Can two non tautologically equivalent propositional-logic statements be quantified into formalisations of the same natural language statement?

Question

Let $a,b,c$ denote elements of some field, say, the real numbers. Consider the statement

If $ac=bc$ and if $c\neq 0$ then $a=b\qquad (\textrm{I})$

Denoting $ac=bc$ by $p$, $a=b$ by $q$ and $c=0$ by $r$, a straightforward formalisation of the above statement is this:

$p\wedge(\neg r)\to q\qquad (1)$

Consider

$\neg p\to (\neg q\vee\neg r)\qquad (2)$

The second statement $(2)$ may be translated to

If $ac\neq bc$ then $a\neq b$ or $c\neq 0\qquad\textrm{(II)}$

and let's examine the truth-table of both formal statements $(1),(2)$:

\begin{array}{|c|c|c|c|c|} \hline p& q & r & p\wedge(\neg r)\to q&\neg p\to (\neg q\vee\neg r)\\ \hline T& T& T&T&T\\ \hline T& T& F&T&T\\ \hline T& F& T&T&T\\ \hline T&F&F&\color{red}F&\color{red}T\\ \hline F&T&T&\color{red}T&\color{red}F\\ \hline F&T&F&T&T\\ \hline F&F&T&T&T\\ \hline F&F&F&T&T\\ \hline \end{array}

Clearly they are not tautologically equivalent. But in the given context, where $a,b,c$ are real numbers (or elements of any field for that matter), the fourth and fifth lines of the truth table are not possible, because the fourth line means that $ac=bc$ and $a\neq b$ and $c\neq 0$, which is impossible, and the fifth line means that $ac\neq bc$ and $a=b$ and $c=0$, which is also impossible.

Question: Which of the formal statements $(1),(2)$ correctly formalises the statement $\textrm{(I)}$? Does the fact that the fourth and fifth lines are impossible imply that both statements $(1),(2)$ are correct formalisations?

Answer 1

Denoting $ac=bc$ by $p,\quad$ $a=b$ by $q\quad$ and $c=0$ by $r,$

$\text{If }ac=bc \text{ and if } c\neq 0 \text{ then }a=b\tag I$

$p\wedge\neg r\to q\tag1$

Correction: $$∀a\,∀b\,∀c\;(Pabc\wedge\neg Rc\to Qab)\tag A$$

$\neg p\to \neg q\vee\neg r\tag2$

$\text{If }ac\neq bc \text{ then }a\neq b \text { or }c\neq 0\tag{II}$

Correction: $$∀a\,∀b\,∀c\;(\neg Pabc\to \neg Qab\vee\neg Rc)\tag B$$

Question: Which of the formal statements $(1),(2)$ correctly formalises the statement $\textrm{(I)}$?

$(\mathrm I)$ is implicitly universally quantified, and is formalised by $(\mathrm A),$ but not $(\mathrm B)$ or $(1)$ or $(2).$

Similarly, $(\mathrm {II})$ is formalised by $(\mathrm B),$ but not $(\mathrm A)$ or $(1)$ or $(2).$

$(\mathrm I),$ i.e., $(\mathrm A),$ is mathematically equivalent, but not logically equivalent, to $(\mathrm {II}),$ i.e., $(\mathrm B).$

in the given context, where $a,b,c$ are real numbers (or elements of any field for that matter),

Given a propositional-logic sentence, each interpretation specifies exactly one row of its truth table.

Regarding your propositional-logic sentences $(1)$ and $(2),$ the triples $(1,1,0)$ and $(3,7,1),$ which specify the first and eighth row, respectively, of your truth table, are two interpretations/scenarios. In these contexts, the two sentences are equivalent.

Does the fact that the fourth and fifth lines are impossible imply that both statements $(1),(2)$ are correct formalisations of $\textrm{(I)}\,?$

No. For a simpler and sillier example, let $A$ and $B$ formalise $\sqrt4=2$ and $\sqrt9=3,$ respectively. Then $A$ and $B$ are both true, yet only $A$ is a formalisation of $\sqrt4=2.$

So, two formalisations being true in a set of contexts/interpretations does not imply that some natural-language sentence that is true in this set is formalised by both of them. What is implied, however, is that in this set of interpretations, their natural-language counterparts are equivalent to each other (that is, have the same truth value).