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PHIL 315/COGS315 Spring 202 2 Jenn Wang 1 Exam 1 This exam is open – notes , but do not consult the internet or other people (in your class or otherwise) . I am happy to answer any purely clarificatory questions. Please complete all of the following. 1. Let S = { 4 , 11 , 13 } and T = { 2 , 4 , 6 }. (a) Find S Ç T. (b) Find S È T. (c) How many elements are in the power set Ã (S)? How many elements are there in the power set Ã (T)? 2 . Let I be an interpretation function such that I(P)= 0 , I(Q)= 1 , I(R)=0, and I(S)=1 . Find the following: (a) V I ( ( ~ P Ú Q ) → (~ Q → ~ P ) ) (b) V I ( ( Q Ù ~ S ) → (Q → ~R)) (c) V I ( (~Q Ú R) ↔ ~ (Q Ù ~R)) 3. Establish the following by giving a counterexample : P→ ~ Q, ~P ↔ Q ⊭ PL Q . 4. Establish the following by giving a semantic proof: (P Ù Q) → (~Q Ú R) ⊨ PL P→(Q→ ( Q→ R)). 5. Fill in the blanks in the proof of the following argument : ~ P → ~ Q ⊢ PL Q →P 1. ___ ___ Premise 2. (~P →~ Q ) →( Q →(~P→~Q)) PL1 3. Q→(~P→~Q) 1,2 MP 4. Q→( ______ ) PL1 5. (~P→~Q)→((~P→Q)→ ______ ) PL3 6. ( (~P→~Q)→((~P→Q)→P) ) → (Q → ( ______ ) ) PL 1 7. Q → ((~P→~Q)→ ((~P→Q)→P)) 5,6 MP 8. (Q → ((~P→~Q)→((~P→Q)→P))) →((Q→(~P→~Q)) → (Q→((~P→Q)→P)) PL2 9. (Q→(~P→~Q)) → (Q→((~P→Q)→P) 7,8 MP 10. ______ 3,9 MP 11. ______ PL2 12. (Q→(~P→Q))→ (Q→P) 10,11 MP 13. Q→P 4,12 MP PHIL 315/COGS315 Spring 202 2 Jenn Wang 2 6. Using only the primitive rules of axiomatic proof theory , g ive a n axiomatic proof of the following: ~~ P →~P ⊢ PL ~P 7. Give a proof of the following. You may use the Deduction Theorem and T oolkit (but not semantic proof). ~( (P →Q) →~ (Q→P) ) ⊢ PL ~P→ ~ Q 8. Determine whether the following is true. Give a proof of its truth or falsity. You may assume the metatheorems of PL that we’ve covered in class. ⊢ PL ~(P →Q)→ ( Q→R) 9. Fill in the blanks in the following proof by induction of: No wff of PL ends with ‘ ~ ’. This is a proof by induction over ______ . The property P = ______ . Proof: (i) Base case: No ______ ends with ‘~’ . (ii) Inductive step: a. Inductive hypothesis (IH): _______ b. Proof that ~ j doesn’t end with ‘~’ : By the IH, ______ . But prefixing j with ‘~’ cannot change this. c. Proof that ( j → y ) doesn’t end with ‘~’ : ( j → y ) does not end with ‘~’, since it ends with ‘ ______ ’. 10. We know that ~P ⊢ PL P→Q. (a) Give an example of a natural language inference that seems to follow this pattern but where the premise seems to be true and the conclusion false. Explain your example in 4 – 5 sentences . (b) Do you think we should thereby give up the inference: ~ φ; therefore φ → ψ in natural language? Defend your answer in 4 – 5 sentences.

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