Tell statements $(p\Rightarrow(q\Leftrightarrow r))$ and $(q\Leftrightarrow(r\wedge p))$ apart

elementary-set-theorylogicpropositional-calculus

I am currently trying to write (and prove) a statement and I am having some trouble figuring out if the statement needs to be of the form
\begin{gather}
p\Rightarrow(q\Leftrightarrow r)
\end{gather}
or of the form
\begin{gather}
q\Leftrightarrow (p\wedge r)
\end{gather}
Using truth tables, one can see that see that the truth values of these statements only differ in the following two scenarios (sorry for the quality of the handwritting):

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Further, suppose that I can show (or know) that
\begin{gather}
\neg p\Rightarrow\neg q\quad\text{ (i.e., }q\Rightarrow p\text{)}
\end{gather}
Can I then rely on this fact to con conclude that my statement is of the following form?
\begin{gather}
q\Leftrightarrow (p\wedge r)
\end{gather}
My intuition is that since $\neg p\Rightarrow \neg q$, the fact that $p\Rightarrow(q\Leftrightarrow r)$ is true when $p$ is false and $q$ is true, the statement $p\Rightarrow(q\Leftrightarrow r)$ cannot be the form of my statement.

More generally, I guess that my question is: how to tell statements $(p\Rightarrow(q\Leftrightarrow r))$ and $(q\Leftrightarrow (r\wedge p))$ apart?

Thank you all very much for your time.

Best Answer

I am having some trouble figuring out if the statement needs to be of the form \begin{gather} p\Rightarrow(q\Leftrightarrow r) \tag1\end{gather} or of the form \begin{gather} q\Leftrightarrow (p\wedge r)\tag2\end{gather}

Further, suppose that I can show (or know) that \begin{gather} \color{red}{\neg p\Rightarrow\neg q}\quad\text{ (i.e., }q\Rightarrow p\text{)}\tag{*}\end{gather} Can I then rely on this fact to conclude that my statement is of the following form? \begin{gather} q\Leftrightarrow (p\wedge r)\tag2\end{gather}

Here are the relevant facts ($\models$ means logically implies): \begin{align} q↔ (p\wedge r) \quad&\normalsize\models\quad \color{red}{\neg p→\neg q}\\ p→(q↔ r) \quad&\not\normalsize\models\quad \color{red}{\neg p→w\neg q}\\ \color{red}{\neg p→\neg q} \quad&\not\normalsize\models\quad \Big(p→(q↔ r) \quad\text{or}\quad q↔ (p\wedge r) \Big)\\ \color{red}{\neg p→\neg q} \quad&\normalsize\models\quad \Big(p→(q↔ r) \quad\equiv\quad q↔ (p\wedge r) \Big). \end{align}

(The brute-force way to verify these four claims is to replace each entailment or non-entailment symbol with $\to,$ then use a truth table to check whether the conditional is a tautology.)

So:

  • $(*)$ is a necessary condition for statement $(2).$
  • However, $(*)$ is a sufficient condition for neither statement $(1)$ nor statement $(2).$
  • Fortunately, given that $(*)$ is true, statements $(1)$ and $(2)$ are equivalent to each other (i.e., they are both true or both false).

More generally, I guess that my question is: how to tell statements $(p\Rightarrow(q\Leftrightarrow r))$ and $(q\Leftrightarrow (r\wedge p))$ apart?

Your truth table reveals that for the valuation $(p,q,r)=(F,T,T),$ statement $(1)$ is true whereas statement $(2)$ is false.

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