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

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?

• @ryang - Your formulation of the question is not what I am asking. Jul 5 at 12:37
• You're basically asking: since 1 and 2 are equivalent in all possible scanarios, then does the fact that 1 formalises I mean that 2 also formalises I? More generally: "If symbolic statement 2 is equivalent to symbolic statement 1, then do they both formalise the same natural-language statement?" [my suggested title that you removed] Jul 5 at 12:41
• @ryang, yes, that's what I am asking, but the point is that prior to quantifying them, they don't have the same truth table. Jul 5 at 12:42
• @ryang thanks again for your interest. “When I use a word,"...., “it means just what I choose it to mean—neither more nor less.” Jul 5 at 12:56

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).

• Thanks for your detailed answer! I am just confused as to how the "given context" specifies rows 4 or 5, where in fact it never does? Jul 5 at 11:27