Consider the following expressions: (i) $false$ (ii) $Q$ (iii) $true$ (iv) $P \vee Q$ (v) $eg Q \vee P$ The number of expressions given above that are logically implied by $P \wedge (P \Rightarrow Q)$ is ________.

An expression $E$ is logically implied by an antecedent $A$ if and only if $A \implies E$ is a tautology (i.e., whenever $A$ is true, $E$ must also be true).

First, simplify the premise:

$$P \wedge (P \Rightarrow Q) \equiv P \wedge ( eg P \vee Q) \equiv (P \wedge eg P) \vee (P \wedge Q) \equiv false \vee (P \wedge Q) \equiv P \wedge Q$$

Thus, the premise is true only when both $P$ and $Q$ are true.

Now test each expression when $P = true$ and $Q = true$:

1. (i) $false$: Evaluates to $false$. Not implied.

2. (ii) $Q$: Evaluates to $true$. $(P \wedge Q) \implies Q$ is valid.

3. (iii) $true$: Evaluates to $true$. Any formula logically implies $true$.

4. (iv) $P \vee Q$: Evaluates to $true \vee true = true$. $(P \wedge Q) \implies (P \vee Q)$ is valid.

5. **(v) $

eg Q \vee P$**: Evaluates to$ eg(true) \vee true = false \vee true = true$.$(P \wedge Q) \implies ( eg Q \vee P)$ is valid.

Therefore, the 4 expressions (ii), (iii), (iv), and (v) are logically implied.