ<<8;04H8O68Bym:es;uF 4mRdF4?0B 2{   Hockey is LiftLogic:  Hockey is Lifthypotheses  jF3  Hockey is LiftIn other words, our logic does not actually define   Hockey is Lifsttruth  Hockey is Lift:   Hockey is Lif`tB  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift. It really defines "  Hockey is Lifsttruth under assumptions  Hockey is Lift":   Hockey is Lif`t...  Hockey is Lift   Hockey is Lif`tB  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift. l^Glq8-Q &Z  <8R: 4T"D 4j/bT,2. 88x^)Q; <@8Kz 8?<|^ < <p <.<`6Z(<|0#!%<,<0hb<D<,24I $4FN<,2<t<4A..[B2h <n4L7 2s24~w : 54R+ZKo<4  .4{|?T* 4A 7S |Yo<:&27^p8',<:4i<<N4@<4?8ۥD8o*x=O*Q@"<|: <$ <\B<4b\ 98̋Lp<"2<" h<< F"Dtw q43x*<(b\F(<@<~t<$x<,8@<2v( <~D %8@{80.zD8 wnG ,8`uP o<D!|   "؀> } <l̀8_G=44<4RvX!4\fB8@?\<4X>!   <8ջ{#LSR<h'2@? /W*b 4Vr| ") >}|Հ& ZN?GyK=2f 8f ~wπZ. Axk# F72L` Ht6)<nP4/(5Q.=( <^| Z8gC N ; 88>QC  \0M F !3}X4]"|47h}d<  4z 4n* 4~ 4%>hd 4>7:4N\ Z 4P 4X\ 0#q?7<7<;%>z4+j%N^`<O 4e> 8*\G eb 8h>5`< r& 4GB<^l$D?4u4R66* *R<|>3W=8|a] 4KG? 8<Z B'݀P P4_Q<j  ='4-J9F&4,;t*vX#@>6πC4YYh@x8@:  8f$&Ie*<@J! 4J AI/Dw<xL< 823{,nS4dTL  ~q%+)[8A^8[m;94sUQ [  8? ̀4خ'8=*4n4r;őy&N8Z8#&.8Al<^8 Ig2_s4jA E %&N 9e80 <&8غ00 wS+j8@zM?> 23  Hockey is LiftLogic: what's it good for? * ``d J8  Hockey is LiftProgramming language researchers are interested in the "  Hockey is Lif`tcomputational content  Hockey is Lift" of   Hockey is Liftproofs  Hockey is Lift. I!%  Hockey is Liftwhat's this??" 23  Hockey is LiftLogic:   Hockey is Liftwhat's it good for? L  8X   Hockey is Liftevery   Hockey is Lif`tproof  Hockey is Lift of a proposition is also a   Hockey is Lif`tprogram  Hockey is Lift.?Y 23  Hockey is LiftLogic:   Hockey is Liftwhat's it good for? L  8X   Hockey is Liftevery   Hockey is Lif`tproof  Hockey is Lift of a proposition is also a   Hockey is Lif`tprogram  Hockey is Lift.V~'$~'$2?@ 23  Hockey is LiftLogic:   Hockey is Liftwhat's it good for?V8ff82 2 Fixedsys Excelsior 2.00"t  Fixedsys Excelsior 2.00s"t#1 Fixedsys Excelsior 2.00"t x Fixedsys Excelsior 2.00jjj"t) k +  Fixedsys Excelsior 2.00jjj"t:  Fixedsys Excelsior 2.00`"tA Fixedsys Excelsior 2.00"t* Fixedsys Excelsior 2.00`"tB  Fixedsys Excelsior 2.00"t-> Fixedsys Excelsior 2.00`"t Ajjjz>b  <L4E<<2Hmp < <6j8ֱS8'lS  $ 4G L <,FH)< 8P8n: >8@g4''#K444-%1s9Hxf8:=8xTc4f&M&<$jN'O'<$<F?Zb84E<`t!( ?"j4hܲ4 ^   4 vj* t 4n^ z9#*= -<8mku8p^<(2x*4C@(@G4Ĺ8 5j־y<p8 PF 8]t < ^ ='<2d,`-[9|+lk4<8\4ڢE)?Rv:TV<0X'4,L44&6"<4}p4``<8Z4~4:4,4;"HXl|<8 c94Y ? 2>  Hockey is LiftLogic:   Hockey is LiftCurry-Howard isomorphism 2] NAX   Hockey is LiftThis is called the   Hockey is LiftCurry-Howard isomorphism  Hockey is Lift.2Q aa///   Hockey is LifjjjtHaskell B.@u ll=1 j  Hockey is LifjjjtWilliam A.?s 2>  Hockey is LiftLogic:   Hockey is LiftCurry-Howard isomorphism 2] NAX ,  Hockey is Lif3tThis is called the Curry-Howard isomorphism.& yH$  Hockey is LiftIt's part of a   Hockey is LiftBERMUDA TRIANGLE  Hockey is Lift of   Hockey is Lif`tthree independent fields3  Hockey is Lift that found out they were all doing the same thing:?X 2>  Hockey is LiftLogic:   Hockey is LiftCurry-Howard isomorphism 2(&]X   Hockey is LiftCategory Theory /(w X   Hockey is Lif tIntuitionistic Logic b."X   Hockey is LiftLambda Calculus17j/ ~o~o,.60  Hockey is Lift(functional programming):l J!<94 &4<r )R(f *<L <^~4< @<^d<ldt  4z  <z(<   ռ+<2|4<Z &   ' &4w4g:~4 4&~rm < xf643x*<(b\F(<@<~t<$x<,8@<2v( <~D %8@{80.zD8 wnG ,8`uP xo<DPxhEv4 < 4ݔ"<|: <$ <\B<4b\ 98̋Lp<"2<" h<< F"DS UR3uрP{L=4.@8V88x^)Q; <@8Kz 8?<|^ < <p <.<`6Z(<|0#!%<,<0hb<D<,2;%>z4+j{#wS+j8@zMx`{W`%N^`<O 4e> 8*\G eb 8h>V <4u4R6x=O*Q@27^p8',<:4i<<N4@<4?8ۥD8o*6* *R<|>3W=8|a] Z 3/rc  A[\q_W+:<Z B'݀P P4_Q<j  ='4-J9F&4,;t*vX#@>6πC4YYh@x8@:  8f$&Ie*<@J! 4J A Z080?&`  <,N>w ."<D4I $4FN<,2<t<4A..[?n4L2h 7J HA Z8gC N xmJ?V8n& K{Cjf`D@8_C&8bU av82mt 8s?PU $7jL 4AJ4-$17m4O.F } <l̀8_G=44<4RvX!4\fB8@?\<4X>7VI26!   <8[T A+ɀ6|TV4E/8 jUCU2TNUaf;o'wN8,Xj-iW\2x_ƒFt%grgx.yHo$Q4?*kgTc={tCTЀ/48w{T`4NTh8 fTR& @TM Te<5m&2Z4G*x{T*x& 8:_ %tPt 1x9|<< @ "_ET$4tҹL; 8OW ׀Q3  44}d<  4z$44]"|W !3}F>0M \ 4%> 4~w N\ 4wX\ 4P ?0ptrf  4d4F!;w4a~9'u4l%8v7X$zr4x76 ^Hd<hp< D & H< rf.v<~z0l,p <l< ;5l  ?7J<(76aL4x72 D 4_W A4@V^ '=K9 1\@<^. 40$W4"4 z.,4z 4<9 7D  4O W*j 4<43~:/tX<j6 4XfL6444Nt  1WFYv <V (&7 $XD ;+41(M Uw3i ! r12e W:4k4trn'844Y4?@Nd  3"7p4\vL?vb4nIJ 7DJ4  c ce) S)$  Hockey is Lif`t Modal Logic   Hockey is Lift  Hockey is Lif}tgood for??6 2  Hockey is LiftModal Logic + y  Hockey is Lift"it is raining" ))  Hockey is Lift"if   Hockey is Lif`tA  Hockey is Lift   Hockey is Liftand  Hockey is Lift   Hockey is Lif`tB  Hockey is Lift   Hockey is Liftthen  Hockey is Lift   Hockey is Lif`tA  Hockey is Lift" !6B  Hockey is Lift"if the   Hockey is Liftgravitational constant  Hockey is Lift is) '6  Hockey is Lif`t6.6842 x 10  Hockey is Lif`t-11  Hockey is Lif`t N m  Hockey is Lif`t2  Hockey is Lif`t kg  Hockey is Lif`t-2 7&r9X   Hockey is Lift then I am   Hockey is Liftpresident of CMU  Hockey is Lift."?  2  Hockey is LiftModal Logic + y  Hockey is Lift"it is raining" ))  Hockey is Lif$t"if A and B then A" !6B!  Hockey is Lif$t"if the gravitational constant is) '6  Hockey is Lif$t6.6842 x 10  Hockey is Lif$t-11  Hockey is Lif$t N m  Hockey is Lif$t2  Hockey is Lif$t kg  Hockey is Lif$t-2 7&r9X +  Hockey is Lif$t then I am president of CMU."  &A 'X   Hockey is Lifsttrue. "it" is raining somewhere.sDDt_'Msg* 4\ Z,:()< )6w*R *_<  <<<"B!2<6>RX   Hockey is LiftBut worlds can have different   Hockey is Lif`tcontingent truths  Hockey is Lift. &,  < Hockey is LiftIn < Hockey is Liftlava world < Hockey is Lift, maybe I  < Hockey is Liftam < Hockey is Lift president of CMU... sy.  < Hockey is LiftIn many worlds like ours, it is  < Hockey is Lif`training < Hockey is Lift in Pittsburgh.8,-I -5j?< 29  Hockey is LiftModal Logic, judgmental styleV  . ~N2  Hockey is LiftOur powerful  Hockey is Lifltjudgmental  Hockey is Lift methodology allows us to   Hockey is Liftdefine  Hockey is Lift this logic easily.?~ 29  Hockey is LiftModal Logic, judgmental styleV  @. ~N2J  Hockey is Lif)tOur powerful judgmental methodology allows us to define this logic easily.  Bp8X   Hockey is LiftInstead of "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift," we'll say: "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue @  Hockey is Lift   Hockey is Lifstw  Hockey is Lift"?-B 29  Hockey is LiftModal Logic, judgmental style  Z"p8X   Hockey is LiftInstead of "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift," we'll say: "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue @  Hockey is Lift   Hockey is Lifstw  Hockey is Lift"  ~:$  Hockey is Lifstw  Hockey is Lift is a   Hockey is Lifstworld expressionB  Hockey is Lift. we need at least one world to get started, like a constant for "  Hockey is Lif`there  Hockey is Lift.",sP!>! &+(<x|<>&4 < O >  "4P[~8h:n4%ew-4no \̧SC4YC 8@ U4_s0hA4-V3 r>qm8 ,k#_h$  4Ig!Ai*dɀ'l4*r'4bg7Jv2#KpRtr<`LZ<* <& ,co**Y4z* )6t$i#V6<+G9΀Po  $ ?H܀7*H8{1ieqwG5m BI78 !܀04w<4PJ8$'B4eI4ѭPw7 Հ-xXO _8f@o 7%yE#1$4:b( &5% O8ʤj$S<ˀb 4+]4[r/8 T9OQ,Qa4s|c/?"#8yb' h,M$A,> )iaEs;>aC@s>jH Ko';ҷ$"8ŵ#~ڮH8 [88@8dkQx!  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Ӏ h x8E-8V@vd4 gӊ8  8 =  M\A8  99+5ȀNf&8! 4z8oo!8oirǀ{m98˜I8#8N[7#of9r8>€98 s_[58N)91 8mzi>=+΀Vy4uUjc8 z*6O<ـ4B|]g4 e)88ٌ4VP4b8[E8 4NIF8ee_ ~ 9 | 68̃B408ʠ;~ƹ<<8;04H8O68Bym:es;uF 4mRdF4?0B 2{   Hockey is LiftLogic:  Hockey is Lifthypotheses  jF3  Hockey is LiftIn other words, our logic does not actually define   Hockey is Lifsttruth  Hockey is Lift:   Hockey is Lif`tB  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift. It really defines "  Hockey is Lifsttruth under assumptions  Hockey is Lift":   Hockey is Lif`t...  Hockey is Lift   Hockey is Lif`tB  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift. l^Glq8-Q &Z  <8R: 4T"D 4j/bT,2. 88x^)Q; <@8Kz 8?<|^ < <p <.<`6Z(<|0#!%<,<0hb<D<,24I $4FN<,2<t<4A..[B2h <n4L7 2s24~w : 54R+ZKo<4  .4{|?T* 4A 7S |Yo<:&27^p8',<:4i<<N4@<4?8ۥD8o*x=O*Q@"<|: <$ <\B<4b\ 98̋Lp<"2<" h<< F"Dtw q43x*<(b\F(<@<~t<$x<,8@<2v( <~D %8@{80.zD8 wnG ,8`uP o<D!|   "؀> } <l̀8_G=44<4RvX!4\fB8@?\<4X>!   <8ջ{#LSR<h'2@? /W*b 4Vr| ") >}|Հ& ZN?GyK=2f 8f ~wπZ. Axk# F72L` Ht6)<nP4/(5Q.=( <^| Z8gC N ; 88>QC  \0M F !3}X4]"|47h}d<  4z 4n* 4~ 4%>hd 4>7:4N\ Z 4P 4X\ 0#q?7<7<;%>z4+j%N^`<O 4e> 8*\G eb 8h>5`< r& 4GB<^l$D?4u4R66* *R<|>3W=8|a] 4KG? 8<Z B'݀P P4_Q<j  ='4-J9F&4,;t*vX#@>6πC4YYh@x8@:  8f$&Ie*<@J! 4J AI/Dw<xL< 823{,nS4dTL  ~q%+)[8A^8[m;94sUQ [  8? ̀4خ'8=*4n4r;őy&N8Z8#&.8Al<^8 Ig2_s4jA E %&N 9e80 <&8غ00 wS+j8@zM?> 23  Hockey is LiftLogic: what's it good for? * ``d J8  Hockey is LiftProgramming language researchers are interested in the "  Hockey is Lif`tcomputational content  Hockey is Lift" of   Hockey is Liftproofs  Hockey is Lift. I!%  Hockey is Liftwhat's this??" 23  Hockey is LiftLogic:   Hockey is Liftwhat's it good for? L  8X   Hockey is Liftevery   Hockey is Lif`tproof  Hockey is Lift of a proposition is also a   Hockey is Lif`tprogram  Hockey is Lift.?Y 23  Hockey is LiftLogic:   Hockey is Liftwhat's it good for? L  8X   Hockey is Liftevery   Hockey is Lif`tproof  Hockey is Lift of a proposition is also a   Hockey is Lif`tprogram  Hockey is Lift.V~'$~'$2?@ 23  Hockey is LiftLogic:   Hockey is Liftwhat's it good for?V8ff82 2 Fixedsys Excelsior 2.00"t  Fixedsys Excelsior 2.00s"t#1 Fixedsys Excelsior 2.00"t x Fixedsys Excelsior 2.00jjj"t) k +  Fixedsys Excelsior 2.00jjj"t:  Fixedsys Excelsior 2.00`"tA Fixedsys Excelsior 2.00"t* Fixedsys Excelsior 2.00`"tB  Fixedsys Excelsior 2.00"t-> Fixedsys Excelsior 2.00`"t Ajjjz>b  <L4E<<2Hmp < <6j8ֱS8'lS  $ 4G L <,FH)< 8P8n: >8@g4''#K444-%1s9Hxf8:=8xTc4f&M&<$jN'O'<$<F?Zb84E<`t!( ?"j4hܲ4 ^   4 vj* t 4n^ z9#*= -<8mku8p^<(2x*4C@(@G4Ĺ8 5j־y<p8 PF 8]t < ^ ='<2d,`-[9|+lk4<8\4ڢE)?Rv:TV<0X'4,L44&6"<4}p4``<8Z4~4:4,4;"HXl|<8 c94Y ? 2>  Hockey is LiftLogic:   Hockey is LiftCurry-Howard isomorphism 2] NAX   Hockey is LiftThis is called the   Hockey is LiftCurry-Howard isomorphism  Hockey is Lift.2Q aa///   Hockey is LifjjjtHaskell B.@u ll=1 j  Hockey is LifjjjtWilliam A.?s 2>  Hockey is LiftLogic:   Hockey is LiftCurry-Howard isomorphism 2] NAX ,  Hockey is Lif3tThis is called the Curry-Howard isomorphism.& yH$  Hockey is LiftIt's part of a   Hockey is LiftBERMUDA TRIANGLE  Hockey is Lift of   Hockey is Lif`tthree independent fields3  Hockey is Lift that found out they were all doing the same thing:?X 2>  Hockey is LiftLogic:   Hockey is LiftCurry-Howard isomorphism 2(&]X   Hockey is LiftCategory Theory /(w X   Hockey is Lif tIntuitionistic Logic b."X   Hockey is LiftLambda Calculus17j/ ~o~o,.60  Hockey is Lift(functional programming):l J!<94 &4<r )R(f *<L <^~4< @<^d<ldt  4z  <z(<   ռ+<2|4<Z &   ' &4w4g:~4 4&~rm < xf643x*<(b\F(<@<~t<$x<,8@<2v( <~D %8@{80.zD8 wnG ,8`uP xo<DPxhEv4 < 4ݔ"<|: <$ <\B<4b\ 98̋Lp<"2<" h<< F"DS UR3uрP{L=4.@8V88x^)Q; <@8Kz 8?<|^ < <p <.<`6Z(<|0#!%<,<0hb<D<,2;%>z4+j{#wS+j8@zMx`{W`%N^`<O 4e> 8*\G eb 8h>V <4u4R6x=O*Q@27^p8',<:4i<<N4@<4?8ۥD8o*6* *R<|>3W=8|a] Z 3/rc  A[\q_W+:<Z B'݀P P4_Q<j  ='4-J9F&4,;t*vX#@>6πC4YYh@x8@:  8f$&Ie*<@J! 4J A Z080?&`  <,N>w ."<D4I $4FN<,2<t<4A..[?n4L2h 7J HA Z8gC N xmJ?V8n& K{Cjf`D@8_C&8bU av82mt 8s?PU $7jL 4AJ4-$17m4O.F } <l̀8_G=44<4RvX!4\fB8@?\<4X>7VI26!   <8[T A+ɀ6|TV4E/8 jUCU2TNUaf;o'wN8,Xj-iW\2x_ƒFt%grgx.yHo$Q4?*kgTc={tCTЀ/48w{T`4NTh8 fTR& @TM Te<5m&2Z4G*x{T*x& 8:_ %tPt 1x9|<< @ "_ET$4tҹL; 8OW ׀Q3  44}d<  4z$44]"|W !3}F>0M \ 4%> 4~w N\ 4wX\ 4P ?0ptrf  4d4F!;w4a~9'u4l%8v7X$zr4x76 ^Hd<hp< D & H< rf.v<~z0l,p <l< ;5l  ?7J<(76aL4x72 D 4_W A4@V^ '=K9 1\@<^. 40$W4"4 z.,4z 4<9 7D  4O W*j 4<43~:/tX<j6 4XfL6444Nt  1WFYv <V (&7 $XD ;+41(M Uw3i ! r12e W:4k4trn'844Y4?@Nd  3"7p4\vL?vb4nIJ 7DJ4  c ce) S)$  Hockey is Lif`t Modal Logic   Hockey is Lift  Hockey is Lif}tgood for??6 2  Hockey is LiftModal Logic + y  Hockey is Lift"it is raining" ))  Hockey is Lift"if   Hockey is Lif`tA  Hockey is Lift   Hockey is Liftand  Hockey is Lift   Hockey is Lif`tB  Hockey is Lift   Hockey is Liftthen  Hockey is Lift   Hockey is Lif`tA  Hockey is Lift" !6B  Hockey is Lift"if the   Hockey is Liftgravitational constant  Hockey is Lift is) '6  Hockey is Lif`t6.6842 x 10  Hockey is Lif`t-11  Hockey is Lif`t N m  Hockey is Lif`t2  Hockey is Lif`t kg  Hockey is Lif`t-2 7&r9X   Hockey is Lift then I am   Hockey is Liftpresident of CMU  Hockey is Lift."?  2  Hockey is LiftModal Logic + y  Hockey is Lift"it is raining" ))  Hockey is Lif$t"if A and B then A" !6B!  Hockey is Lif$t"if the gravitational constant is) '6  Hockey is Lif$t6.6842 x 10  Hockey is Lif$t-11  Hockey is Lif$t N m  Hockey is Lif$t2  Hockey is Lif$t kg  Hockey is Lif$t-2 7&r9X +  Hockey is Lif$t then I am president of CMU."  &A 'X   Hockey is Lifsttrue. "it" is raining somewhere.sDDt_'Msg* 4\ Z,:()< )6w*R *_<  <<<"B!2<6>RX   Hockey is LiftBut worlds can have different   Hockey is Lif`tcontingent truths  Hockey is Lift. &,  < Hockey is LiftIn < Hockey is Liftlava world < Hockey is Lift, maybe I  < Hockey is Liftam < Hockey is Lift president of CMU... sy.  < Hockey is LiftIn many worlds like ours, it is  < Hockey is Lif`training < Hockey is Lift in Pittsburgh.8,-I -5j?< 29  Hockey is LiftModal Logic, judgmental styleV  . ~N2  Hockey is LiftOur powerful  Hockey is Lifltjudgmental  Hockey is Lift methodology allows us to   Hockey is Liftdefine  Hockey is Lift this logic easily.?~ 29  Hockey is LiftModal Logic, judgmental styleV  @. ~N2J  Hockey is Lif)tOur powerful judgmental methodology allows us to define this logic easily.  Bp8X   Hockey is LiftInstead of "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift," we'll say: "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue @  Hockey is Lift   Hockey is Lifstw  Hockey is Lift"?-B 29  Hockey is LiftModal Logic, judgmental style  Z"p8X   Hockey is LiftInstead of "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue  Hockey is Lift," we'll say: "  Hockey is Lif`tA  Hockey is Lift   Hockey is Liflttrue @  Hockey is Lift   Hockey is Lifstw  Hockey is Lift"  ~:$  Hockey is Lifstw  Hockey is Lift is a   Hockey is Lifstworld expressionB  Hockey is Lift. we need at least one world to get started, like a constant for "  Hockey is Lif`there  Hockey is Lift.",sP!>! &+(<x|<>&4 < O >  "4P[~8h:n4%ew-4no \̧SC4YC 8@ U4_s0hA4-V3 r>qm8 ,k#_h$  4Ig!Ai*dɀ'l4*r'4bg7Jv2#KpRtr<`LZ<* <& ,co**Y4z* )6t$i#V6<+G9΀Po  $ ?H܀7*H8{1ieqwG5m BI78 !܀04w<4PJ8$'B4eI4ѭPw7 Հ-xXO _8f@o 7%yE#1$4:b( &5% O8ʤj$S<ˀb 4+]4[r/8 T9OQ,Qa4s|c/?"#8yb' h,M$A,> )iaEs;>aC@s>jH Ko';ҷ$"8ŵ#~ڮH8 [88@8dkQx!  O >  "4P[~8h:n4%ew-4no \̧SC4YC 8@ U4_s0hA4-V3 r>qm8 ,k#(  4Ig!Ai*dɀ'l4*r'4bg7Jv2#h$ `LZ<* <& 946`87l]͗b,c V 8?! +;̗+a  *|J&[Yd 8B4(8Z3ji 8y*k7E/80u<8W' @m2R"7-y/ /"!ڀJy0`>,co**Y4z* )6t$i#V6<+G9΀Po  $ ?H܀7*H8{1ieqwG5m BI78 !܀04w<4PJ8$'B4eI4ѭPw7 Հ-xXO _8f@o 7%yE#1$4:b( &5% O8ʤj$S<ˀb 4+]4[r/8 T9OQ,XK4s|c/ 88#8yb' h,M$A,> )i946`84<}#4U+~( ,&Ha25 ,5  l '2o 0sE g) 7]͗b,c V 8?! +;̗+a  *|J&[Yd 8B4(8Z3ji 8y*k7E/80u<8W' @m2R"7-yC& 8AxH 8sk`v]/8ZHd8N@8A/8:%T8kd! rx"_ ) N" z ZrP0,!!S 82os\8Yw 8 ~~8O^,B < Jn;MC& 8AxH 8sk`v]/8ZHd8N@8A/8:%T8kd!{Z p:ܛ43Cd%VQ r ʱ4KbO04%x~퀈  . 0\ 8 P 58XQ)YF  41PXc2Q_4^4Lc 9\'W @#݀!+0U8,"S6 }d44 8^k8J%(| b*L4S4(%2,t :!%2,t *L4S4(?f  29  Hockey is LiftModal Logic, judgmental style$ ssnZI"  Hockey is LiftJust like gave us , our new   Hockey is Lifltjudgmental machinery  Hockey is Lift gives us new  Hockey is Liftconnectives  Hockey is Lift.wlsr.@<Z<*Z< ?NLf <rv?2$h<? :t2_4B&4 h )4f%M ?@t\<&'<(7m42B BGKc)4> % 4I$0 ' A&@ +=N 0D <Z4f 4Z<<Z@ <~4I?X<N&<`p<.TZ<@l<" !h>}|%< ><! #38z 8r3 4B<\xJ=/88 ipz?@(\!8UP*84I8k8ER<8`L~ @<8ZQ4|8K  +]  48 *8L-n8=c>4@4p׹8ZK?WF 2!  Hockey is LiftModal Logic: rules !S&   Hockey is Liftintro.Vo!o!//G IO  Hockey is LiftIn order for   Hockey is Lif`tA  Hockey is Lift to be   Hockey is Liftnecessarily true,  1 L<X   Hockey is Liftwe   Hockey is Lifstassume  Hockey is Lift a world   Hockey is Lifstw'  Hockey is Lift exists, about which we  Hockey is Liftknow nothing  Hockey is Lift..ls`s-@>24E  4NTVC4<  <h % 4B 9: 4s  3 & <$4 Xs%j 8`<2D  $38QOB8 f=r4.8m8gw4t%T8     4GLTt <4X-X5!%E82?2" *~>2 j! T4 4M L"b !< F!8+2<&.v    8.8 ghR <"*LD<^NL&  88x^)Q; <@8Kz 8?<|^ < <p <.<`6Z(<|0#!%<,<0hb<D<,24I $4FN<,2<t<4A..[B2h <n4L7 2s24~w : 54R+ZKo<4  .4{|?T* 4A 7S |Yo<:&27^p8',<:4i<<N4@<4?8ۥD8o*x=O*Q@"<|: <$ <\B<4b\ 98̋Lp<"2<" h<< F"Dtw q43x*<(b\F(<@<~t<$x<,8@<2v( <~D %8@{80.zD8 wnG ,8`uP o<D!|   "؀> } <l̀8_G=44<4RvX!4\fB8@?\<4X>!   <8<~z $4x"?Ht6)<n/(5Q.=4b <^| Z8gC N ; 88>QC  \0M F !3}X4]"|47h}d<  4z 4n* 4~ 4%>hd 4>7:4N\ Z 4P 4X\ 0#q?7<7<;%>z4+j%N^`<O 4e> 8*\G eb 8h>5`< r& 4GB<^l$D?4u4R66* *R<|>3W=8|a] 4KG? 8<Z B'݀P P4_Q<j  ='4-J9F&4,;t*vX#@>6πC4YYh@x8@:  8f$&Ie*<@J! 4J AI/Dw<xL< 823{,nS4dTL  ~q%+)[8A^8[m;94sUQ [  8? ̀4خ'8=*4n4r;őy&N8Z8#&.8Al<^8 Ig2_s4jA E %&N 9e80 <&8غ00 wS+j8@zM?/ 2+,  Hockey is LiftModal Logic: application 2 7  Hockey is LifGGtLogic  Hockey is Lift gives us   Hockey is Lif`tprogramming languages  Hockey is Lift through the   Hockey is LiftCH isomorphism  Hockey is Lift.?il 2+,  Hockey is LiftModal Logic: application 2 7@  Hockey is Lif.tLogic gives us programming languages through the CH isomorphism.  7X "  Hockey is LiftCool, this thing is good for like  Hockey is Lifǎt1,000,000  Hockey is Lift theses!!?( 2+,  Hockey is LiftModal Logic: application  JV&  Hockey is LiftModal logic  Hockey is Lift lets us talk about   Hockey is Lifgttruth  Hockey is Lift from   Hockey is Liftmultiple perspectives  Hockey is Lift. 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