# Interpreting negation of quantifiers.

Let F(x,y,t) mean Person x can fool person y at time t.

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

I can write it as

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

I am having doubt of how to interpret the negation outside. Does it apply only to x ?

Should the predicate be " There does not exist x who can fool everyone all the time " or " There does not exist x who can fool someone at some time" ?

In first interpretation, I am applying negation to only x (not exists). Is it allowed ?

• ∀x∃y∃t(¬F(x,y,t)) means: "for every person (x) there is someone (y) that he (x) cannot fool sometime". – Mauro ALLEGRANZA Jan 3 '18 at 19:04
• Because the un-negated formula is "there is..." thus its negation will be "there does not exist ...". – Mauro ALLEGRANZA Jan 3 '18 at 19:15
• If we have "There is a cat on the table that is eating a fish" its negation will be "there is not a cat on the table that is eating a fish" and not "there is not a cat on the table that is not eating a fish". – Mauro ALLEGRANZA Jan 3 '18 at 19:17
• Which textbooks have you looked at Sagar P? Standard presentations of quantificational logic typically have clear explanations of just this sort of thing. Maybe you can tell us what you've read and not understood?? – Peter Smith Jan 3 '18 at 19:44
• "It is not true that for every x there is an y such that P(x,y)". – Mauro ALLEGRANZA Jan 3 '18 at 19:49