Question

it is about the classical logic in the subject of formal method： the question is shown as the picture

Question 1: Classical Logic [25 marks) a) Answer the following questions briefly but precisely. i. State what it means for an argument to be valid in Predicate Logic. [3 marks ii. Suppose you use resolution to prove that KB = a. Does this mean that a is valid? And why? [3 marks b) Consider the following three English sentences: Sl: If I write the test and do well, I will get a passing grade. S2: I did not get a passing grade. S3: Thus, either I did not write the test or I did not do well. Using the three propositional symbols: W denotes “I write the test", D denotes “I do well", and F denotes “I get pass grade”, answer the following: i. Convert Si, S2, S3 into three sentences in Propositional Logic. [3 marks ii. Construct the proof: S1, S2 F 53 (Hint: build a truth table). [3 marks] c) Prove the validity of the logic expression below (using Natural Deduction): (p/q)V(Ar)+pA(qVr) [4 marks d) Using quantifiers and the given translation keys, symbolise the sentences below. C(x) S(x) A{x) 1(x) H(x) x is clever. x is a student. x is ambitious. x fails the exam x is happy i. Every clever student is ambitious. ii. No student is both clever and ambitious. iii. If anyone fails the exam, then no one will be happy. [3 marks [3 marks [3 marks

Question 1: Classical Logic [25 marks)

a) Answer the following questions briefly but precisely.

i. State what it means for an argument to be valid in Predicate Logic. [3 marks

ii. Suppose you use resolution to prove that KB = a. Does this mean that a is valid? And why? [3 marks

b) Consider the following three English sentences: Sl: If I write the test and do well, I will get a passing grade. S2: I did not get a passing grade. S3: Thus, either I did not write the test or I did not do well. Using the three propositional symbols: W denotes “I write the test", D denotes “I do well", and F denotes “I get pass grade”, answer the following:

i. Convert Si, S2, S3 into three sentences in Propositional Logic. [3 marks

ii. Construct the proof: S1, S2 F 53 (Hint: build a truth table). [3 marks]

c) Prove the validity of the logic expression below (using Natural Deduction): (p/q)V(Ar)+pA(qVr) [4 marks d) Using quantifiers and the given translation keys, symbolise the sentences below. C(x) S(x) A{x) 1(x) H(x) x is clever. x is a student. x is ambitious. x fails the exam x is happy

i. Every clever student is ambitious.

ii. No student is both clever and ambitious.

iii. If anyone fails the exam, then no one will be happy. [3 marks [3 marks [3 marks

Answer 1

d) Every clever student is ambitious

(for all x) ((C(x)&S(x))-> A(x))

No student is clever and ambitious

~(some x)(S(x)&(C(x)&A(x)))

If anyone fails the exam, then no one will be happy

(for all x) I(x) -> ~(some x) H(x)

a) <ii> A valid sentence is one that is true in all models. If KB|=(alpha), then we know that (alpha) is true in all models for which the KB is true. However, this doesn't include all models, so no, this does not mean that (alpha) is valid.

<i> An argument is valid in predicate logic if and only if whenever an interpretation makes the premises true, the conclusions are also true.

b)

b) 1: (WAD) → F 520 N 83% (ww VND) NF (NW VND) FWD (WAD)F T T TIL tttttt LL T ttttttt & n t t t n t tt Ttttttttt & 4 ttttttt 7 - - .. 55,82 9=53/