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I am trying to solve this question using predicate logic and natural deduction but could only do so using propositional logic.

The cops have three suspects for the murder of Dave: Adam, Ben, and Charlie. Adam, Ben, and Charlie each declare that they did not kill Dave. Adam says that Dave was friends with Ben and that Charlie did not like him. Ben says that he didn’t know Dave and that he was not in town the day Dave was murdered. Charlie says that he saw both Adam and Ben with Dave on the day of the crime and that Adam or Ben must have killed Dave. Find the murderer if 2 of the 3 men are honest and one of them is guilty of the murder.

I am running into a lot of trouble forming appropriate propositions and could only form these two :

$T(x)$ - True if $x$ is telling the truth, false otherwise

$K(x)$ - True if $x$ murdered Dave, false otherwise

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  • $\begingroup$ You have to identify all the predicates: Honest, Guilty, Cop, etc. as well as the appropriate individual constants: Adam, Ben, Dave, Charlie. $\endgroup$ Dec 6, 2022 at 14:21
  • $\begingroup$ I dont know what quantifiers have to be applied here, what i did is looking exactly like my propositional logic solution, instead of a proposition saying "Adam is truthful" its just T(Adam) $\endgroup$ Dec 6, 2022 at 14:25
  • $\begingroup$ Not sure what advantage predicate logic would give you here given a domain of 3 people with specific information for each. Even if you use quantifiers, the proof would immediately drop those quantifiers and effectively become the propositional logic proof you already have. $\endgroup$
    – Bram28
    Dec 6, 2022 at 14:40

1 Answer 1

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$M(x)$ : $x$ Murdered Dave

$K(x)$ : $x$ Knows Dave

$S(x)$ : $x$ Saw Dave

$A=Adam , B=Ben , C=Charlie$

(0) "Adam, Ben, and Charlie each declare that they did not kill Dave."
By Default , that is not useful because 1 is lying & 2 are honest.

(1) "Adam says that Dave was friends with Ben and that Charlie did not like him."
$K(B) \land \lnot K(C)$

(2) "Ben says that he didn’t know Dave and that he was not in town the day Dave was murdered."
$\lnot K(B) \land \lnot S(B)$

(3) "Charlie says that he saw both Adam and Ben with Dave on the day of the crime and that Adam or Ben must have killed Dave."
$(S(A) \land S(B)) \land (M(A) \land M(B))$

Out of (1) & (2) & (3) , 2 Statements are true & 1 Statement is not true :

$(1 \land 2 \land \lnot 3) \lor (1 \land \lnot 2 \land 3) \lor ( \lnot 1 \land 2 \land3)$

Plug in the Statements & Simplify.

I hope it will indicate that Ben is lying & hence is the Murderer.

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