1 Prawitz style vs. Fitch style. 2 First examples. Here is an example of a derivation ofa B,B C,A C:

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1 Math 4680, Toics i Logic ad Coutatio, Witer 2012 Lecture Notes 5: Fitch style atural deductio Peter Seliger 1 Prawitz style vs Fitch style I revious lecture otes, we have used the Prawitz style resetatio of atural deductio: a deductio is a certai kid of tree whose leaves rereset hyotheses, ad whose root reresets a coclusio These trees are colicated by two facts: first, soeties hyotheses are discharged ( crossed out ) durig a roof, ad secod, the freshess coditios i the quatifier rules require that certai variables do ot occur freely i the hyothesis ad coclusios of certai subtrees, excludig hyotheses that have already bee discharged at the tie the quatifier rule is alied There is also a certai ractical disadvatage to writig roofs as trees: large roofs ted to be uch wider tha they are high, so oe quickly rus out of sace Here, we briefly describe a alterative otatio for atural deductio derivatios, called the Fitch style otatio It is ore liear, i the sese that a roof is essetially a list of forulas, oe o each lie, ad forulas o later lies are eat to be cosequeces of forulas o earlier lies Istead of crossig out hyotheses, the Fitch style otatio uses idetatio to idicate a subderivatio usig a teorary hyothesis 2 First exales Here is a exale of a derivatio of, C, C: 1 2 C 3 4 E,1,2 5 C E,2,4 Note that lies 1 3 cotai hyotheses; each subsequet lie is justified by a rule Soeties a derivatio cotais a subderivatio that deeds o a hyothetical, or teorary, assutio Such subderivatios are ideted ad arked with aother vertical lie For exale, here is a derivatio of, C C: 1 2 C 3 4 E,1,2 5 C E,2,4 6 C I,3 5 O lie 3, we teorarily assue Lies 4 ad 5 are cosequeces The subderivatio eds o lie 5 with coclusioc; therefore, oe has roved C at the ext level u o lie 6 I its silest for, a Fitch style atural deductio is just a list of ubered lies, each cotaiig a forula, such that each forula is either a hyothesis (searated fro the rest of the roof by a horizotal lie), or else follows fro revious forulas (idicated by a rule ae ad lie ubers of relevat forulas) The very last lie i the derivatio cotais the coclusio 1 2

2 3 The rules of Fitch style atural deductio Cojuctio itroductio ( I) I,, I,, Cojuctio eliiatio ( E) E, E, Disjuctio Itroductio ( I) I, I, Disjuctio Eliiatio ( E) ϕ q r ϕ s ϕ E,,,q r Ilicatio Itroductio ( I) + 1 I, Ilicatio Eliiatio ( E) E,, E,, 3 Negatio Itroductio ( I) + 1 I, Negatio Eliiatio ( E) E,, E,, Cotradictio Eliiatio ( E) C E, Proof by Cotradictio (C) + 1 C, Reetitio (R) R, Forall-itroductio ( I) u [u/x] + 1 x I, Forall-eliiatio ( E) x [t/x] E, Exists-Itroductio ( I) [t/x] x I, Exists-Eliiatio ( E) x u [u/x] ϕ + 1 ϕ E,, 4

3 4 Rearks The bicoditioal ( ) To silify our foral roof syste, we do ot itroduce ay secial rules for the coective Istead, we sily regard the forula as a abbreviatio for( ) ( ) Falsity ( ) The sybol stads for cotradictio or falsity The forula is always false, ad it is used i the rules for egatio ad cotradictio above Negatio ( ) s we have doe before, it is ossible to regard egatio as a abbreviatio for I this case, the egatio itroductio ad eliiatio rules are sily istaces of the ilicatio itroductio ad eliiatio rules Reetitio (R) Let be a forula writte at liek (either as a hyothesis, or as a forula already rove) The oe ca reeatat lie if: (1)k <, ad (2) every vertical fro lie k cotiues without iterrutio to lie Exales of reetitio: Quatifiers I the rules for quatifiers: ut ot this: i E ad I,tis ay ter k R,k i I ad E, u is a fresh variable Here fresh eas that this variable does ot occur aywhere else i the derivatio It ay oly occur i the subderivatio fro lies The u that is writte betwee the vertical lies o lie is called a guard it serves as a reider that u ust be fresh i this subderivatio I articular, this eas that o forula cotaiig u ca be iorted (reeated) ito lies fro outside lies lso, this eas that u caot occur i the forula ϕ i lies ad +1 of E This is ok: k R, k This too: k R,k 5 6

4 5 Loger exales Without usig the logical equivalece rule, we derive oe directio of Morga s law for disjuctio, ( ) 1 ( ) 2 3 I,2 4 ( ) R,1 5 E,3,4 6 I, I,7 9 ( ) R,1 The ext two exales use quatifiers 10 E,8,9 11 I, I,6, 11 1 x((x) (x)) 2 y (y) 3 u (u) 4 x((x) (x)) R, 1 5 (u) (u) E, 4 6 (u) E,3, 5 7 y(y) I,6 8 y(y) E,2,3 7 9 y(y) y(y) I,2 8 6 No-exales No-exale 1 1 x((x) (x)) 2 y (y) 3 u (u) 1 xp(x,x) 2 u xp(x,x) R, 1 3 P(u, u) E, 2 4 zp(u,z) I,3 5 y zp(y,z) I,2 4 6 xp(x,x) y zp(y,z) I,1 5 4 x((x) (x)) R, 1 5 (u) (u) E, 4 6 (u) E, 3, 5 7 (u) E,2,3 6 8 y (y) (u) I, 2 7 No-exale 2 1 xp(x,x) 2 P(u, u) E, 1 3 u P(u, u) R, 2 4 zp(u,z) I,3 5 y zp(y,z) I,2 4 6 xp(x,x) y zp(y,z) I, 1 5 WRONG, because u is ot fresh i lies 3 6 (u ust ot occur i lies 6,7,8) WRONG, because u is ot fresh i lies 3 4 (u caot be reeated ast the guard fro lie 2 to lie 3) 7 8

5 No-exale 3 1 x((x) y(x,y)) 2 (y) 3 (y) y(y,y) E,1 4 y(y,y) E,2, 3 Exale WRONG, because the substitutio i lie 3 iroertly catured the variable y i the scoe of a quatifier Γ Γ C ( E) Γ, Γ (C) Γ [u/x] Γ x ( I) Γ x Γ [t/x] ( E) Γ [t/x] Γ x ( I) Γ x Γ,[u/x] ϕ Γ ϕ ( E) where the rules( I) ad( E) are subject to the coditio thatuis ot free i the coclusio of the rule, ie, i Γ,, adϕ 1 x((x) y(x,y)) 2 (y) 3 x((x) z (x, z)) reae boud variables, 1 4 (y) z(y,z) E,1 5 z(y,z) E,2, 4 CORRECT, because ow the variable y does ot get catured i the substitutio i lie 4 8 Practice robles You do t have to do all these They are just for ractice Proble 1 Prove the followig i atural deductio: (a) Q xp(x) x(q P(x)) assue thatxdoes ot occur i Q 7 Equivalece of Fitch style ad Prawitz style For the urose of logic, what atters about a roof syste is the derivability relatio, ie, Γ ϕ, eaig there is a derivatio of ϕ fro a set of hyotheses Γ lthough Fitch style ad Prawitz style atural deductio looks quite differet, they both describe the sae derivability relatio The rules of both systes ca be traslated ito rules about the derivability relatio i idetical ways I both cases, it is easy to rove by iductio that the derivability relatio is the sallest relatio satisfyig: Γ Γ ( I) Γ Γ Γ ( I) Γ Γ ( E) Γ Γ ( E) Γ Γ ( I) Γ Γ, ϕ Γ, ϕ Γ ϕ ( E) Γ, Γ ( I) Γ Γ ( E) Γ Γ, Γ ( I) Γ Γ ( E) Γ (b) xp(x) y P(y) (c) xp(x) xq(x) x(p(x) Q(x)) (d) xp(x) xq(x) x(p(x) Q(x)) (e) x yp(x,y) y xp(x,y) (f) x yp(x,y) zp(z,z) (g) xp(x) xq(x) x(p(x) Q(x)) (h) x(p(x) Q(x)) xp(x) xq(x) (i) xp(x,x) y zp(y,z) (j) x((x) (x)) x (x) x (x) (k) x((x) (x)) x((x) (x)) (l) x yp(x,y,x) x y zp(x,y,z) () x(p(x) yp(y)) () x( yp(y) P(x)) 9 10

6 (o) xp(x) xp(x) () x((x) (x)), y((y) C(y)) z((z) C(z)) (q) x(x), x((x) (x)) x((x) (x)) (r) x(x), x((x) (x)) x((x) (x)) (s) x((x) (x)) x (x) x (x) (t) xp(x) yq(y) x y(p(x) Q(y)) Proble 2 Prove the followig by atural deductio Note: each of these robles requires the -eliiatio rule (u) Q xp(x) x(q P(x)) assue thatxdoes ot occur i Q (v) xp(x) y P(y) (w) x((x) (x)) x((x) (x)) (x) x( yp(y) P(x)) (y) x((x) (x)) x (x) x (x) (z) xp(x) yq(y) x y(p(x) Q(y)) Proble 3 I Proble 1 (d), (e), (f), (h), (i), (j), (l), (o), (), (q), (r), rove that the coverse directio does ot hold by givig a couterexale, ie, a structure where it is false 11