I need help with a logic question, or rather i need somebody to explain me the actual problem:
Let F(Q) be a formula. if F(T) and F($\bot$) are tautologies, prove that $F(Q)$ is as well.
I got this problem for homework but it doesn't make much sense in my opinion. Because if I replace $Q$ for $T$ so that $F(T) = 1$ or replace $Q$ for $\bot$ so that $F(\bot) = 1$ then of course $F(Q) = 1$ since $Q$ can only be $1$ or $0$ itself. If somebody could explain me what I misunderstand it would be much appreciated!
Since $\top$ is not quite the same as $True$, and $\bot$ not quite the same as $\bot$ ($\top$ and $\bot$ are claims, while $True$ and $False$ are truth-values, you can make it a bit more precise like this:
$F(\varphi)$ is some logical formula that contains zero or more instances of the sentence variable $\varphi$.
We are told that $F(\top)$ is a tautology. That is, when we substitute $\varphi$ with $\top$ we obtain a tautology. But given that $\top$ always evaluates to $True$, that tells us that when we evaluate the truth-value of $F(\varphi)$ by setting $\varphi=True$, we get $True$. We can write this as $F(True) = True$
Analogously, since $F(\bot)$ is also a tautology, we know that when we evaluate the truth-value of $F(\varphi)$ by setting $\varphi=False$, we get $True$. So $F(False) = True$
So, what is the value of $F(Q)$? Well, if $Q$ is set to $True$, then $F(Q) = F(True) = True$. And if $Q = False$, then $F(Q) = F(False) = True$. So, $F(Q)$ wil always evaluate to $True$. So, $F(Q)$ is a tautology.
More technical yet, we can consider interpretations $I$ that assign truth-values to all variables involved, and where a statement $\varphi$ is defined to be a tautology iff for all interpretations $I$: $I(\varphi) = True$ (written as $I \vDash \varphi$)
So, take any interpretation $I$ for $F(Q)$. Since $F(\top)$ is a tautology, we know that $I \vDash F(\top)$. But since $I(\top) = True$ for any interpretation $I$, we thus know that $I(F(True)) = True$. Likewise, $I \vDash F(\bot)$, and since $I(\bot) = False$ for any interpretation $I$, we thus know that $I(F(False)) = True$. Since for any interpretation for $F(Q)$ we have that $I(Q) = True$ or $I(Q) = False$, we have that either $I(F(Q)) = I(F(True)) = True$ or $I(F(Q)) = I(F(False)) = True$. So, we definitely have $I(F(Q)) = True$ for any interpretation $I$, making $F(Q)$ a tautology.