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

Answer:

Step-by-step 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 $\neg$ 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: $\neg T$

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: $\neg J$ 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 $\neg b \implies a$

Coming back to the problem, we have the following premises

$J\lor L \implies C, \neg T, \neg T \implies \neg C, \neg C \implies \neg(J\lor L)$

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

$\neg( a \lor b ) = \neg a \land \neg b$ where $\land$ is the and logical operator.

USing this, we get the premises

$J\lor L \implies C, \neg T, \neg T \implies \neg C, \neg C \implies \neg J\land \neg L$

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