Question

A says “The two of us are both knights” and B says A “ is a knave.”

Short answer

A is a knave and B is a knight.

Step-by-step solution

Introduction

In response to such directions, truth-tellers frequently supply more comprehensive information in order to show their innocence. Liars, on the other hand, want to keep their mistakes hidden.

Explanation

From the given data,

Consider $\mathrm{p}$ denotes $\mathrm{A}$is a knight.

$\mathrm{q}$denotes $B$is a knight.

Thus, $-\mathrm{p}$denote “$\mathrm{A}$is a knave” and $-\mathrm{q}$denote “ $\mathrm{B}$is a knave”.

The statement can be written as $\mathrm{p}\wedge \mathrm{q}$and $\mathrm{q}$.

Consider the statement that $\mathrm{A}$is a knight to begin with. Then $\mathbf{p}\mathbf{\wedge }\mathbf{q}$is true. It implies that $\mathrm{B}$is likewise a knight. If $\mathrm{B}$is a knight, however, $\mathrm{B}$tells the truth and $\mathbf{-}\mathbf{p}$is true, indicating that $\mathrm{A}$is a knave. As a result, there is a discrepancy, and $\mathrm{A}$cannot be a knight. Another option is that $\mathrm{A}$is a knave. Then $\mathbf{p}\mathbf{\wedge }\mathbf{q}$is untrue, and neither of them is a knight. However, when $\mathrm{B}$states "$\mathrm{A}$is a knave," the assertion is correct. It denotes that $\mathrm{B}$is a knight who tells the truth, indicating that $\mathrm{A}$is a knave.

As a result, is a knave and is a knight.