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Unformatted text preview: 8: Sequent and Theorem Introduction 127 8: Sequent and Theorem Introduction 127 ! Exercises ! Exercises I Some sequents exhibited below are substitution-instances of sequents in I Some sequents exhibited below are substitution-instances of sequents the list on page 123. For each such sequent below, identify the sequent in the in the which it is substitution-instance. Justify your answer in every case by list of list on pagea123. For each such sequent below, identify the sequent in the list of which it is a substitution-instance. Justify whichanswer in everymade,by stating, for each sentence-letterOctober 24, 2008 i in the sequent in your substitution is case stating, for each sentence-letter i in for (refer to the definition of `substiwhich formula uj has been substituted the sequent in which substitution is made, i which formula u has 121). Homework Assignment to the definition of `substitution-instance' onj pagebeen substituted for i (refer #4 Homework Assignment #4 Answer tution-instance' on page 121). the following six (6) exercises from pp. 127128 of the text. You may 2007 (but, you do(1) ~~(R & S)to)~T, T Theorem26, 2007 Sequent Introduction on any of not have use October&26, and/or S) NK ~(R October (1) ~~(R & S) ~T, T ~(A ~(RB) S) NK & these Answer (2) following six ~C exercises from pp. 127128 of the text.are those listed on problems. The only (6) NK the (A B) C, Theorems/Sequents you can use You may (2) following ~(~R NK ~(A ~~(R S) Answerdo ~(~(R B) C, ~C exercises from pp. & ~~(~R of ~S) text. You may the (A S) use Theorem B) (but, of (3)(3)not have S)six (6) ~S)) and/or 127128 ~S) Sequent ~~(~R the page 123 youthe((P~(~(R ) to) ~(~R Q ) NK NK ~~(R S) )&Introduction on any of text. (4) have to) ((P Theorem (P Q (R & S) (but, you do notQ R) & use ~S)) S) and/or Sequent Introduction on any of these problems. The onlyR) & ((P Q ) S) NK NKyou Q )use are S) Theorems/Sequents (P can (R & those listed on (4) ((P Q ) ~(M The these*(5) the text. only Theorems/Sequents you can use are those listed on problems. N) (W & U) NK (M N) (W & U) page 123 *(5) ~(M N) (W & U) NK (M N) (W & U) of [NOTE: page 123 of the text. Two (2) extra credit points (for each proof) will be awarded to those of Homework Assignment #4 II Below (2) extra credit points (for each proof) will be awarded to list of you who find the shortest lists of sequents,class (allsequent in the forthoseis a will get [NOTE: Twothere are two proof in the and each those tied firstshortest II Below (2) extra credit points (for second. Say which in the to those [NOTE: Twothere are two lists of in theeach proof) sequent shortest will get of substitution-instance of sequentsequents, and eachwill be awarded first list is a class extra you who find the shortestaproof in thein your(all those tied forsequents are subcredit).] substitution-instance of a justifying the second. Say which sequentsin I. substitution-instances of which, sequent answer in the same way as are you who find get extra credit).] the shortest proof in the class (all those tied for shortest will I. stitution-instances of which, justifying your answer in the same way as in extra credit).] 1. III.3 List 1: 1. III.3 List 1: 1. III.3 2. III.5 III.5 ~(R & S) ~~(~T & S), ~W ~~T, ~(R & S) ~~W 2. (i) (~T & S) ~T 2. III.5 ~(R & S) ~~(~T & S), ~W ~~T, ~(R & S) ~~W NK NK (~T & S) ~T (ii) (i) ~~(R & S) ~(~T & S), ~~W ~T, ~~(R & S) ~~~W 3. 3. III.7 III.7 (ii) ~~(R & & S) ~T & S), ~~W ~T, ~~(R & S) ~~~W S) ~(~T 3. III.7 NK ~(~T NK ~(~T & S) ~T ~~W ~~T, ~~(R & *(iii) ~(~T & 4. III.9 ~~(R & S) S) ~(~T S),S), ~~W ~~T, ~~(R S) ~~~W *(iii) ~~(R & & & S) ~~~W 4. III.94. III.9 NK ~(~T & S) ~~T NK ~(~T & S) ~~T 5. III.14 5. III.14 III.14 List 5. 2: 6. IV.3 List 2: 6. IV.3 6. IV.3 (a) A ~B, C ~D, A ~C NK ~B ~D Just to make sure we're on the same~C ~D, ~~A page (some people have an earlier printing ~D (b)(a) A ~B,~~C on D, A same ~C ~B B D ~~A B, C the NK we're an earlier of Just text), I ~~Aincluded the same ~C NK(some D the textook, below: the to make sure B, ~~C entire page (some people have have an printing printing Just to of the (c)(b) have~~B, on theD,~A problemBset from the textook, below: maketext), I have included the ~~A page NK from sure we're C ~~D, entire problemBset people earlier ~A ~C NK ~D (c) ~A ~~B, C ~~D, ~A ~C NK B ~D of the text), I have included the entire problem set from the textook, below: 128 Chapter 4: Natural Deduction in Sentential Logic III Show the following. Wherever you apply Sequent Introduction, be sure to III Show the following. Wherever you you Sequent indicate which previously proved sequentapplyare using.Introduction, be sure to indicate which previously proved sequent you are using. (11) ~(A & ~B)~B ~(~D & B) ~(E B), C (~E (~D & A)) NK ~C (1) ~A NK ~(A ~E), (12) (1) B) (C & ~(A B) ~C), ~B G) H], (2)(A (B ~C),D), (~E (B [(F NK ~C A ~A NK ~B ~A C) A (B ~C), (~I NK J) [G & B) (B NK K (~A ~I) (3)(2) A (A (H ~A~K)] ~C), ~B NK ~C *(13) (3) A (A B) NK (A C) (B D) NK A *(4)(A ~B B) (C D) A) A NK (B ~B (14) *(4) B) B) NK (B A) ~A] & D) (C D) ~B) C) (5)(A NK (A& A [(ANK (B A (A (15) (5) (A~(B C)] NK (B ~B) NK ~(C A) ~A] (6)~(A NK (B B) C),[(A C) A ~[A B), (7)(6) B, (B ~A) (C D), A NK ~C A ~[A (~B C)] NK (B C) ~D *(8)(7) A B, (~B arguments (state your ~C (A following ~A) (C D), ~D IV Symbolize theB) (A C) NK A (B C) NK dictionary explicitly). Then *(8) (A C, ~(C ~A) A (B C) NK 128 (9) of the4: Natural argument-forms. C) Logic Chapter B) B) (A Deduction in Sentential NK ~B give128 Chapter 4: Natural Deduction in Sentential Logic proofs (A & resulting ~A) NK ~B 128 (9) (B 4: Natural Deduction (A Sentential Logic (10)Chapter B) C) C, ~(C(A B) in C) A (A & NK (10) A (B C) NK (A B) (A C) (1) If God is omnipotent then He can do anything. So He can create a stone which is too heavy to be ~E), ~(E B), means He can't & it, (11) ~(A & ~B) ~(~D & lifted. But that C (~E (~D liftA)) so there's ~C (11) ~(A & ~B) ~(~D & ~E), ~(E B), C (~E (~D & A)) NK ~C NK something & can't ~(~D & C) isn't G) (~E (12) (A ~(A He~B) do. Therefore,~(E[(F B),omnipotent. (~D & A)) H], (11) B) (C & D), (~E ~E), He C ~C (~I is J) [G & way ~K)] (~A ~I) (2) If there(~I an [G (H(H of (A NK K D) ~I) *(13) (A B) empiricalD) NK distinguishing between absolute rest and ab J) (C ~K)] C) (B (~A NK *(13) (Amotion, NewtonD) right to think K (B D) is absolute, not relative. B) (C & solute B) & (C D)was NK (A C) & D) that space (14) (A (A B) (C D) (B C) C) (B D) *(13) NK NK (A (A (14) (A B) & (C D) space, C) (Areal difference between absolute Also, if there is absolute NK (B there is a & D) (15) ~(A B), ~(B C), ~(C A) C) (A & D) (14) (A B) & motion--whether D) NK (B not (15) and absolute (C C), ~(C A)or NK they are empirically distinguishrest ~(A B), ~(B NK (15) ~(A B), ~(B C), ~(C cannot NK able. So if, as some argue, there A) really be a difference between absolute rest and absolute motion unless they are empirically distinguishable, IV Symbolize the following arguments (state your dictionary explicitly). Then IV Symbolize the following arguments (state your dictionary explicitly). Then an empirical way of distinguishing between give proofs of thethe following arguments (state your dictionary explicitly). Then IV Symbolize resulting argument-forms. absolute rest and absolute mogive proofs is necessary and sufficient for the existence of absolute space. tion of the resulting argument-forms. (1) If God is omnipotent then He can do anything. So He can create a stone (1) If God is omnipotent then Heiscan do to do so, So He can create a stone (3) If God is willing toheavy to evil lifted. unable anything. He is impotent. If God which is is prevent be but (1) If God tooomnipotent then HeBut that means He So He lift it, so there's can do anything. can't canit, so there's which is too to means He can't If create a stone is able to prevent heavy do.be lifted. But that He is malevolent. liftHe is neisomething Heevil but unwilling to do so, omnipotent.can't lift it, so there's can't Therefore, But that which is too heavy to be lifted. He isn't means He ther something He can't do. Therefore, He isn't omnipotent. able nor willing, then he is both impotent and malevolent. Evil exists if something He can't do. Therefore, He isn't omnipotent. and only if is an unwilling or of distinguishing it. God exists only if He is (2) If thereGod isempirical wayunable to prevent between absolute rest and ab(2) If there is an empirical way of distinguishing betweenevil does rest and ababsolute neither impotent nor malevolent. Therefore, if God exists absolute, not.relative. solute motion, Newton (2) If there is anNewton was right to think that space is absolute, not relative. solute motion, empirical way of distinguishing between absolute rest and abwas right to think that space is not Also, if there is absolute space, there think that space is absolute, not relative. solute motion, Newton space, there is a real difference between absolute was right to is a real difference between absolute Also, if there is absolute rest and if there is motion--whetherthere is they are empirically distinguishAlso, absolute motion--whether or not a real difference between absolute absolute space, or not they are empirically distinguishrest and absolute able. So if, as some argue, there cannot really be a difference between abrest and as some not they are able. So if, absolute motion--whether or really be a empirically distinguish9 Alternativeable.rest andasforargue, thereunless they really be difference between abformats some argue, there cannot are empirically distinguishable,abproofs cannot are empirically distinguishable, solute rest if, absolute motion unless a difference between solute So and absolute motion they an solute restway of distinguishing between absolute rest and distinguishable, empirical and absolute motion absolute moan empirical way of distinguishing unless they are empirically absolute mobetween absolute rest and tion isempirical way of distinguishing between absolute rest and absolute moan necessary and sufficient for the existence of absolute space. tion is necessary and sufficient for the existence of absolute space. The format in which we have been setting out our proofs, taken from Lemmon, tion is necessary and sufficient for the existence of absolute space. sits midway between two other formats, known respectively as tree format (this God (3) If God is willing to prevent evil but is unable to do so, He is impotent. If (3) original format) and sequent-to-sequent format. Lemmon format God If God is willing to prevent evil but is unable to do so, He is impotent. If was Gentzen's is able to is willing evilprevent evil but isdo so, He is malevolent. If He is nei(3) able to prevent evil but unwilling to do so, He is malevolent. If He is If God unable to do so, He is impotent. is If God prevent to but unwilling can be regarded either as a willing, evil but of bothto to do so, Hemalevolent. EvilIf He neiformat or as is malevolent. exists neither able norprevent then he unwilling is able to linearization is tree impotent and a notational vari- is if ther able nor willing, then he is both impotent and malevolent. Evil exists if ant of sequent-to-sequent nor is unwilling or unable to prevent it. are revealing, we is if format. Since he is other formats God exists only if He and only if God willing, then both both impotent and malevolent. Evil exists ther able and only if God is unwilling or unable to prevent it. God exists only if He is neither only present them briefly here.if Godnorunwilling or unable to prevent exists evil does not. He is and impotent nor malevolent. Therefore, if God it. God exists only if neither impotent is malevolent. Therefore, if God exists evil does not. By contrast with parse treesnor malevolent. Therefore, ifproof trees are not neither impotent and semantic tableaux, God exists evil does not. (12) (A B) (C & D), (~E C) [(F G) H], (~I J) [G & (H (~E (12) (A B) (C & D), ~K)] C) (~AG) H], NK K [(F ~I) NK give proofs of the resulting argument-forms. inverted: leaves are at the top and roots at the bottom. But like other trees, construction proceeds downward. A proof begins with a listing of premises and ...
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