in Mathematical Logic edited by
4,645 views
30 votes
30 votes

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?

  1. $\forall x: \text{glitters} (x)\Rightarrow \neg \text{gold}(x)$
  2. $\forall x:\text{gold} (x)\Rightarrow \text{glitters}(x)$
  3. $\exists x: \text{gold}(x)\wedge \neg \text{glitters}(x)$
  4. $\exists x: \text{glitters}(x)\wedge \neg \text{gold}(x)$
in Mathematical Logic edited by
by
4.6k views

3 Comments

option B is not true because it says that " All gold glitters " which cannot be extracted from "Not all that glitters is gold"

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
2
2

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.

1
1

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.

2
2

4 Answers

28 votes
28 votes
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. 

edited by

2 Comments

moved by

 "Not all that glitters is gold” 

6
6

i guess according to ur diagram , option b also becomes true!!!

for any metal,if it is gold=> it glitters

0
0
15 votes
15 votes

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"

 

edited
by

4 Comments

x(glitters(x)^gold(x) I think it's meaning is all things are glitters and it is gold.They said to represent all that glitters is gold

0
0
yaar sometimes mathematical logic seems to be difficult ...

??
0
0
0
0
1 vote
1 vote
answerv is d
by
0 votes
0 votes
The statement “Not all that glitters is gold” can be expressed as follows :

¬(∀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) .

1 comment

Complex answer ...!!
0
0
Answer:

Related questions