GATE CS 1995 Q2.19 — Propositional and First Order Logic

We are given that the proposition $\lnot p \to q$ is True.

First, let us simplify the given premise:

So, we have:

Now, let us simplify the target proposition, say $E$:

Using the implication identity $p \to q \equiv \lnot p \lor q$:

Using the associative and idempotent laws of disjunction:

To determine if the truth value of $E$ is uniquely fixed when $p \lor q$ is True, let us test different truth assignments for $p$ and $q$:

1. Case 1: Let $p = \text{True}$ and $q = \text{True}$.
   • Premise: $p \lor q = \text{True} \lor \text{True} = \text{True}$ (Satisfied)
   • Target: $E = \lnot p \lor q = \text{False} \lor \text{True} = \text{True}$
2. Case 2: Let $p = \text{True}$ and $q = \text{False}$.
   • Premise: $p \lor q = \text{True} \lor \text{False} = \text{True}$ (Satisfied)
   • Target: $E = \lnot p \lor q = \text{False} \lor \text{False} = \text{False}$

Since the target proposition $E$ can be either True or False depending on the specific truth values of $p$ and $q$, its truth value cannot be determined.

Therefore, the correct option is D.