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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,

$\qquad∀x∃y∃t(¬F(x,y,t))$

1. Everyone can fool some person at some time
2. No one can fool everyone all the time
3. Everyone cannot fool some person all the time
4. No one can fool some person at some time
edited | 2.7k views

F(x, y, t) = 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)) = ¬∃x∀y∀t(F(x, y, t)) [By applying de morgan's law.]

Now converting ¬∃x∀y∀t(F(x, y, t)) in english is simple.

¬∃xyt(F(x, y, t)) = There doesn't exist a person who can fool everyone all the time.

Which means No one can fool everyone all the time.

So option B is correct.

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after applying de morgan's law why negation sign is with only there exits and not with for all
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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))

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The last line should be  ¬∃x∀y∀t(F(x,y,t)).

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@vineet.ildm negation sign is just propagated outward by successively applying de Morgan's law.

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.

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

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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.

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.

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What would be the predicate logic for option C : "Everyone cannot fool some person all the time".
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∀x∃y∀t(¬F(x,y,t))
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Then what will be the meaning of this logic: ∀x∃y(¬∀t(F(x,y,t)))
I simply took negation outside the function F.
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1. 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
2. ∀x∃y(¬∀t(F(x,y,t))) = ∀x∃y∃t(¬F(x,y,t)) same meaning of above.
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∀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.
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∀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 .
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for all t also negate na u cannot ignore it.
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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...
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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).

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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
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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 ALL time).

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.

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@arjun sir..... So the answer could be C also?
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Nopes. Then "all the time" in C should be changed to "some time".
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Thanks sir