42 votes

Suppose the predicate $F(x, y, t)$ is used to represent the statement that person $x$ can fool person $y$ at time $t$.

Which one of the statements below expresses best the meaning of the formula,

$\qquad∀x∃y∃t(¬F(x,y,t))$

- Everyone can fool some person at some time
- No one can fool everyone all the time
- Everyone cannot fool some person all the time
- No one can fool some person at some time

71 votes

Best answer

$F(x, y, t) \implies$ person $x$ can fool person $y$ at time $t.$

For the sake of simplicity propagate negation sign outward by applying De Morgan's law.

$∀x∃y∃t(¬F(x,y,t)) \equiv ¬∃x∀y∀t(F(x, y, t))$ [By applying De Morgan's law.]

Now converting $¬∃x∀y∀t(F(x, y, t))$ to English is simple.

$\color{blue}{¬∃}x\color{green}{∀}y\color{red}{∀}t(F(x, y, t))\implies \color{blue}{\text{ There does not exist }}$$ \text{a person who can fool }$$\color{green}{\text{everyone}} $$\color{red}{\text{ all}} \text{ the time.}$

Which means **No** one can fool everyone all the time.

So, **option (B) is correct.**

23

Negation sign propagates like this.

∀x∃y∃t(¬F(x,y,t))

= ∀x∃y¬∀t(F(x,y,t))

= ∀x¬∀y∀t(F(x,y,t))

=** ¬∃x∀y∀t(F(x,y,t))**

0

if we don't want to bring negation outside, then is it possible to answer?

What does it mean by $\neg F(x,y,t)$ ?

What does it mean by $\neg F(x,y,t)$ ?

0

Does the negate sign only apply with the leftmost (all values of x) or negate for the entire expression.

0

@Verma Ashish it says if there is a group of people and you call anyone from the group, for every person being called there will always exist a person who will not get fooled by him.

EDIT: I option c, a specific person is targeted by all which is not implied by the formulae. "There exist y" y need not be unique, just for each and every person there exist a person y.

44 votes

B is the correct answer. The trick is to bring the negate sign to the extreme left. Form a sentence without using negate and just negate that.

$\forall x \exists y \exists t(\neg F(x,y,t))$

$\equiv \neg(\neg\forall x \neg \exists y \neg \exists t) (\neg F(x,y,t))$

$\equiv \neg (\neg\forall x \neg \exists y \neg \exists t ( F(x,y,t)))$

$\equiv \neg (\exists x \forall y \forall t ( F(x,y,t)))$.

$\forall x \exists y \exists t(\neg F(x,y,t))$

$\equiv \neg(\neg\forall x \neg \exists y \neg \exists t) (\neg F(x,y,t))$

$\equiv \neg (\neg\forall x \neg \exists y \neg \exists t ( F(x,y,t)))$

$\equiv \neg (\exists x \forall y \forall t ( F(x,y,t)))$.

0

I'm not getting this: by your logic: ! ∃x∀y∀z(F(x,y,z) without negation: *some person can fool everyone all the time. *

If you negate the sentence: ** some person can't fool everyone all the time. ** so I'm not able to arrive at option b. pls help.

1

¬(∃x∀y∀t(F(x,y,t)))≡¬(∃x∀y∀t(F(x,y,t))). means that "it is false that there exists at least some x,which can fool all persons (y's) at all times". That means no 'x' exists which do it i.e "No one can fool everyone all the time"

10 votes

Without negation the statement is like: Everyone can fool Someone at some time.

So symply, with negation it will be like: No one can fool everyone all the time.

0

Then what will be the meaning of this logic: ∀x∃y(¬∀t(F(x,y,t)))

I simply took negation outside the function F.

I simply took negation outside the function F.

0

- No one can fool everyone all the time = No (one can fool everyone all the time) = ¬[∃x∀y∀t(F(x,y,t))] = ∀x∃y∃t(¬F(x,y,t)) got it
- ∀x∃y(¬∀t(F(x,y,t))) = ∀x∃y∃t(¬F(x,y,t)) same meaning of above.

0

I am just asking that...

∀x∃y(¬∀t(F(x,y,t))) should be equal to "Everyone cannot fool some person all the time".

English tranlation of this logic (∀x∃y(¬∀t(F(x,y,t)))) could be: For all x there exist a Y such that x can not fool y all the time.

∀x∃y(¬∀t(F(x,y,t))) should be equal to "Everyone cannot fool some person all the time".

English tranlation of this logic (∀x∃y(¬∀t(F(x,y,t)))) could be: For all x there exist a Y such that x can not fool y all the time.

0

∀x∃y(¬∀t(F(x,y,t))) should be equal to Everyone cannot fool some person not all the time

which means Everyone cannot fool some person all the time .

which means Everyone cannot fool some person all the time .

0

Ya ok...so

For all x there exist a Y such that x can not fool y not all the time.

What would the meaning of above statement in simple english?

And sorry of these silly doubts...

For all x there exist a Y such that x can not fool y not all the time.

What would the meaning of above statement in simple english?

And sorry of these silly doubts...

1

Actualy asking doubt is not bad think but asking same thing in another form is bad :)

What would the meaning of above statement in simple english :

For all x there exist a Y such that x can[ **not** fool y **not** all the time].

= For all x there exist a Y such that x can not (fool y some time).

0

For all x there exist a Y such that x can not (fool y some time).

So my actual doubt was can it be equal to "Everyone cannot fool some person all the time"(option c)....:P

So my actual doubt was can it be equal to "Everyone cannot fool some person all the time"(option c)....:P

1

No. I guess a "not" came extra in beween and you meant

For all x there exist a Y such that x can not (fool y

ALLtime).

Now

Everyone cannot fool some person all the time

Both are same. Both mean, a "person" exist who can not be fooled by everyone all the time- this person can be different for each person.