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?
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.