Is "true" implied by $P∧(P⇒Q)$? Consider the following expressions:
$(i)$ false
$(ii)$ Q
$(iii)$ true
$(iv)$ P∨Q
$(v)$ ¬Q∨P
The number of expressions given above that are logically implied by $P∧(P⇒Q)$ is ___________.

My attempt:
My doubt is regarding "true". Can we logically imply "true" from "p and p implies q" ? Per my understanding "true" and "false" are the "truth values" that can be assigned to propositions. But they are not propositions themselves.

Can you explain in formal way, please?

 A: True is implied by everything. You can take it to be a/any tautology, for example $(p\lor\neg p)$. Sometimes it's a propositional constant $\top$ defined to be equivalent to a tautology. 
Similarly, False implies everything. You can take it be either a shorthand for a contradiction such as $(p\land \neg p)$, or a propositional constant $\bot$ defined to be equivalent to a contradiction.
A: Truth ($\top$, tautology) is entailed by any proposition.
$$\begin{align}p &~\vdash~ p\wedge \top \\p\wedge \top & ~\vdash~ \top \\ \hline p & ~\vdash~ \top \end{align}$$
"If anything is true, then true is true."

Likewise falsity ($\bot$, contradiction) entails anything.
$$\begin{align}\bot &~\vdash~ p\wedge \bot\\p\wedge \bot & ~\vdash~ p\\ \hline \bot & ~\vdash~ p\end{align}$$
"If false is true, then anything is true."
A: Given, expression is : 
$$=P\land(P\implies Q)$$
$$=P\land(P'\lor Q)$$
$$=(P\land P')\lor(P\land Q)$$
$$=F\land(P\land Q)$$  
$$=(P\land Q)$$ 


*

*$(P\land Q)\implies F = (P\land Q)'\lor F = (P'\lor Q')\lor F=(P'\lor Q')= \text{Contingency but not Tautology}$ 

*$(P\land Q)\implies Q = (P\land Q)'\lor Q = (P'\lor Q')\lor Q=P'\lor (Q'\lor Q) =P' \lor T =T =\text{Tautology} $

*$(P\land Q)\implies T = (P\land Q)'\lor T = T = \text{Tautology}$

*$(P\land Q)\implies P\lor Q = (P\land Q)'\lor P\lor Q = (P'\lor Q')\lor P\lor Q =(P'\lor P)\lor( Q\lor Q') = T \lor T = T = \text{Tautology}$

*$(P\land Q)\implies Q'\lor P = (P\land Q)'\lor Q'\lor P = (P'\lor Q')\lor Q'\lor P =(P'\lor P)\lor( Q'\lor Q') = T \lor Q' = T = \text{Tautology}$



Therefore, statements $(2), (3), (4)$ and $(5)$ satisfies.
Hence, answer is $4$
