Prove or refute the following conjecture about wffs of the system 𝒫 of propositional calculus. F

Step 1: Understand the problem statement. We are given a wellu002Dformed formula (wff) $u005Cmathbf{A}$

Question

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 $\vDash \mathbf{A} \equiv \sim \mathbf{q}$.

   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 $\vDash \mathbf{A} \equiv \sim \mathbf{q}$.

An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Computer Science & Applied Mathematics)

Peter B. Andrews 1st Edition

Chapter 4, Problem 5

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Step 1

We are given a well-formed formula (wff) $\mathbf{A}$ of propositional calculus such that for any truth assignment $\varphi$, the truth value of $\mathbf{A}$ under $\varphi$ is the negation of the truth value of $\mathbf{A}$ under $\varphi'$, where $\varphi'$ is Show more…

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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 $\vDash \mathbf{A} \equiv \sim \mathbf{q}$.

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