Is the set of Valuation Functions finite?

Question

I've been learning about interpretation functions and valuation functions. Intuitively, it seems as though there are an infinite number of Interpretation Functions, which I will denote as $I_n$.

i.e. given the set of all possible atomic well-formed formulas (which is presumably infinite), there must be a similarly infinite number of $I_n$...this belief is motivated by the idea that if I have an infinite list of atomic well-formed formulas (wffs), then I have a correspondingly infinite way of assigning them Truth/False values...each assignment strategy results from one specific $I_n$

However, for Valuation Functions, it seems as though I only have 6, which are limited to the 6 basic complex wffs.

1. the valuation function that operates on a propositional letter
2. the valuation function that operates on a negated proposition
3. the valuation function that operates on a conjunction proposition
4. the valuation function that operates on a disjunction proposition
5. the valuation function that operates on a conditional proposition
6. the valuation function that operates on a biconditional proposition

Does this mean that the set of valuation functions is effectively finite?

Answer 1

A valuation function maps formulas to truth values. There is not a different valuation function for disjuctions vs. propositional letters. The valuation function is defined as an extension of a truth assignment for propositional letters, by induction on formula structure and the standard truth tables.

Since each valuation function is determined by a truth assignment on propositional letters, there are as many valuation functions as there are truth assignments on propositional letters. And this depends how many propositional letters there are: if there are $n$, then there are $2^n$ truth assignments.

So if there are infinitely many propositional letters, then there are infinitely many truth assignments. For example, if there are a countably infinite number of propositional letters (which is usually the case), then there are $2^{\aleph_0}$ (cardinality of the reals) many truth assignments, and hence that many valuation functions.