we can only extract from this statement is "Some that glitters is not gold "

We don't know about the gold that all of it glitters or not

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Consider the statement

"Not all that glitters is gold”

Predicate glitters$(x)$ is true if $x$ glitters and predicate gold$(x)$ is true if $x$ is gold. Which one of the following logical formulae represents the above statement?

- $\forall x: \text{glitters} (x)\Rightarrow \neg \text{gold}(x)$
- $\forall x:\text{gold} (x)\Rightarrow \text{glitters}(x)$
- $\exists x: \text{gold}(x)\wedge \neg \text{glitters}(x)$
- $\exists x: \text{glitters}(x)\wedge \neg \text{gold}(x)$

Understand the Difference between this & this.

If it says "None of that glitters is gold" then $\sim (\forall x(glitters(x)\rightarrow Gold(x)))$ will not work because this is saying **some of that glitters is not gold** (or)** all of that glitters is not gold, ** not **None of that glitters is gold**.

Think **domain** as **all the metals in the world**.

Note :- glitter means “shines”

**Understand meaning of each option in english below :-**

**Option A** :- All the metals which glitter are not gold. (false because word “all” means gold are also included but gold always glitter)

**Option B** :- All the metals which are gold are glitter (although this statement is true but it not saying same thing as this statement “Not all that glitters is gold”)

**Option C** :- there exist a metal x which is gold but not glitter (this statement is always false)

**Option D** :- there exist a metal x which glitter but not gold (this is same statement that we have given in question)

So, option D is correct.

Best answer

**"Not all that glitters is gold”** can be expressed as :

$\neg(\forall x(\text{glitters}(x)\implies \text{gold}(x)))$

*(as restriction of universal quantification is same as universal quantification of a conditional statement.)*

**"Not all that glitters is gold" **means **"some glitters are not gold"** which can be expressed as

$\exists x(\text{glitters}(x)\wedge \neg \text{gold}(x))$

*(as restriction of an existential quantification is same as existential quantification of a conjunction.)*

**So option (D) is correct. **

Option D is correct .

"Not all that glitters is gold”

can be expressed as :

⇒∼(∀x(glitters(x)⇒gold(x))

⇒$ ∃x\neg(glitters(x)⇒gold(x))$

⇒$∃x(\neg(\neg glitters(x) \vee gold(x))$

⇒$∃x(glitters(x) \wedge \neg gold(x))$

can be expressed as :

" some glitters are not gold"

I think this will help you

“For every person x, if person x is a student in this class then x has studied calculus.”

If S(x) represents the statement that person x is in this class, we see that our statement can be

expressed as ∀x(S(x) → C(x)). [Caution! Our statement cannot be expressed as ∀x(S(x) ∧

C(x)) because this statement says that all people are students in this class and have studied

calculus!]

“For every person x, if person x is a student in this class then x has studied calculus.”

If S(x) represents the statement that person x is in this class, we see that our statement can be

expressed as ∀x(S(x) → C(x)). [Caution! Our statement cannot be expressed as ∀x(S(x) ∧

C(x)) because this statement says that all people are students in this class and have studied

calculus!]

¬(∀x(glitters(x)⇒gold(x)) ... (1)

Where ∀x(glitters(x)⇒gold(x) refers that all glitters is gold. Now ,

∃x¬(glitters(x)⇒gold(x)) ... (2) , Since we know ¬∀x() = ∃x¬()

(Where ∀ refers to -> All and ∃x refers to -> There exists some).

As we know, A⇒B is true only in the case that either A is false or B is true. It can also defined in the other way :

A⇒B=¬A∨B (negationA or B ) ... (3)

From equation (2) and (3) , we have ∃x(¬(¬glitters(x)∨gold(x))

⇒∃x(glitters(x)∧¬gold(x)) ... (4) , Negation cancellation ¬(¬) = () : and ¬(()∨()) = (¬()∧¬()) .

So Answer is (D) .

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