Question

Using propositional logic to prove that each argument is valid.If Jose took the jewelry or Mrs. Krasov lied, then a crime was committed. Mr. Kraso was not in town. If a crime was committed, then Mr. Krasov was in town. Therefore Jose did not take the jewerly. Use letters J, L, C, T.So for this question, I am very confused and would appreciate any help offerd.

Answer 1

Explanation:

We will first translate the situation to propositional logic. First, some notation is needed:
`\lor` is the or logical operation and
`\implies` is the symbol for logical implication. Define the following events:

J: Jose took the jewelry. L: Mrs Krasov lied, C: a crime was committed. T: Mr Krasov was in town.

We will symbol the propositions in logical symbols. Recall that
`\\eg` means negation

If Jose took the jewelry or Mrs. Krasov lied, then a crime was committed:
`J\lor L \implies C`

Mr. Krasov was not in town:
`\\eg T`

If a crime was committed, then Mr. Krasov was in town:
`C\implies T`

We want to check if the conclusion Jose did not take the jewerly:
`\\eg J` can be deduced from the premises.

First, recall the following:

- if
`a\implies b` and a is true, then b is true.

-
`a\implies b` is logically equivalent to
`\\eg b \implies a`

Coming back to the problem, we have the following premises

`J\lor L \implies C, \\eg T, \\eg T \implies \\eg C, \\eg C \implies \\eg(J\lor L)`

where the equivalence for the logical implication was applied. REcall that the negation of an or statement is g iven by

`\\eg( a \lor b ) = \\eg a \land \\eg b` where
`\land` is the and logical operator.

USing this, we get the premises

`J\lor L \implies C, \\eg T, \\eg T \implies \\eg C, \\eg C \implies \\eg J\land \\eg L`

Since
`\\eg T`, by having
`\\eg T \implies \\eg C`, then it must be true that
`\\eg C`. Since
`\\eg C \implies \\eg J\land \\eg L`, then it must be true that
`\\eg J\land \\eg L`. This final conclusion implies that it is true that
`\\eg J` which is the statement that Jose did not take the jewelry.