Q1 what is proof?
Suppose that you proved this:
Does it mean:
a) Premises 1 to n are true
b) Conclusion is true
c) Premises 1 to n are not a contradiction
d) Conclusion isn't a contradiction
e) None of the above.
Premises: w The argument is valid iff:
x [w ∧ x ∧ y] → z
y is a tautology
Conclusion: ∴ z
w ∧ x ∧ y means all premises must be true and they are not contradiction.
(say if x and w are contradiction, then x = ~ w. Then w ∧ x ∧ y ∧ ... = w ∧ (~ w) ∧ y ∧ ... = F ∧ y ∧ ... = F)
And Conclusion z is also true.
Q2 invalid rule example
Let's consider an invalid rule:
What can we say about the truth value of p?
a) p is true
b) p is false
c) p might be either true or false
d) p can be neither true nor false
For all of the premises
(p→ q) = ~ p v q
since q is true form premise ~ p v q = ~ p v T = T
We can't tell what p is since ~ p v T is always true.
So the rule is invalid.
Q3 proof Example
~ (q v r)
(u ∧ q) ↔ s
~ s → ~ p
1. ~(q v r) Premise
2. (u ∧ q) ↔ s Premise
3. ~ s → ~ p Premise
4. ~q ∧ ~r De Morgan’s (1)
5. ~q Specialization (4)
6. ((u ∧ q) → s) ∧ (s → (u ∧ q)) Bicond (2)
7. s → (u ∧ q) Specialization (6)
8. ~u v ~q Generalization (5)
9. ~(u ∧ q) De Morgan’s (8)
10. ~s Modus tollens (7, 9)
11. ~p Modus ponens (3,10)
Q4 translate problem and then prove example(Onnagata)
Problem: Critique the following argument.
Premise 1: If women are too close to femininity to portray women then men must be too close to masculinity to play men, and vice versa.
Premise 2: And yet, if the onnagata are correct, women are too close to femininity to portray women and yet men are not too close to masculinity to play men.
Conclusion: Therefore, the onnagata are incorrect, and women are not too close to femininity to portray women.
First, make appropriate notation for the statement:
w = women are too close to femininity to portray women;
m = men are too close to masculinity to portray men;
o = onnagata are correct
Then using the notation to translate the story
Premise 1: w ↔ m (the key work "vice versa" indicates a notation ↔ here)
Premise 2: o → (w ∧ ~m).
Third, base on the two premises, we want to check if o is true
1. w ↔ m Premise
2. o → (w ∧ ~m) Premise
3. (w → m) ∧ (m → w) Bicond (1)
4. w → m Specialization (3)
5. ~w v m Implication: [IMP] p → q ≡∼p ∨ q (4)
6. ~ (~ (~w v m)) Double Negation: [DNEG] ∼(∼p) ≡ p (5)
10.~ (w ∧ ~m) De Morgan’s: [DM] ∼(p ∨ q) ≡ (∼p) ∧ (∼q)
11 ~o Modus Tollens: [M.TOL] (2,10)
As we can prove, onnagata is not correct.
Hercule Poirot has been asked by Lord Rumpd Dalton to find out who closed the lid of his piano after dumping the cat inside. Poirot interrogates two of the servants, Meece Pink and Jhyl Klone. One and only one of them put the cat in the piano. Plus, one always lies and one never lies.
Jhyl Klone: I did not put the cat in the piano. Ayul Parn gave me less than $60 to help her study.
Meece Pink: Jhyl Klone did it. Ayul Parn paid him $50 to help her study.
Who put the cat in the piano?
notes(Feel free to correct if you find mistakes):
j: Jhyl Klone lies
m: Meece Pink lies
c: Jhyl Klone put the cat in the piano
l: Ayul Parn paid Jhyl Klone less than $60 to help her study
e: Ayul Parn paid Jhyl Klone exactly $50 to help her study
1. j ⊕ m Premise(one always lies and one never lies)
2. j ↔ ~l Premise(if Jhyl Klone lies, Ayul Parn paid Jhyl Klone more than $60 to help her study, vice versa)
3. m ↔ ~e Premise(if Meece Pink lies, Ayul Parn didn't paid Jhyl Klone exactly $50 to help her study, vice versa)
4. e → l Premise(if Ayul Parn paid Jhyl Klone exactly $50 to help her study, we can say she pay less than $60)
5. m → ~c Premise(if Meece Pink lies, Jhyl Klone didn't put the cat in piano)
6. j → ~l Implication: [IMP] (2)
7.~l → ~e contrapositive: ∼q →∼p ≡ p → q (4)
8.j → ~e Transitivity: [TRANS] (6, 7)
10.~e→m Implication: [IMP] (3)
11.j → m Transitivity: [TRANS] (8, 10)
12.~j v m Implication: [IMP] p → q ≡∼p ∨ q(11)
13.~(j ∧ ~m) De Morgan’s: [DM] ∼(p ∧ q) ≡ (∼p) ∨ (∼q) (12)
14. (j ∧ ∼m) ∨ (∼j ∧ m) Exclusive Or [XOR]: p ⊕ q ≡ (p∧ ∼q) ∨ (∼p ∧ q)(1)
15. ∼j ∧ m Elimination: [ELIM] (13, 14)
16. m Specialization: [SPEC](15)
17. ~c Modus Ponens: [M.PON] (16)
As we can prove, Jhyl Klone didn't put the cat in piano. Then Meece Pink did.