Section 1
Duality
Complete the proof of Proposition 4000 .
Give a semantic proof of 4001 (using Gödel's Completeness Theorem) by using 4002 to prove that $\mathscr{V}_\phi^{A t} \mathbf{A}^{\mathrm{dd}}=\mathscr{V}_\phi^{\mathscr{A}} \mathbf{A}$ for all $\mathscr{M}$ and $\varphi$.
Give a completely syntactic proof of 4003 as follows: Let $\mathbf{A}_1, \ldots, \mathbf{A}_n$ be a proof of $\mathbf{A}$; show that $r \sim \mathbf{A}_i^{\mathrm{d}}$ for $1 \leq i \leq n$ by complete induction on
Give examples to illustrate Corollaries 4003, 4004, and 4005.
Prove or refute the following conjecture about wffs of the system $\mathscr{P}$ of propositional calculus. For any assignment $\varphi$ of truth values to propositional variables, let $\varphi^{\prime} \mathbf{p}=\sim \varphi \mathbf{p}$ for all p. Suppose $\mathbf{A}$ is any wff of $\mathscr{P}$ such that $\mathscr{V}_{\varphi} \mathbf{A}=\sim \mathscr{V}_{\varphi^{\prime}} \mathbf{A}$ for every assignment $\varphi$. Then there is a propositional variable $\mathbf{q}$ such that $\vDash \mathbf{A} \equiv \mathbf{q}$ or $\vDash \mathbf{A} \equiv \sim \mathbf{q}$.