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If the proposition $\lnot p \to q$ is true, then the truth value of the proposition $\lnot p \lor \left ( p \to q \right )$, where $\lnot$ is negation, $\lor$ is inclusive OR and $\to$ is implication, is

1. True
2. Multiple Values
3. False
4. Cannot be determined

Cannot be determined.

From the axiom $\lnot p \to q$, we can conclude that $p \vee q$.

So, either $p$ or $q$ must be TRUE.

\begin{align} &\lnot p \lor (p \to q)\\[1em] \equiv& \lnot p \lor ( \lnot p \lor q)\\[1em] \equiv&\lnot p \lor q \end{align}

Since nothing can be said about the Truth value of $p$, it implies that $\lnot p \lor q$ can also be True or False.

Hence, the value cannot be determined.

@Pragy Agarwal Thank you for correcting me. I understood the mistake I made.

I get it that it can be either T or F, but I was wondering what is meant by multiple valued (option B)
It is for people like me who drew the truth table and thought as the truth table has multiple values, B must be the answer.

given that $\lnot p \to q$ is true

A proposition can have only 2 possible values namely true or false. Here  I feel “cannot be determined”(option D) is a more suitable answer than “multiple values”(option B) because propositional variables are similar to variables in programming which can have only one value at a time.

by

We need to look only for the GREEN case as in that only P’ → Q is true .

And for these three (GREEN) cases we have truth value for some cases and false for some cases .

So the value can’t be determined .

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I get it that it can be either T or F, but I was wondering what is meant by multiple valued (option B)