Question 34

Let:

• $H$ be the proposition: "The result of the toss is head".
• $T$ be the proposition: "The person is telling the truth" (meaning the person is Type 1).
  If the person is Type 2, the person is lying, so $T$ is false.

The person asserts statement $S$: "The result of the toss is head if and only if I am telling the truth", which translates to:

Now consider the two cases:

1. If the person is Type 1, the person always tells the truth, so statement $S$ must be True:
   $$(H \leftrightarrow T) \equiv \text{True} \implies (H \leftrightarrow \text{True}) \equiv \text{True} \implies H \equiv \text{True}$$
   So the result is head.

2. If the person is Type 2, the person always lies, so statement $S$ must be False:
   $$(H \leftrightarrow T) \equiv \text{False} \implies (H \leftrightarrow \text{False}) \equiv \text{False}$$
   For a biconditional to be false when one operand is false, the other operand must be true:
   $$H \equiv \text{True}$$
   So the result is also head!

In both possible cases, $H$ is true: the result of the toss is head (Option A).