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Post by Eugene 2.0 on Jan 17, 2021 14:33:14 GMT
An old detective solved one task, and he made a conclusion about some facts; he wasn't satisfied though. Help him to check the task is the conclusion correct or not?
Here are premises and a conclusion:
(Pi) If Jones didn't meet Smith this night, then either Smith was a murderer, or Jones was lying.
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight.
(Piii) If the murder case had happened before the midnight, then either Smith was a murderer, or Jones was lying.
(C) Therefore, Smith was a murderer.
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Post by Eugene 2.0 on Mar 24, 2021 6:32:08 GMT
(Pi) If Jones didn't meet Smith this night, then either Smith was a murderer, or Jones was lying.
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight.
(Piii) If the murder case had happened before the midnight, then either Smith was a murderer, or Jones was lying.
(C) Therefore, Smith was a murderer
Let's write it symbolically:
1. p = Ajs = John approaches Smith
2. q = Ms = Smith is a murderer
3. r = Lj = John lies
4. s = Hmb = Murder case happened before
(Pi) ~p → (q v r)
(Pii) ~q → (~p & s)
(Piii) s → (q v r)
(C) q
Let's solve this task using conjunctive normal form (CNF):
((~p → (q v r)) & (~q → (~p & s)) & (s → (q v r))) → q
~((~~p v q v r) & ((~~q v ~p) & (~~q v s)) & (~s v q v r)) v q
(~(p v q v r) v ~((q v ~p) & (q v s)) v ~(~s v q v r)) v q
(~p & ~q & ~r) v ~(q & ~p) v ~(q & s) v (s & ~q & ~r) v q
(~p & ~q & ~r) v ~q v p v ~q v s v (s & ~q & ~r) v q
(~p & ~q & ~r) v (s & ~q & ~r) v p v q v ~q v s
((~p & ~q & ~r) v p v q v ~q v s) & ((~p & ~q & ~r) v p v q v ~q v s) & ((~p & ~q & ~r) v p v q v ~q v ~r v s)
((~p & ~q & ~r) v p v q v ~q v s) & ((~p & ~q & ~r) v p v q v ~q v ~r v s)
(p v ~p v q v ~q v s) & (p v q v ~q v s) & (p v q v ~q v ~r v s) & (p v ~p v q v ~q v ~r v s) & (p v q v ~q v ~r v s) & (p v q v ~q v ~r v s)
(p v ~p v q v ~q v s) & (p v q v ~q v s) & (p v q v ~q v ~r v s) & (p v ~p v q v ~q v ~r v s)
As we see all the conjunctive members of the last derivation have a pair or contradiction element, therefore the detective's puzzle was solved correctly.
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Post by karl on Mar 24, 2021 6:56:04 GMT
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight.
Pii means Pi has to be changed to: "If Jones didn't meet Smith this night, then Jones was lying."
Pii also changes Piii to: "If the murder case had happened before the midnight, then Jones was lying."
Then one getsL
"If Jones didn't meet Smith this night, then Jones was lying."
And:
"If Smith wasn't a murderer, then Jones was lying."
We'd have to know whether Jones was lying to know if Smith was the murderer.
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Post by Eugene 2.0 on Mar 24, 2021 8:01:06 GMT
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight. Pii means Pi has to be changed to: "If Jones didn't meet Smith this night, then Jones was lying." Pii also changes Piii to: "If the murder case had happened before the midnight, then Jones was lying." Then one getsL "If Jones didn't meet Smith this night, then Jones was lying." And: "If Smith wasn't a murderer, then Jones was lying." We'd have to know whether Jones was lying to know if Smith was the murderer. Must say I didn't get some points of how you solved it. For instance, why to change Pi? Probably I wrote roughly what the detective wanted to find – in the puzzle. The task appealed to present how popular styles of thinking could be non-obvious, and that the standard method in logical calculus couldn't handle this. So, CNF method came out to be a really powerful tool. |
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Post by karl on Mar 24, 2021 8:30:49 GMT
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight. Pii means Pi has to be changed to: "If Jones didn't meet Smith this night, then Jones was lying." Pii also changes Piii to: "If the murder case had happened before the midnight, then Jones was lying." Then one getsL "If Jones didn't meet Smith this night, then Jones was lying." And: "If Smith wasn't a murderer, then Jones was lying." We'd have to know whether Jones was lying to know if Smith was the murderer. Must say I didn't get some points of how you solved it. For instance, why to change Pi? Probably I wrote roughly what the detective wanted to find – in the puzzle. The task appealed to present how popular styles of thinking could be non-obvious, and that the standard method in logical calculus couldn't handle this. So, CNF method came out to be a really powerful tool.
(Pi) If Jones didn't meet Smith this night, then either Smith was a murderer, or Jones was lying.
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight.
Pi claims that at least one out of two claims is true:
1) If Jones didn't meet Smith this night, then Smith was a murderer
2) If Jones didn't meet Smith this night, Jones was lying.
Pii means that it can't be so that if Jones didn't meet Smith this night, then Smith was a murderer. So this means 1) can't be true. Which only leaves 2).
(If it's true what Pii claims, that If Smith wasn't a murderer, then Jones didn't meet Smith this night, then it can't also be true that If Jones didn't meet Smith this night, then Smith was a murderer. If both are true then you'd get: If Smith wasn't a murderer then Smith was a murderer.)
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Post by Eugene 2.0 on Mar 24, 2021 9:26:28 GMT
Must say I didn't get some points of how you solved it. For instance, why to change Pi? Probably I wrote roughly what the detective wanted to find – in the puzzle. The task appealed to present how popular styles of thinking could be non-obvious, and that the standard method in logical calculus couldn't handle this. So, CNF method came out to be a really powerful tool.
(Pi) If Jones didn't meet Smith this night, then either Smith was a murderer, or Jones was lying.
(Pii) If Smith wasn't a murderer, then Jones didn't meet Smith this night, and the murder case had happened before the midnight.
Pi claims that at least one out of two claims is true:
1) If Jones didn't meet Smith this night, then Smith was a murderer
2) If Jones didn't meet Smith this night, Jones was lying.
Pii means that it can't be so that if Jones didn't meet Smith this night, then Smith was a murderer. So this means 1) can't be true. Which only leaves 2).
(If it's true what Pii claims, that If Smith wasn't a murderer, then Jones didn't meet Smith this night, then it can't also be true that If Jones didn't meet Smith this night, then Smith was a murderer. If both are true then you'd get: If Smith wasn't a murderer then Smith was a murderer.)
Oh yeah, excellent! |
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