Is the Cardinality of the Set of Contingent Propositions the Same as the Cardinality of the Set of Tautologies?
By a "contingent proposition" I mean a proposition which is neither a tautology or contradiction. Or in other words, there exists at least one valuation of the variables such that the formula evaluates to true and one valuation of the formula such that the formula evaluates to false.
It seems straight forward to prove that the set of contradictions has the same cardinality as the set of tautologies. How does one prove that the set of contingent propositions has the same cardinality as the set of tautologies?
Let there be $\kappa$ propositional variables, and let $p$ be such a variable. Let us investigate the case that $\kappa$ is infinite, for finite cardinalities the argument below shows both sets are countably infinite.
Then we have $|{\sf Prop}| = \kappa$, as it consists of finite constructions from the variables.
From $p \lor \neg p$ being a tautology, we infer that:
$$|{\sf Var}| \le |{\sf Taut}| \le |{\sf Prop}|$$
meaning $|{\sf Taut}| = \kappa$.
Also, since $p \land \phi$ is contingent for any $\phi \in {\sf Taut}$, it follows that:
$$|{\sf Taut}| \le |{\sf Cont}| \le |{\sf Prop}|$$
and we conclude $|{\sf Taut}| = |{\sf Cont}| = \kappa$.