I. Complete These Proofs. There Are No Additional Assumptions. (16)

Philosophy 211

Sample Exam 1

In-class portion

I. Complete these proofs. There are no additional assumptions. (16)

A. (P®Q)®(S®R), P®U, W®Q B. ((S®T)®U)®V

├ S®((~W® ~U)®R) ├ ~V®((~S&~U)v(T&~U))

1 (1) (P®Q)®(S®R) A 1 (1) ((S®T)®U)®V A

2 (2) P®U A 2 (2) ~V A

3 (3) W®Q A 3 (3) ~((~S&~U)v(T&~U)) A

4 (4) S A (4) ~(~S&~U)&~(T&~U)

5 (5) ~W® ~U A (5) ~(~S&~U)

6 (6) P A (6) SvU

(7) U (7) ~(T&~U)

(8) W (8) ~TvU

(9) Q (9) ~((S®T)®U)

(10) P®Q (10) (S®T)&~U

(11) S®R (11) S®T

(12) R (12) ~U

(13) (~W® ~U)®R (13) ~T

(14) S®((~W® ~U)®R) (14) ~S

(15) S

(16) (~S&~U)v(T&~U)

(17) ~V®((~S&~U)v(T&~U))

II. Consider these alleged proofs. If they are correct, say so. If they are not correct, indicate where they are incorrect and why. (12)

A. Q®R, S® ~R, P®S ├ ~P B. (P&Q)®R, ~P®P, R®S ├ S

1 (1) Q®R A 1 (1) (P&Q)®R A

2 (2) S® ~R A 2 (2) ~P®P A

3 (3) P®S A 3 (3) R®S A

4 (4) P&Q A 4 (4) ~P A

4 (5) P 4&E 2,4 (5) P 2,4 ®E

4 (6) Q 4&E 2 (6) P 4,5 RAA(4)

1,4 (7) R 1,6®E 1 (7) P®R 1&E

1,2,4 (8) ~S 2,7 MT 1,2 (8) R 6,7 ®E

3,4 (9) S 3,5 ®E 1,2,3 (9) S 3,8 ®E

1,2,3 (10) ~P 8,9 RAA(5)

III. Which of the following are adequate paraphrases of these English sentences? By an “adequate paraphrase”, I mean a “direct translation”, or something equivalent to a direct translation. There may be 0,1,2, or 3 correct answers. M, C, and A represent the obvious sentences for each part. (27)

1. Mary won’t attend unless at least one of Alice or Carol do not attend.

A. M®(~Av~C)

B. (A&C)® ~M

C. ~M® ~(AvC)

2. At least one of Carol or Mary attended, but not both.

A. (MvC) & (~Mv~C)

B. (M&~C) v (C&~M)

C. (M® ~C) & (C® ~M)

3. Exactly two of Mary, Alice, and Carol attended.

A. ((M&A)v(M&C))v(C&A)

B. ((~A®(M&C)) (~C®(A&M))) (~M®(A&C))

C. (M® ((AvC)&~(A&C))) & (~M®(A&C))

IV. Determine whether the following sequents are valid or not by either producing an invalidating assignment or giving some argument that there is no such assignment. (15)

1. ~P®Q, ~(RvQ) ├ P&R

2. P®(RvS), S® ~Q├ P®(Q®R)

3. ~R®(~S&~Q) ├ ~(P®Q)vR

V. Prove these sequents. You may use any rules you wish. (30)

1. (P&~Q)®(S&T), ~R® ~Q ├ ~R®(P®S)

2. ~Pv(QvR), ~P®R, (QvT)®S ├ ~R®S

3. QvP, ~R®P ├ (P®(Q&R))®(Q&R)