What is the difference between "faithful" and "induced" in Priest (2008)? I have been going over the soundness and completeness proofs for classical propositional logic in Priest (2008) An Introduction to Non-Classical Logic, and I'm having a bit of trouble to see the relevant difference between the definitions of "faithful" and "induced".
Here is his definition of "faithful":

Let $v$ be any propositional interpretation. Let $b$ be any branch of a tableau. Say that $v$ is faithful to $b$ iff for every formula, $A$, on the branch, $v(A) = 1$. (p. 16)

And here is his definition of "induced":

"Let $b$ be an open branch of a tableau. The interpretation induced by $b$ is any interpretation, $v$, such that for every propositional parameter, $p$, if $p$ is at a node on $b$, $v(p) = 1$, and if $\lnot p$ is at a node on $b$, $v(p) = 0$. (And if neither, $v(p)$ can be anything one likes.) This is well defined, since b is open, and so we cannot have both $p$ and $\lnot p$ on $b$. (p. 17)

These definitions are pretty similar. It almost look like faithfulness and inducement are the same notions. The difference I can see is that the definition of faithfulness involves quantification over all formulas in a branch, while the definition of inducement involves quantification only over the propositional parameters. Is this the only difference here? Is there something important I'm missing?
 A: They are indeed similar definitions. Indeed, we can show that $v$ is faithful to $b$ if and only if $v$ is induced by $b$.
Let $v$ be an arbitrary propositional interpretation, and let $b$ be an arbitrary branch. First, $b$ must either be open or closed. Suppose it's closed. Then $b$ contains $A$ and $\neg A$ for some formula $A$.  $v$ is not induced by $b$, by definition ($v$ is induced by $b$ only if $b$ is open). And $v$ is not faithful to $b$, since we cannot have $v(A)=1$ together with $v(\neg A)=1$. So not every formula on $b$ equals 1 under $v$, hence $v$ is not faithful to $b$.
Suppose $b$ is open. Now, suppose $v$ is faithful. Let $A$ be an arbitrary formula on $b$. By faithfulness, $v(A)=1$. If $A$ is propositional parameter $p$, then $v(p)=1$. And if $A$ is $\neg p$, then $v(\neg p)=1$, so $v(p)=0$. So $v$ is induced by $b$. Now suppose that $v$ is induced by $b$. By Completeness Lemma (p. 17), for any formula $A$ on $b$, $v(A)=1$, certifying that $v$ is faithful.
I think Priest provides the two different definitions simply because they streamline his soundness and completeness proofs, respectively. When I read the definition of faithfulness, I get the intuition that I start with the interpretation function, and I can then sort through the branches it's faithful to. I move "from" the semantics, "to" the proof-system. He proves Soundness by contraposition, which involves moving "from" the semantics, "to" the proof-system; and indeed, this proof employs faithfulness. And when I read the defn. of inducement, I get the intuition of starting with the branch, and building up an interpretation from that. In other words, I move "from" the proof-system "to" the semantics. And indeed, he uses inducement to prove completeness. He proves completeness by contraposition, which involves moving "from" the proof system "to" the semantics. So I think, ultimately, he is trying to forge definitions that help to make soundness/completeness proofs more intuitive. But this is just my diagnosis.
