“All that glitters is not Gold” Propositional Logic.

Question

Found this in Pugh’s Real Mathematical Analysis.

The problem says to “Recast the following English sentences in Mathematics, using correct mathematical grammar. Preserve their meaning”.

(c) All that glitters is not gold.

My thought ( could be way off) was to create functional notation notating that “element $x$ glitters” and “element $x$ is gold”. That is $g(x)$ and $Au(x)$ respectively. Then I would say something like...

It is not tru that for each $x,$ $g(x)$ implies $Au(x)$,

which then translates to $$\sim (g(x) \Longrightarrow Au(x)) \equiv g(x) \wedge \sim Au(x).$$ Is this even close? Any hints or direction is appreciated.

Answer 1

Your proposed sentence is close, but isn't quite there yet. $\ g(x) \wedge \sim Au(x)\ $ says that the this particular $\ x\ $ glitters but isn't gold. This would be true for some values of $\ x\ $ (cellulose sequins, trout scales, etc.) and false for others (gold, matte paper, etc.). To make it synonymous with the target sentence you need to precede it with an existential quantifier: $$ \exists x: \ g(x) \wedge \sim Au(x)\ , $$ which says that there exists an $\ x\ $ which glitters but isn't gold. Equivalently, you could place a universal quantifier between the negation sign and the remainder of the first version of your two equivalent formulations: $$ \sim\forall x:g(x)\implies Au(x)\ . $$ This says it's not true that everything which glistens must be gold.

Answer 2

You are very close to translating this correctly. You need only add a quantifier to the statement.

Propositional logic alone will not suffice for translating this statement into anything more than a single proposition. For that, you will need first order logic.

If the formal translation of a statement is ever tricky, it generally helps to translate the statement recursively, making it slightly more formal each time while being careful to preserve meaning...

"All that glitters is not gold."

"Not everything that glitters is gold."

"It is not the case that, for any thing $x$, if $x$ glitters, then $x$ is gold."

Now the quantifier and predicates are more clear. If the domain of discourse, S, is the set of all things, then we have the following

$\neg \forall x [G(x) \rightarrow A(x)]$

where $x \in S$

$G(x):x$ giltters

$A(x):x$ is gold

And of course, this is logically equivalent to...

$\Leftrightarrow \exists x \neg [G(x) \rightarrow A(x)]$

$\Leftrightarrow \exists x \neg [ \neg G(x) \vee A(x)]$

$\Leftrightarrow \exists x [\neg \neg G(x) \wedge \neg A(x)]$

$\Leftrightarrow \exists x [G(x) \wedge \neg A(x)]$