in Mathematical Logic recategorized by
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35 votes
35 votes

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
in Mathematical Logic recategorized by
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3 Answers

59 votes
59 votes
Best answer

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.

edited by

4 Comments

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

 

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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)
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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.
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3
5 votes
5 votes

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.

0 votes
0 votes

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 .

Answer will be D.

1 comment

I get it that it can be either T or F, but I was wondering what is meant by multiple valued (option B)
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0
Answer:

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