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Unformatted text preview: Problem Set 2 Elementary Logic
Due: 21 November 2007 Name ’71 £4“ . 5/,14:\ Student ID Number email Mark —%
Due 21 November 2007 by 4:00PM.
Submit your problem set to Ms. Loletta Li in Main Building 302. Make sure your problem
set is timestamped. Do not submit assignments by email. Late penalty: 10% for each day late. This problem set will not be accepted after 23 November. Answer the questions on the problem set itself. Write neatly. If the grader cannot read your
handwriting, you will not receive credit. Be sure that all pages of the assignment are securely stapled together.
Check the course bulletin board for announcements about the assignment. Do your own work. If you copy your problem set, or permit others to copy, you may fail the
course. 1. (15 marks) Tme or false? [5 @ Circle ‘T’ if the statement is true.
Circle ‘F’ if the statement is false. @ F Any inductive argument can be made valid by adding one extra premise.
T @ If a set of MPL WFFS is inconsistent and go is a member of that set,
then go is inconsistent with every other member within the set.
T ® “Vm(G‘x —» Go)” is a valid MPL WFF.
T @ “It is possible that” is a truth functional connective. T @ The following argument 'can be shown to be valid in MPL: “If someone is here, we can start. We cannot start. So, someone is not here.”
® F If an SL WFF go is a tautological conjunction, then each conjunct of (p is consistent.
T ® There is no interpretation under which “ Va:(H:c&Ga:)” is false
and “ Vzv(Ha: —> Gm)” is true.
T @ If X is an inconsistent set of MPL WFFs, then some member of X is inconsistent.
@ F If X is a consistent set of MPL WFFS, then every member of X is consistent. T a “332(Fac —> G55)” is a valid MPL WFF.
x /15 2. (16 marks) _ 3—7075 C43
For each of the following: Circle “valid” if it is a valid SL sequent.
Circle “invalid” if it is an invalid SL sequent. Otherwise, don’t cirrdean'yihing. 7
(A H (B A» A)) l: A And
(A4 B),(A—>C') I: (Ia—+0)
valid @ ((A —> B) A 0) B, ~o I: A army/Bel . invalid
(A& ~ A), (B V NB) I: (0' V NC)
invalid 1 (~A&B) B ~(A&B) @l invalid (B—~>(A—> @=( (B—u‘l)
valid (A‘&~ B) @(AvB) @ invalid ((ABBH ~ —>~A))I=C‘
valid invalid fiﬂrLﬁ‘t’é Ssh @ 3. (21 marks) In Mel PMI , ”I ﬁr My! WWW/{£1
Translate the following statements and argaments into MPL. y ﬂaw/ﬂ ﬂ’lmiéeﬂrj,
Preserve as much structure as possible. . Use the following translation scheme: Domain: The set of all human beings.
‘ m: Mary p: Peter Cx: x is clever. Hx: x is happy. he x is a student in logic class. PX: x is a professor. (a) Either Mary is clever or Peter is happy, and Peter is clever if and only if he is a
student in the logic class. «Cm V HP)?>~’ (CPéalpp (b) If any of the students in the logic class is clever, then no professor will be unhappy.
(Bax $CL)"> 4 9X ( PX 3’AHXD Jr
(9). (bezﬁkj —> Vx ( Px—> my) (c) All students in the logic class who are happy are clever. Vxlf 1.x 9? W 5 C26) ((1) N o professor will be happy unless not all students in the logic class are happy.
(A it (863’ Hx) V " folxé H20)
(VX(PX enllx) ‘/ 3%(b'RlK’AH/KD (e) All and only students in the logic class are happy.
Vii L): ‘97 Hit) N (Vx (DU; HA) gr VXCH} ﬁLX»
4 (f) Provided that Peter is a student in the logic class, Mary will be happy if and only if she is clever.
(lp '> (lime49 Cm» (g) If every person is either clever or happy, then Peter is not a student in the logic ‘ 
class. Both Peter are Mary are students in the logic class. Therefore, someone is both
not clever and not happy. 0/va Hoe ALPLW mm) ]= {lumen/111x) /21
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4. (10 marks) 2 PL C Give an MPL WFF that is logically equivalent to each of the following WFFs. Your
answer must include an existential quantiﬁer if the original WFF contains a universal
quantiﬁer, and vice versa. (MPL WFF (p is logically equivalent to MPL WFFiD if and
only if (p entails 1b, and 71b entails go.) (a) NElre(Ha: —> Gas)
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F)?” ”WWIVK/Id ”PL UFF 407A 61 made/36% gin/m. /77wem (d) ~Vm(~Ha: V Gm) 
Eli/\(Al/zaVé'rx) :7X"(FIX*>6X) 9’4H’345") (e) V$(Hm&~ G33)
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5. (20 marks) Determine whether the following sequents are valid. If a sequent is valid, write “valid”.
If not, gwe an mterpretatwn whzch shows that the sequent 25 not vahd. A”)! m t9?” 51:911er ' } 330332846233), ~Pa [2 «£251 “A” _
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31(61)’ 75 XIX/3 4 /20 6% [IX/3 4W @JmM/T r? M p [A a?) = [adj Avon; a ma A07 ‘ @JW 15m MN? 61 6. (20 marks) 4 til/Q For each of the following, circle either “Yes” or "No“.
Is there an interpretation under which all the following MPL WFFS are true? Vx(B:L' u» Orr)
N‘s/:1: N (Ax&(Dx&C;c))
39(By86 N 0'9) 33:(A3:&('~C:c V 13113)) Yes ® Is there a consistent MPL WFF which is false under every‘interpretation containing
more than 17 elements in its domain? 7W5: J Sad/x 5‘ UH: 771241 IY/ /J 77466 Wcér am #7“ (W‘Wy‘ 43"? 7%” ’léémwﬁr. Ye @2) fl} zém‘wbi'? 719 7%: WI Wﬂ/ﬂﬁmﬂdi 2:277 210 My, . Me F ADV/f? )1
Is there a set of 7 MPL WFFs such thattzgch air in the set i: consisfealit/b & 7;“? ' —5 /:JM 3% 0/39; amt szr/wém/IL . but the entire. set is inconsistent? ego Pm ( ARM/ﬂ a)
620* 1 ("l’a Vega)
No ( lav A 6%“) A 0k
C Pal/ & a ' Is there an interpretation under which “ V2:(K:c&B:c)” is false
and “ V56(K:c —: B33)” is true? ' No Is there an SL WFF which contains no sentence letters other “A” and “B”, and which
entails “((A&D) —> B)”, and which is entailed by “((CV A) <—> (B V A))”? a Yes No /20 ...
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